Fundamentals of Spatial Information Systems (Apic Studies in Data Processing)

I I II' !\ .p,l.e. S(~li("; The.; A.P.I.c. SERlliS (;('I/I' /(/{ I~di/()/s: M . .I. R. SII;\VEand I. C. WAND N l llld ...

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I I II' !\ .p,l.e. S(~li(";

The.; A.P.I.c. SERlliS (;('I/I' /(/{ I~di/()/s: M . .I. R. SII;\VEand I. C. WAND

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undamentals of patial Information ystems

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xv XIX

l' j\lU ONE INTRODUCTION TO THE SPATIAL CONTEXT

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2

Geomatics: Introduction to spatial information systems

3 3 10

Spatia l data o rgani zat ion Heteroge neity o f uses o f spa ti a l informa tion ~ys tem s J .2, I Uses of spat ial info rmati o n sys te ms 1.2 .2 Examples o f da ta re quirem e nts Somc co mpon e nts of spatial information syste ms 1,3. J The toolbox vi ew 1.3,2 Th e physical co mpone nts T he role of a utom atio n: geoma tics B ibliography

10 12 16 16 19 19 24

Needs: Purposes and types of spatial problem

27

' . I Problem s to be so lved; tasks to be performed

'2 Loca ti o n and c har acte r

,'.3 Measure me nts
2,3, I Spa tial prope rtie s

2 ,3,2 Spatial relati o nships

,'.4 Categories of spa tial problems 2.4 ,1 Types of spatial pro blem 2.4 ,2 Oth er as pects l ) Som e exa mpl es of multi-faceted needs 2,5. J E xa mpl e of flow s over landsca pes 2.5,2 R eso urces inve nto ry 2,5, 3 Predi ctin g th e lo cati on of mine ral ore deposits 2,5,4 Engineering netwo rk simulation .' () Main catego ries of too ls in a spa tial infor mati o n system ,;,7 Some implications fo r th e d esign of s patial inform a ti o n systems .'.. K Bibliogra ph y

27

33 38 39 41 43 43 47 48

48 49 49 53 54 58 59

V

( (J II II ' III>

VI

t 1" 11 "111 ',

VII

Chapter 3

Semantics: Objects, surfaces, data

3. 1 The info rma tio n in 'a s["atial inform a tio n sys te m 3. 1.1 Spati a l e nti ties 3. 1. 2 Catego ries o f info rmati o n 3. 1.3 Me tainfo rm a ti o n 3.2 No n-s pati al a tt ributes 3.3 Spa ti al cha racte ris ti cs o f e ntities 3.3. 1 Dime ns io na lit y of e ntities 3.3 .2 G eo me tri c e le me nts 3.4 Co mbin a tio ns o f e ntit y types 3.4 .1 Combina ti o ns of spa ti a l units 3.4.2 Substi tuti o ns of spa ti a l units 3. 4.3 Mi xed uses 3.5 Co ntinu o us va ri atio n ove r a nd in space 3.5. 1 A fie ld view o f spatia l varia ti o ns 3.5.2 Iso tro picit y 3.5.3 Disc re te and co ntinu o us vie ws 3.6 Spa ti a l a nd non -s pati al pro pe rti es toge th e r 3.7 An introduction to the mecha ni cs o f spa ti a l dat a o rga ni za ti o n 3.7 .1 Ta bles a nd matrices 3.7.2 Ma ps 3.7.3 D ata mode ls 3.8 Pe rso na l spa ti a l se ma nti cs 3.9 So me o th e r as pects o f spat ia l info rm atio n 3.9. 1 D ata qu a lit y 3.9.2 The time e leme nt 3.9.3 In te nsio na l a nd ex te nsio n
61

·I..~.) SIII()() tilul

61

·1. S ..l SO I\l C n:; i/i lll's () f lil lC , 11 1(/ pol ygo ll rc p rcse nta ti o n

62

""t'.'

·I S 4 Intensional il ii d CXlc ll sio nal re prese nta tion of o bj ects

4 .1 4.2

Geometries: Position, representation, dimensions

Diffe re nt geo metri es Pos itio nin g objec ts in spati al re fe re ncing sys te ms 4.2. 1 Co ntinu o us space refe re ncing 4 .2. 2 Re fe re nc ing fo r disc rete e ntities 4.3 G lo ba l reference sys te ms 4 .3. 1 G loba l re fe re ncing .1:. 4.3.2 Ma p projec ti o ns 4.3.3 So me exa mpl es of g lo ba l sys tem s 4.4 Th c funcl a me nt a l e le me nt o f dista nce 4 .') ( 'oordinat es and splines: th e represe nt a ti o n o f lines ,I. ~ . I I .iil c silllplili c;ltion

151 152 152 155 159 159 161

1,' I'il Cl ilb : a W, I)' to rcprcscnt n<J turaJ o bj ec ts ·I.() . I C reati o n o r fra ct al obj ects 4.(,. 2 Stochas ti c fracta ls I. / Sp;lcc-filling curves and dim ensio nali ty 4.7 .1 Pa th s thro ugh space 47 .2 Space-filling cur ves '~.7 . 3 Dime nsiona lity

I S Summary

I 'I ili bliog ra ph y

I I,

63 67

67 72 72

76 81 81

84

167

170 172

85

87 87 90

I 11",,1,·,. :,

Topology: Graphs, areas, ordering

Ne two rks and graphs

'i. I.1 G ra phs

'i 1.2 Pro pe rties of graphs

', ' ( j ra ph s a nd a reas

S.2 .1 Digita l line gra phs 'i .22 Topo logical co nsis te ncy

'\ \ I ~ ITo r ide ntifi ca ti o n

S.3 . I Possible co nditions in di gita l ma ps S.3.2 So me procedures fo r chec king fo r e rro rs Po lygo ns and a reas

SA. I T ypes o f areal spa ti al unit

S.4 .2 Co nta inme nt a nd coi ncid e nce ~., Data fo r spati a l re lati o nships

~ I, So me o th e r co nside ra tio ns a nd summ a ry

... I Ili bliog ra ph y

175

. I

91 92 95 95 96 98 99

102 103

104

106 106 108

175

176 180 183 183 J86

190 190 192 197 198

200 206 211

214

PART TWO GEOMETRIES FOR SPATIAL DATA Chapter 4

144

J49

113 114 116 117

122 127 127 128 131 134

140 141

I

It 11"'" () Tessellations: Regular and irregular cells, hierarchies r. I Mosa ics, tesse lla ti o ns and lat ti ces h .l .l Tesse lla tio ns h. I. 2 La ttices (,. 1.3 Sca le and resolutio n

(I.' The geo me try of reg ul a r tesse lla tio ns

/ , I, I,'ixed spa tia l reso luti o ns: reg ula r ce ll grids h . .1. I D a ta e ncodin g (, .1 .2 Spa ti a l pro pe rti es

(1.\ .3 Surface mode llin g fro m latti ces

(>. .\.4 Structures fo r g rid-ce ll da ta

I. I V
I,. Il icra rchi ca l tesse llatio ns fo r a sp he re

217

217

218

219

22 1

222

226

226

230

231 232 235 243

( () /Ilc 'llh

f

IX

'IfJI I'IIt-.

VIII

6.6

6.7 6.8

Irn.:gul;iI· ll'ssl' ll;lli'lIl <'; h;lSl"
. Chapte~ 7 ) Manipulations: Interpolations, geometric operations, . transformations

r

7.1

Inte rpo lation and ex trapolation 7.1.1 The inte rpol ation a nd ex trapo lati o n co ncept 7.1.2 Some prac tica lities 7.2 Basic o perations on lines and points 7.2.1 Line inte rsec tion s 7.2.2 Segment intersecti o ns 7.2.3 Point-in-polygon procedure 7.2.4 Ce ntroid d e finiti o n 7.2.5 So me spatial statistics based on po int d a ta 7.3 So me operations fo r polygons 7.3.1 Intersec ti o n of lines with polygo ns 7 .3.2 Union a nd intersec tion of po lygo ns 7.3.3 Area computation 7. 3.4 Areal inte rpol a tion 7.3.5 Shape mea sures fo r po lygo ns 7.3 .6 po lygon clipping

_,.k; 7.3.7 Buffe r zo nes

7 .3.8 Polygo n overlay process 7.4 Spa tial data tra nsfo rm at ions 7.4. \ Ch a nges in dimensionality 7. 4.2 Changes in position 7.4.3 Co ntl at ion 7.4.4 Changes in to pology 7. 5 Transform a ti o ns between regul a r cells and entities 7.5.1 C hange to regular cells 7 .5.2 C hange from regular ce lls to vecto rs 7.6 Access to spatial d ata 7. 6.1 Access by identifiers and by locators 7.6.2 Rectangles a nd strip trees 7.6.3 Shee ts and til es 7.6.4 Diffe rent form s of spatial addrc" 7 .7 7 .'1:\

Summary

Bibliograph y

24(1 246 247 252 254

( h.lpter 8 K.I

:--;.2

K. :' K ~4

257 258 258

261 264 264 265 267 269 270 27 1 272 272

273 275 276 277 279 280

284 2'1:\5

286 290 29 \

291 292 293

295 295 296

299 300

302 303

," .S

.".Il

K.7 K.t:

Spdli
.\l1.IIy~is:

!\\lrihule ddtd, modelling, integration

306

Intcgr;ll illg llil' ;lllrihllll' <1 ;1 1<1 Somc operation s lor pl;lnar nctwork e ntities Some oper;.iliolls for grid-cell based ma p ove rlay mode Jling '1:\.3. I Bas ic oper;ltiUIlS for grid-ce ll data 8.3. 2 Spatial modelling with grid-ce ll dat a Operations fo r quadtree tesse lla tions Ope rati o ns for irreg ular polygons a nd for gr::lphs '1:\.5. \ Crea ting regi o ns 8. 5. 2 Location probl e m solving 8.5.3 M a p overlay mode lling and analysis Integration and multiple represe nt a tion 8.6 .1 Multiple re prese ntatio n '1:\.6.2 Integr::l tion ~.6.3 Some exa mples o f public and co mm e rcial spatia l dat a organization S umm a ry and co nclu sio ns Bibliogra ph y

306 310 314 314 3 18

321 324 325 32~

329 333 334

334 336 343 345

l'Aln rHREE CONCEPTUAL MODElLING FOR SPATIAL DATA ( ".'pler 9

Design for Information Systems; Methodologies, issues

351

lJ . 1

352

'12

358

D a taba se ma nage ment systems The ANSI-SPARC desig n me thod o logy 'I .1 Co nce ptu a l mod e lling: the e ntity-relation ship approach ') 4 Logica! mode lling : re lational datab ases <J .S Transformin g e ntity-rela tionship mo del s into re lati o nal mod e ls ' I.() Logi ca l modelling: CO DASYL databases 9.6.1 The mod e lling 9.6.2 The d::lta definiti o n and manipulation lan guages ') 7 Some issues in e ntity-rel a tions hip and logical modelling 9.7.1 Impli ed re la tion s hips 9 .7.2 Pe rson-mad e and natural rules 9.7.3 T a ble organization I) .i-) The process for the design of spatial informatio n systems 9,S,I The inform a tion discove ry 9.8.2 Info rm a tion sys te m desig n ') -'I Summary <J. I () Bi bliography

362 366 369

374

374

378 383 384 3~6

388

391 391

393 396 396

( ""/V""

x

Chapter 10 Spaghelti: Conceptudl modelling of line-oriented

objects

10 .1 Represent ation o f ,C gIll C llt~, po lylines a nd mi xtilin es 10.1.1 Segm e nts 10.1.2 Polylines 10.1 .3 Rep rese nta tion of a mixtilin e 10.2 One-dimensio nal repres en ta tion o f pol ygo ns a nd areas 10.2.1 Isolate d polygons 10 .2.2 Sets of polygons 10 .3 Mod e lling for graphs 10.4 Conceptual m o delling of terrains 10.4.1 G radi e nts, grid s and co ntours

10.4 .2 Triangulated irreguhH ne two rks

_ - 10.5 R e prese ntatio n of polyhedra

10 .5.1 S impl e polyh ed ra 10.5.2 Co mpl ex polyhe dra 10 .6 Some exa mples of vector o rient e d geoma ti c models 10 .7 Summary 10 .8 Bibliography

, ) Chapter 11

Pizza: Conceptual modelling for areas and volumes

11 .1 R egular cell grid represe ntati on 11. 2 Quadtrees 11. 2. 1 Review o f th e concep t of quadtrees 11.2 .2 Mo dellin g polygons and terra ins by quadtrees 11.2. 3 Exte nded q uadtrecs 11.3 Pyramid models

I J.4 Mo dellin g via oc ttre es

11.4 .1 Hi e ra rchi ca l and linear oc ttrees 11.4.2 Exten ded octtrees LI.5 Example: modelling of geological objects 11 .6 Summary

11 7 Bibli og raphy

Chapter 12 Spatial Object Modelling: Views, integration, complexities 12 . 1 Se lec tion crite ri a for a good rep rese ntati o n 12 2 Exte rnal mo dels : synthesis with diffe re nt rep resen ta tions 12. 2. 1 Standardi za ti o n of geom e tric represe ntation 12. 2 2 ('oexistence o f seve ra l geometric re prese nta tions 1',1 : ..\ 1\11 :Idditional step in co nceptual mo de lling I' \ WOI kllll~ willi complex fea tures

t

1' li lr 'l lf'.

xi

1.' 1 SI' /Ildlll il' dal ;1 111(l(k-I ~

456 460 I>!.S . I C()nllIl CIT i;d sjlali:d informati on sys te ms software exa mpl es 461 1'.) :;.2 N,iI ili nal ca rtographi c dat abases 467 1 ' .(' Iss lI e,' in rcprese ntati o ns and conceptual mo dellin g 468 12,(,. I Met
399 399 401 402 403 403 405 406

409 411 411 414 416

416 417

419 424 425

426 427 428 428 430 432 436 436 438 438 441 442 443

444 444 445 447 447 448

453

111\1(1 lO UR SPATIAL DATA RETRIEVAL AND REASONING I

h.'pll'r 13 Algebras: Relational and Peano tuple

479

I \ I I :c, ltures of re la tion s

479

1.\. I . I Some prope rti es of tuples 1.\.1.2 The Ca rt esia n prod uct for rel a ti o ns I I ' I{ c latio nal ope rators a nd re la tion a l algeb ra 11.2. 1 Inte rsect io n 1.\ .2 .2 Union 1.1.2.3 Differe nce 1.1.2.4 J o in 1.\.2.5 Re la tion
480 481 483 483 485 486

486 487 488 488

490 490

492 493

495 496 496 497

507 507 509

512 512 515

516 51 8

519 520

520 523

( ( 1/ 1/('11/\

I "' , / tl flf o. XIII

XII

\3.7.3 Rehlliollal operalors 13.7.4 Exa mplc ~ o r l'eallo- luplc ,dge bri:l 4u c ri es

13 .8 Summary 13 .9 Bibliograph y

Chapter 14 Spatial Queries: Types, algorithms 14.1 T he process for spatial queries 14 .2 Point-in-polygon queries 14.3 Regio n queri es 14.4 V aca nt place queri es 14. 5 Distan ce and buffer zone queri es

\4.6 Path queries

14 .7 Exa mples of multimedia queries

\4 .~ Impli cations for spatial informatio n systems

14 .9 Bibliograph y Chapter: 15 Access and Quality: Spatial indices and integrity constraints

I S. I Index ing \ 5. 1. \ Index ing in file man ,lge mcll l sys tems 15. 1.2 Ind ex ing in relational dat abases

\ 5.2 Spatial 15.2. 1 15.2.2 \ 5.2.3 15 .2.4 \ 5.2.5

index in g Index in g by space-filling curves Index ing b y qu adtrees lndex ing by R- and R +-trees 1nd ex in g by other kind s of trees

Some practical aspects of spatial index ing

15,3 1ntegrity constraints

\5.3. \ B<:sic integrity constr aints

\5.3.2 Spalial data checking

15.3.3 Examp le of a cadastre \ 5.4 T he use of topology in creatin g integrity checking mechanisms 15 .4 . \ Th e topology of tessell ation s 15 .4 .2 Th e topology of network s 15. 4.3 Th e topology of digit al terrain models 15.5 An exa mple of consiste ncy checkin g for a terrain model IS S I T ri angulated irregu lar network represe ntation 15.5.2 Regular ce ll represe ntati on I.~, (' ( 'ol\l'llI sion s ,Ibnut spa tial indexing and integrit y I ~ . 7 Ilil'liogr;qlhy

52('

530

53 1

532

534

534

537

538

54 1

543

546

548

553

556

I

''''lIkr I h "yperllH'di,l : Mull inwdiil SI)atial information syslems .lnd hYflerm.lps II, I II Y Il~' rd ()cu l1l cll t s

U,. I . I Muitillledia sp;ltial data 1(' . 1.2 Til l', hypc rmap concep t j(, .' 1\!I 1Iltilllcdi ,1 image data 11> .2. 1 IIll .2.3 D ynami c image models 11> .2.--1 I' ieture object mode lling fo r retrieval If, (lrgdniZ,lti o n of collection s o f maps and images If" Il ypl'f'lll aps 11> ,-1 . I Spatial referencin g of hyperd ocuments I h -1 ,2 Spati al queries for ret ri ev ing h yperm ap nodes 11> .-1 ..1 E ncoding hyper map spatial references by Pea no relati o ns I il.4 .4 R-trees and map pyramid s

Ifl .'.! .) Navigation in hypermaps If, " ,') '"111n ;lry I" I, I tihl iogr'lphy

557

S58 559

562

563

507

569

57 1

573

575

576

576

577

578

581

582

583

584

586

586

589

590

592

'l.Iplc ' l 17

I'

Spat ial Knowledge: Intelligent spatial information

systems

I 'j'nwdrLis intellige nt spatial info rmation sys tems I I j I, "III record-orient ed to obj ec t-oriented databases 1/ ..2. I Rati ona le and object i ves 1 7 , ~ . 2 Classes , subclasses and instances 17.2 .3 Attributes and data types 1/.2.4 Inh eritancc 11. 2..) Links between cla sses and instance s 2.(, Methods I lIi1i za tion for geomatics i ' ( )h iccl-ori ented databases and spatial information sys tems I J , \ II ificial intelli ge nce and expert sys tems 11. -' . 1 F , 1 Inference engin e

I / S -I Metarules 1/ (, " 1':lli;Ji knowledge representatio n I I /) , I Spat ial facts

I / 11 , .2 Spatial relations 11 .. 1 Spatia l meta rule s I I (" ,1 Fuzzy spat ial knowl edge I I II , ~ Spa ti al knowledge from logica l deduction

1/ (,,(, Spat ial knowledge derived from numerical formulae

594

594

595

597

600

600

601

604

604

607

6 12

6 12

6 13

6 14

614

616

617

6 18

620

621

62 1

022

624

626

628

630

63 1

632

635

639

639

641

642

644

644

645

645

646

646

646

648

( (l1I/"/,h XIV

17 .6.7 I-:x'llnples 0\ spdi"d pi ()cess represe ntation 17 .6.K Visual k'Hlwlctil,!.e enc()ding 17.6.9 Examples in s paii ~d kn()wledge e nginee ring 17.7 Spatial reasoning in spa lial infurrnation sys tems 17.7.1 Learning possibilities 17.7.2 Logico-ded uctive and spatial reasoning 17.7.3 Example of districting 17.8 Summary 179 Bibliography

AFTERWORD i NDEX

reface

649

651 653 657 658 659 660 665 667

671

673

II " ,. IHHlk came about because the physical separation of the two authors lit e Atlantic Ocean was a surmountable space in the pursuit of a IrI ,"
,'rd ,/( (' XVI

base me nt with a perso ll ,d l'O llli lut c r to crea te th e latest pl a n fo r po litica l di stricts , can loo k a t a so ft w
!'n '/d( ('

XVII

geographe r living in lil l' [ IS/\ W .l ~ p,lrt 0 1 th e dfort ill pu([ill g lil e book toge th e r . Our ap proach i:-: L" lll L'''Il lu;ti , having a st ro ng pedagogic o ri e nta tio n with many exa lllp k s jlre se nlcd lor ed ificati o n a nd assimilation. Mo reover, the va riety of c X'll11plcs ,ti so provides a framework for eva luatio n , and a re alization th at th e re ca n be a so lid syste ma tic conceptual organization fo r many co ncevfs , too ls and pe rspectives. We have conce ntrate d o n show in g how to organ ize and use spatia l information rather than on how to process or manage spa tial d a ta. Just as a n und e rstanding of a rt is faci litated by look ing at ma ny pictures, so we think spa ti a l d a ta m gani zat io n ca n be be tte r unde rstood by see in g man y exa mpl es. The ~' a riet y is inh ere nt in the cultura l a nd na tura l realms o f thi s Earth ; it is also p rese nt in th e m a ny view po ints a nd needs th a t use rs o f spa ti a l data have . We see thi s boo k as he lpin g o th ers to a ppreciate th e varie d world of spa ti a l informa tion syste ms. We do not know exactl y whom 'ot he rs' mi ght be; therefore we have Ilot oriented th e book to Just one gro up . For exa mpl e, it is like ly th a t IIl fo rmat icia ns a nd softwa re engineers with no backgro und in a spa tia lly 'lr ic nted scie nce ca n be nefi t from the fi rst two pa rts o f the book; for t'cngrap hers a nd ca rtog ra phe rs much of the first two parts may be a I ~'\'i ew . Fo r so me spatial scie ntists, Part Three co uld be the startin g po int . We have tried to accom mo date the reader who ma y wish to de lve o nl y '"10 ce rta in topics by hav in g some repetitio n of mate rial. Pe rh a ps th e pl
It ti ll 1

I 'w/.l( ('

xviii The synthesis in Chapter 8 leads into, in Part Three, Conceptual Modelling for Spatial Data, the presentation of the techniques for conceptual modelling of spatial situations, chiefly the entity-relational modelling approach (Chapter 9, on design methodologies for information systems). Many examples for point and line entities (Chapter 10, Conceptual Modelling of Line-oriented Objects), and for areas and solids (Chapter 11, Conceptual Modelling for Areas and Volumes) are then provided as tastes of spaghetti and pizza, respectively. Chapter 12, Spatial Object Modelling: Views, Integration , Complexities, concludes the third part by a review of the process used in synthesizing different models , and then raises some issues to be dealt with in the fourth part of this book. Part Four emphasizes techniques and principles for accessing , retrieving and using information from databases containing a variety of spatial entities. The basis of relational algebra is presented (Chapter 13 , Algebras: Relational and Peano Tuple), followed by examples of queries (Chapter 14, Spatial Queries). After a discussion in Chapter 15 of the challenging topics of spatial indexing and integrity constraints, the book then moves on to discussions of currently developing fields, at first looking at multimedia and hypermedia concepts (Chapter 16, Multimedia Spatial Information Systems and Hypermaps), and then presents some ideas about intelligent spatial information systems (Chapter 17). These last two chapters (perhaps these are the dessert!) establish how the stage is changing, pointing the way to the anticipated developments of the next few years. While we have presented this book in traditional form, we wish we could have had the time and resources to prepare it as a hyperdocument. We should be embarrassed about this - Chapter 16 talks about bypermaps, and there already are several 'Hypermap' products in the marketplace. But addressing the medium as part of the message will have to wait until next (?) year. Robert Laurini Villeurbanne, France Derek Thompson Greenbelt, Maryland, USA

March 1991

Acknowledgements

"" 1111 \1 pe()pic have helped with the preparation of this book. Large in 11 111 ill(' r. st udcnts and professionals in several places have sat through II ' i 1(1 v ~ ;t11(1 less formal presentations of much of the material contained (I IlI l' IHI()k , albeit under a different label. We appreciate their interest, till II ' .I 1!~ 1!.L'stions , and their enthusiasm about spatial information systems. W" (' specially wish to acknowledge the intellectual contributions made !.\ 1.1' Idl y colleagues, partly as collaborators on projects, as co-authors of 11' 11 1, I" (Ir presentations, or generally via discussions: in Lyon, Fran<;oise 11 11I· 1t·1 I~affort and Jean Marie Pinon both with the Computing I II jI,I IIIIIL'lIt of Institut National des Sciences Appliquees (INSA-Lyon); 11 ( l"I"I',c Park. PohChin Lai, currently associated with the Department if r " tl("'lic Science at the Ohio State University, was an enthusiastic " 1'1 >I II L'I'-oriented colleague in the Department of Geography at the UIIl \ I' I, it y ()r Maryland . In particular, we thank these three colleagues for i II I)' willing to spend hours reading earlier drafts of the manuscript. \ t.-w (It her colleagues have helped in particular ways. Fred Broome .j I III' liS Census Bureau , Washington, DC, and John Townshend, I : !I II'"LIII ()f the Geography Department of the University of Maryland, jlllll l(' lIiL'il on an earlier draft of this work; Lisa Wolfisch Nyman created , ) Ilpt· IHll11ber of figures (with electronic aid , of course), and Guinn III IH' I Ilroduced photographic versions for some figures, and executed ill 1011' , ' 1 illustration. The latter was created by the cartographer, Joseph Ii ,10- 1, ;1 proJessor of geography at the University of Maryland , College !'I III f'l 1. lry\;tlld, USA. We appreciate his assistance. We are indebted to iii' ( II \' of I'auova, Italy for the air photograph used in that illustration. I II I" k Th()mpson also wishes to acknowledge the extensive hospitality I I 1, 1111';[ hy various institutions which aided his efforts to learn directly 1i ' 1I1 1 "I' ;lii:t! information systems teaching and practice in Western I Ii l li l" Tlic Department of Geography, University of Edinburgh, .11111Id . [11< ; the Laboratory for Urban Studies of the Universite de ill " ;11 Nanterre , and L 'Ecole des Hautes Etudes en Sciences it:ill ll , 110111 in Paris, France; and the Instituto de Economfa y

A( /, II() w/I'dg( 'III('/lI ,

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XXI

xx

Geografia Aplicad(\~, M;ldlid, Sp;lill. He is especially grateful for the financial support lrom the Ccntrc National de la Recherche Scientifique for his three-Olonth visit to Francc, and to the Consejo Superior de Investigaciones Cientificas of the Ministerio de Educaci6n Y Ciencia for his three-month sojourn in Spain. While most figures have been created by th e authors specially for this book, some illustrations appear by the grace of others. Therefore we acknowledge the following individuals and organizations for granting permission to use their material: the Association of American G eo graphers, Washington , DC, USA for Figure 4.31; the Cartographic L aboratory of the University of Wisconsin at Madison , USA for Figure 3.24; the Bureau of the Census of the United States Department of Comm e rce (and to Fred Broome, Robert Marx and Alan Saalfeld) for Figures 5.8, 5.16, 8.14 and 12.14; the Washington Post Company, Washington, DC, USA for Figure 1.7; the Environmental Systems Resea rch Institute, Inc., Redlands , California, USAfor Figures 1.9 , 1.10 , 2.14 , 8. 15 and 9.24; The Maryland Office of Planning , Baltimore, Maryland , USA, for Figure 6.6; The National Center for Geographic Information and Analysis, Santa Barbara, California, USA , for Figure 4.10; the Maryland Department of Natural Resource s, Annapolis, M aryland , USA , and Dr. pohChin Lai, Ohio State University , Colum bus , Ohio , USA, for unpubli shed material used as Figures 1.6, and 2.15; The Cambridge University Press a nd Dr. Mahes Visvalingham, Th e University of Hull, Hull, England, UK, for Figures 5.20 and 5.24; to Mr. Jan Van Est of IRIS International, Voorburg, The N e therlands, for Figure 4.6; Dr. Marc Armstrong, the University of Iowa , Iowa City, USA, and Taylor and Francis, Ltd., London, UK for Figure 12.7; Dr. John Herring, The Int e rgraph Corporation , Huntsville, Alabama, USA, and the American Society for Photogrammetry and Remote Sensing, Bethesda, Maryland , USA, for Figures 8.16 and 12 . 12; Dr. Ram ez Elmasri and th e Bejamin/Cummings Publishing Company Inc., Redwood C ity, California, USA, for Figure 13.2, and Dr. Pete r Taylor, Newcastle University , Newcas tle-upon-Tyne, UK, for Figure 7.20. Several illustra tions are loose ly based on the work of others; in this case we re ly on source citations within figure captions as our acknowlcdgement. We also point out that numerOIJS figures in this book have appea led earlier in the collection of materials prepared for the S('llIill;11 Oil Multimedia Urban Information Systems presented at Lyon, 1'1 :1\\1'(', N()v~' Ill\Jcr 13-17 , 1989, and distributed as A Primer on MuIIIlIW,\i :1 1111l :1I1 Inlurmation Systems Concepts (edited by Robert I :11111111 :11111 1'1: 1((("I~l' Millcll'l -Kaffort) , through the Urban and Regional

'1',ill :d All:dys is N,,' lw(l!'k I'll l:dllC;lli()1I ;lI1d Training (URSA -NET), , " "l\llIll'I'> ill 1'1;1111(111).', Sl'Ii ,' , > . /\lhcl1s , Gre ece : Figures 1. 13,9.6, 10.6, I(I H. \(11(1 , 10\7, 1() 2() , IlJ n . 11.3 , 11.4, 11.llb , 12.9 , 12.10 , 13.]5, I I :' , 11.1 , 14.7 , 14.11 , 14.1 2, 15 . 13, 15.14,16. 7, 16 .8, 16 .]],16. ]2, II, 1 \ 1(, IS, 1(1.17 . 16.19, and 16.20. Additional figures appeared in the 1111 111",/',1 :Iph for a later sem inar held at Patras , Greece, Jun e, 1990: 1 1I'I II l''> 172, 173, 17.4,17 .5 , 17.6, 17.7, l7.8 , 17.9, 1725,17.26, and 1/ ;, ', Tilese se minars were deliv e red under the auspices of the European , " "lllIlIlily Programme on Cooperation between Universiti es and III.!II',II V (COMETT) , and organized by Nicos Polydorides of the I 1 II I ': I ~ il y of Patras , Greece. 1IIIlllIgllOlit this book we refe r to several commercial products and 11 1111 I,d go ve rnment products. Our mention does not constitute an I lill" l ~ l' IlICl1t, or, indeed, the opposite, a refutation ; we refer to the ite ms dlil ~ lldti()ns of different a ppro ac hes or concepts. Even though some . 1" .111 ', IIl c l1tioncd may soon be out of date, we arc not concerned with 11 !' ld, 11I~ 'l l(;llion speciflcs of the so ftware; we offer the exa mples so that I . ,,1" 1, ,': 111 relate somewhat more easily to the practical world. We I Iii I t' lil:lt the progress mad e by many software , and other, companies I I ill" p:lc.t decad e is one of the cxciting aspects of spatial information I f 111 \.

\ 1'1( ' is d trademark for th e software of the company APIC Systemes, .1 .., uilsidiary of Lyonnaise des Eaux, Pari s, France. \ I, ( I I N N ) is a rcgistered trademark , and ARC/INFO NETWORK ,lI ld ;lI?CIINFO TIN arc trademarks of the Environmental Systems I ~\'''' ~'drch Institute, Inc., Redlands, California , USA. ESRI is the 1' ) 'i slCred co mpany name. I II (. i.., thc abbreviation for th e official name of the Digital Line (;I :lpll data of the US Geological Survey, R esto n, Virginia, USA. I I, 'I!H·..,d:IY Disks are a reso urce produced by the British Broadcasting ( "I poration , London, UK. i. ii I I 11 )( M C is the acronym for the official name of the G eographic II .",' h ie /Dual Indepe ndent Map Encoding compute r system and ,I,iI:lil;ISC of the US Census Bureau , Washington, DC , USA. f t! \/\'{ 'ONE is the name of software marke ted by Se rviologic, i '''').',(1I1, USA. 11 ( )/ ,, ( ) t , is a trad e mark for software of G e neration 5 Technology, III, , We stminster, Colorado, USA. f i I ( I\' /I '.'W is the name of a database design created at th e I I I 1',llllIlc nt of Geography, University of Edinburgh, UK. I iI~ I I\ I S i,s a trademark for software of GIMMS Ltd. , Edinburgh, '., "11,11((1 , UK.

XXII

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\If., jI )\ \ Ji 'I 1.1:1 ' II/I 'J! [',

xxiii

GIRAS is the official )WIlIC or tlte (ieographic Information Resources Analysis System or the US (;cologieal Survey, Reston , Virginia. GRASS is the official name for the Geographic Resources Analysis Support System created by the US Army Construction Engineering Research Laboratory, Champaign , Illinois, USA. GTV is a trademark for the videodisk resource produced by the National Geographic Society, Washington, DC, USA HBDS is a data structure devised by Fran<;ois Bouille, Universite de Paris, France. HYPERTIES is a trademark of Cognetics Corporation , Princeton Junction , New Jersey, USA. HYPERCARD is a registered trademark of Apple Computer Inc., Cupertino, California, USA. IDRISI is the official name of software developed by Ron Eastman, at the Department of Geography, Clark University, Worcester, Massachusetts, USA. INGRES is the trademark of a database system product line of the Ingres Corporation (now a division of ASK Computer Systems, Inc., Alameda, California, USA. MAGI is the official name of the Maryland Automated Geographic Information System, of the Office of Planning, Baltimore, Maryland , USA. MAP is the name for the software and map algebra, Map Analysis System , developed at Harvard University, Cambridge, Massa chusetts, USA by Dana Tomlin . O 2 is a trademark of O 2 Technologies, Paris, France. ODYSSEY is the name of software developed by the Laboratory for Computer Graphics, Harvard University, Cambridge, Massachusetts , USA. ORACLE is a trademark of a database system product line of the Oracle Corporation, Belmont, California, USA. ORION is a product marketed by MMC , Austin, Texas, USA. POSTGRES is the name for a relational database base extension concept developed by Dr Michael Stonebraker, Stanford Univer sity, Palo Alto, California, USA. QUILT is the name for an experimental quadtree geographic information system deve loped at the University of Maryland, College Park, Maryland, USA by Hanan Sam et. SALADIN is a trademark of the Institute for Urban and Regional Information Systems, Inc., Voorburg, The Netherlands. SICAD is a trademark of Siemens AG, Munich, Germany.

Si\I : i 1.1. W( )IUI> is ;, II ;1<1"111:11 k of lite SIll;tllworld SysIl'IllS LilllltcU. ('dlllhridge . 1': llgLllld. {IK .

,')I'ANS is

;1

Irademark of TyddC Technologies, [nc .. Ottawa, Ontario,

( ';111;1(.1;1.

,\1'( )'f' is it Irademark for the SPOT Image Societe, Toulouse, France. ,') ()I . is tlte name for a standard computer query language adapted by lite National Bureau of Standards , USA. S 1'.\"ILM<Jisa trademark of Prime Wild GIS , Inc., Natick , Massachusetts, ()SA.

SYMA P is a software product created by the Labora tory for Computer

(iraphics , Ha rvard University, Cambridge, Massachu se tts , USA.

II( iLR is an acronym for the Topologically Integrated Geographic

":Ilcouing and Referencing System of the US Bureau of the Census, Washington, DC, USA.

,'JOS is a trademark of th e Intergraph Corporation , Huntsville , Alabama, USA. Intergraph is a registe red company name. I RANSCAD is a trademark of Caliper Corporation, Newton , Milssachuse tts, USA. f/(

I N II'TIK is a trademark of the Ameri ca n Automobile Association,

Washington, DC.

\ ( ; 1':M2 is an official name of a software system develo ped at the ( 'S [RO, Canberra, Australia. World Data Bank is the official name for two cartographic databases developed by the Central Inte llige nce Agellcy , Washington , DC.

o

1 Ceomatics Introduction to spatial information systems Today in Redlands, California, a group of people i.s building a digital data equivalent ul' the world land map represented on almost three hundred sheets of aeronautical charts at a scale of approximately one-to-onc million, showing cultural and physical I'c
( Jill co ncep tu al view of spati al info rm ation sys tems begins with three \ iI ;qllcrs empha sizing th e comprehensive nature of spati al data use and ' OI /':llli /.a tioll . W e start with an introdu ct io n to the ge neral co ntext of the l li '(' d~ for and nature of autom ate d inform atio n systems f o r aiding the .• dlilillil of rra ctica l and resea rch prob lems in m any fi eld s. We move from III,' sl ;,gc-se ttin g first chapter to a more detail ed discuss ion o f functions oI l lti ('olllll()llent parts in C hapter 2. In the third chapter w e provide some ti l' l. J1h III' the co nce pts an d practicalities involved in wo rkin g in the d. OII1 :lill or spati al co ncepts.

, I I.,I'/\TI/\I . DATA ORGANIZATION \; \~( . III()V(, loward s th e l: nu of the twenti eth ce ntury , we are positioned II II 1IIIIl' "I' ('xcii ill )!. deve lopm ell ts ill Ih e ability to o btain data abo ut the I dl \"dl',Ji :llld n dlllr:d wo rld s. :111<1 I" II."e Iilose da!;l to do researc h o r to li lll 1II ;I\'lil ';d llJ'()ilil-lll s. I)il,if ;d ;llld ;In;Ji' )I: ,' kclrOllic dev ice s fa cilitate Ili l IIlII ' lil Il IV (II' Jl"Ollln' ~ , III\' til " IlI :I\' .. r rlilori sli ,' s('l'Il;lrios ;llld th e

4

Ceomatics

rapid execution of many arithm e tic or logical operations. Conte mporary computer-based tools for working with d ata fo r phenomena on, above o r below th e ea rth's surface (known as spatial information systems) are num e ro us; they can utili ze dat a o f many forms a nd are still und e rgoing much improvem e nt. In a period o f increasing uses of digital computers, we still use ph o tographi c film ca m e ras , filing cabinets, paper m a ps and mathe matical to ols develo pe d cen turi es ago. But we a re now able much more rapidly to cre ate , manipulate, store a nd use spati a l data by mea ns of info rm a tio n sys te ms. Spatial is a te rm used he re to refe r to located data , for objects pos itioned in a ny space , not just geographical , a te rm we will use for the world space. An information system , a collective of data a nd tools tor working with those data, contains data in analog fo rm, for exa mpl e, ha ndwritte n no tes or ph o togra phic slides; o r digital form, for example, by compute r bin a ry e ncoding, a bout th e ph e no mena in th e rea l world . Our pe rceptions of the world, inv o lving cognitive pro cesses like se le cti o n, ge ne ra li za tion a nd sy nth esis, g ive us informatio n which can be shared amo ng and use d by peo ple other th a n the creators (Figure 1.1). The physic a l re presen ta tio ns of this information, that is , th e d a ta , const itute a mode l o f th ose ph e no me na. So th e co llection o f d a ta , the databank or datahase , is a ph ys ica l repository of varied views of th e rea l world re prese ntin g o ur knowl ed ge at one point in tim e. In th e d at abase context th e te rms data , info rm a tion a nd kn o wl e dge are diffe renti a ted . Information is d e rive d fro m th e individual da ta e le ments in a data base - inform a tion th a t is no t directl y a ppare nt. In a se nse, th e inform a tion is pro duced from data by use of o ur th o ught processes , intuitio n o r whate ve r, based on our know ledge . Fo r exa mple, a bi cycle can be re prese nted by circles fo r th e two wheel s, va riou s lines fo r th e frame , whee l spok es, a nd so on . Th e geo me tric entity of th e bicycle is th e asse mbly of the d a ta elements; hum a n knowledge is need ed to deduce th e rea lity of a bicycle as a mea ns o f tra nspo rta ti o n a nd the ste ps necessary for the physical object to fun cti o n . In a less comp lex case, knowl edge about world la titud e a nd lon gitud e ean a llo w us to make se nse of some numbers in a d a ta base . An x o r y coordinate is a d ata e le me nt. Extra data are needed to ma ke sense of th e magnitud e of th ose num bers. We know that hi gh pos itive valu es for x must re fe r to a place nea r th e North P o le, provided th a t we have info rm a tion a bo ut th e sca ling o f th e axes a nd how the y fit th e rea l wo rld . H av ing the 'extra d a ta' a llows LI S to produ ce th e inform ation from th e d a ta. Inte rpret a ti o n of discrete , apparent ly unre lat ed , pi eces of data is akin to ha ving ' add e d v;tlue ' ; IS we pn> g ress ['rolll (\.lIa , to inlorm ' ltioll, to knowkd gl' .

\ patial data organization MATERIAL

5 LEVELS OF APPRECIATION

PROCESSES

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London Madrid Moscow New York Paris

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Figure 1.] Different interes ts about the 'Rea l W orld' .

Th e m aterial in thi s me la nge of 'data' has many conce ptu a l origins and [mms (Figure 1.2) . The d a ta may be any of the foll ow ing: I . Real, for exa mple the te rra in conditions or buildings .

) Ca pture d , tha t is, record e d by physica l d ev ices like e lectronic se nsors ,

and film ca m e ras; fo r exa mple, seismic signals o r la ndscape pictures.

\ . Int e rpre te d, that is, inVOlving so me hllm ~ n inte rve ntio n as in field

ske tches or lanu sca pes, a questionnaire, or writing in books.

,I. E ncoded , ;IS in paper maps , dig il;t\ d, lliI for d ep th s of oceans or

sUllllllary SI
'> . StruclUI\'d III (11").'.< 1II i/.t"II ill .' (lIII \" wil y. s li r li ; 1:-' !;Ihl es in census reports ()r t!;11 1I III 1'.l·"I',llIplii (· in[prIILIII
6

Ceomatics Field sketch

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7

zoning conditions, landowners , and improvements to property. This layered data set approach is associated with photogrammetry, computer
.

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Figure 1.2 Various forms of data.

It is the structuring for this variety of information in an environment of digital computers that is the subject of this book. As you might expect, just as there are many ways to organize furniture in a house, so there are many ways systematically to arrange information about the world. Various commercial software packages are available to aid this struc turing; there are some dominant forms or models for working with spatial data, and there is a growing body of fundamental principles and formalisms to guide us in the process of information and data organization. By principles we mean a conceptual framework for information organization; and by formalisms we refer to mathematically based representations and organization of spatial data. One common organization (Figure 1.3) is that of layers of sets of maps, Each layer, representing a thematic approach to a particular purpose or set of needs , may contain one or several different kinds of information. For example, for studies of natural resources, bedrock geology, subsurface conditions, land use, soil types, drainage , elevation above sea level, slope and aspect, and transportation features, may constitute the data thcmcs. For urhan planning , an information sct may contain data for ~(rc"'s, Illilily lilies, tr;llIspo('LII;OIl k;illires , (;IX and

Geodetic Control

Composite (excluding elevation) (. I)

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Figure 1.4 Layers and tiles for map data . (a) M ap sheets and th e mat ic layers. (b) T iles and thematic layers. Figure 1.5 The object-based form of o rga ni zation.

A structuring into themes and pieces of space (Figure 1.4) , perhaps counties o r map sheets or special units called til es, is very common in analog inform atio n systems as r epresented by photographic film positive plates for different ca tegories of informatio n gathered by the British Ordnance Survey, or the pl ates for the 55,000 topographic quadrangles at a sca le of 1: 24 ,000 of th e United States Geological Survey . This organization is also co mmon for automat ed spatial informatio n systems, refl ectin g the fact that these commercial products were first developed as digital equiv alents to the pape r map collections. Another arran gement (Figure 1.5) works with a singl e geographic area containing a variety of objects, some, or all, of which may be of inter est at any one time . These entities are imagined to be in one lay er o nly , thereby allowing the tbird space dimensio n to represe nt the world vertical variations as well as providing a latitudello ngi tude reference frame . This object orientation is associated with human cognitio n and reasoning , but is not so good for dealill/-', with surLI(;es or re
It \,' I L"1 1I1;lt ch with a conceptual laye rin g, a concept very useful fo r use r ,; ,"tl il il . Th e details of forms of systemat ic represe ntation are as much a 1',, 11 "I" thi s hook as the general conceptual st ru cturing into layers o r 1I 1 111"~ W e Ircat Ihe foundati o ns for this topic exten sively in Part Two . IIHkpL: lld L: nt of th e wo rld of autom ated mapping , use rs of spatial data 1tillT IOll g worked in a particul ar mode: th e world of the paper m ap . Such I 1I 1I ' lltod or rL:co rdill g and lI sing data for the earth, which we will ca ll th e '"IIP lIIockl , is OIlL: ()f thc steps alon g th c way to th e modern-day digital 11\ II "llIll,-' II\, ;llId is see n in Ihe replical ion o f manual procedures, use of :"111.11 IL" t"llliliol ngy, ;IIHI I"r;lI11 C of milld ;Isso~' i ; tl l' d w ith so me co mputer11(1ti( d " Il;tli ;t\ (\al ;1 pr()("(" ssillg . as I~ dl"lliOIlS11 :1II" d ill slIhse qu ellt chapters.

Ceomatics

10 ,

1.2 HETEROGENEITY OF USES OF SPATIAL INFORMATION SYSTEMS

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z 1.2.1 Uses of spatial information systems Today, looking at the worlds of business and government, we see spatial information systems applications, activities and tools for dealing with

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Computer-aided design

Thematic mapping software

Marine cartography

Remote sensing systems

Surface modelling

Environmental modelling

Resource management

Transportation planning

Emergency response

Geomarketing

Geotechnics

Archaeology Military exercises Written in no particular order, this deliberately mixed bag of terms conveys descriptive characteristics of: type of subject matter (e.g . land), purpose or activity (e.g. city planning), and instrumentation (e.g. remote sensing). In the large realm of practical uses of spatial information systems, the general purposes include: inventory, a dat
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Because the different domains of use of automated spatial information systems have particular needs, it appears at a general level that there are many, perhaps too many, commercial software systems or theoretical models to establish principles or order. The language confusion that can, and does, arise from the use of similar tools in different fields, in part a reflection of the development of the contemporary information systems from different roots over a twenty-five year history, adds to the difficulties in understanding, evaluating, and discussing these automated

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utilities infrastructures (Figure 1.7); and administrative record -keeping, as for property-based taxation. In these domains , typically large quantities of real world objects are mapped at large scales. Whether or not analytical or querying procedures arc used. spatial information systems are often employed as support resources for decision making, as in bus route planning or determination of market areas. There is also an increasing use of such systems for educational purposes. either in schools or at the individual's home. Knowledge of particular requirements is important for appropriate and effective design of the entire system of tools for processing data for earth phenomena. For example, the layered style, also well demonstrated by Figure 1.6, reflects the manual procedures used in composite mapping and map modelling of the late 1960s and early 1970s. This design type may not be the most efficient for all needs. We illustrate this topic of information use tasks at this time by only two examples, simply to establish the general content and orientation for this book. More extensive treatment follows in Chapters 2 and 3.

1.2.2 Examples of data requirements Way-finding is a basic human need. Knowing how to navigate through a city street system, how to find the best route from home to the park , or giving directions to strangers who show up lost in one's neighbourhood, are a few varieties on this basic theme, assuming that persons are involved (for they do not have to be, as seen in the current developments of autonomous land vehicles). If we imagine. though, way-finding in a car, we can imagine that we need the following: 1. 2. 3. 4.

To know the current location of the vehicle. A process to find a destination. The retrieval of a map to help in the location and way-finding task. The presentation of information to the navigator.

Figure 1.7 Utilities networks-an artist's view of subterranean Washington. Public utility companies worldwide are beginning to invest huge sums of money to create computer databases for pipes. poles, and pathways, cables, and conduits , sewers, valves , and tracks, etc ., below, on, and above ground. (This caricature appeared in The Washington Post ncwsp;qx: r , Novemher II, I')SS. under the caption 'What is under Ihe strcch 01 W;lshill.~i"ll'l ' The piciure is used with the pc:rIlHssi()1l ,,1'1'11( ' W:lshilll',loll I'w,\ 111(' , W;lShilll',1()Il, DC.)

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Ceomatics 14

Assuming that national legislation allows ve hicles to carry visual display devices to aid in thi s task , then imagine a small television display screen at the front of the vehicle. The monitor shows a detailed map of part of the city, with a blinking cursor identifying the vehicle location (Figure 1.8). The screen also has a small-scale map inset for a much larger area. A destination address or eve n place name is entered by some means , and the on-board computer determines a good route to get to that d estination . Assuming that national laws allow drive rs to activate video displays while vehicles move , as the vehicle moves a long streets , the map changes, showing only features necessa ry to help in the driving task, for example by signalling when a turn should be made, and identifying temporary barri e rs like road works. Rather than co nsider the electronic d ev ices needed - a nd th ey all exist today _ let us focus on the data require ments (White , 1987). We need coordinate inform a tion to draw detailed maps revealing the shapes of the roads along which a journey might be m ade . We need to know street addresses , names of locations , possibly important features that can be used in a landmark oriented guidance system, and street names (recognizing that in some countries one street can have many names). Some ancillary data, like the existence of traffic works, might be helpful in choosing a route, or knowing the aver age trave l times. Also , it will be important to know if and how particular streets are connected , in order to indicate the sequence from origin to destination o r for determining how

15

and where to make a turn. Some filtering tools will be necessary for prod uctidn of the right kinds of map for assisting the navigator , for not all dcta ils are needed for following a pa rticular route. Many data ite ms, appropriat~ s tructured, are thus needed for effective and efficient (timely responses to the user) vehicular based way-finding. Consider, nex t, a research problem: analysing the impact o f the pattern of land uses of a landscape on th e amount of chemical nutrients Ih a t are disc hargcd into streams (Fi gure 1. 9). Sin gle event o r long-te rm p,ltterns of precipitation create conditions of oversurface wash , channel i/,ltion , seepage and underground flow . The amount and speed of exodu s 111/0 the river o r stream channe ls is affected by many factors includin g ~ I ()pe angle , soil perm ea bility , underground hydrology and type of wgctation (for exa mple, forested or not) . It is therefore importa nt to 1i ;lvc information abo ut re la tive elevations, surface g radients a nd aspects , ;IIHI the sequence of land uses along the rive r courses. It may also be Ill"Ccssary to know what is under the surface , in case so me underground d r: linage causes flows to bypass the study area . To dea l effectively with Ilie ,lJl a lysis may necessitate certain kinds of data structuring; not simply Illl' Clcquisition of information.

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Ceomatics

Some components 17

The common elements in the many domains of the use of spatial information systems, as demonstrated by the two more detailed examples are, howeve r, not extensive. Wh e ther we are navigating through city streets or following chemical laden water over a la ndscape; or whether we are inventorying city telephone poles, or counting forest fires ; or designing the layout of EuroDisneyworid near Paris or forecasting the weather for the Commonwealth Games in Auckland, N e w Zealand; or planning the extension of a sewer system underneath the streets of Moscow, Russia, we deal in a general sense with just the following few features like: l. 2. 3. 4. 5.

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In other words , we deal with properties of spatial entities positioned in a spatial refere nce system.

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1.3 SOME COMPONENTS OF SPATIAL INFORMATION SYSTEMS An automated spatial information system is a toolbox for represe nting views of the real world via data about locations (Figure 1.10). Spatial information systems are a technology for process ing spatial data. The tools, whieh may be activated by pushing a button or typing a command, represent processi ng functions or operations for example, drawing a map, or measuring the distance from Paris to Madrid . The tools work on so me or all of the information stored in some systematic way in a database. The information system requires data, software, hardware , 'brainware' and other reso urces, and exists within some institutional setting as a resource to solve problems.

1.3.1 The toolbox view The support for d ecision making, practical or academic, may consist of virtually instantaneous answers to questions, a query type of system. For example, the best route for crossing a city is disrlayed in map form for the vehicle na viga tor, or Ihe urhan geog rapher Gill quickly find which citics in l ': uropl.: :Irc k:s ~ 111:111 IO() kilonH; lrcs 1i'(>l1I ;IIIY lIalion;d frontier.

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1'(' lll aps in th e form of maps or tables, for subsequent study. PrOducts

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Geomatics

18

systems must, at a general level of detail , fulfil the following:

1. Provide tools for the creation of digital representations of the spatial

phenomena, that is, data acquisition and encoding.

2. Handle and secure these encodings efficiently, by providing tools for

editing, updating, managing and storing; for reorganization or

conversion of data from one form to another, and for verifying and

validating those data. 3. Foster the easy development of additional insight into theoretical or applied problems, by providing tools for information browsing, querying, summarizing and the like: that is to say, facilities for

!orester, politician). Then the information subsystem itself has a physical component (the hardware and data encodings), the documentation component describing the characteristics of the spatial information system , alTd the guid a nce component (the person who looks after the database and tools , and solves malfunctions). Consequently , beyond a general level of the domain of the use of spatial information systems, we can begin to appreciate differences in lerms of General purposes Themes of information Types of data Particular tools Specific processing requirements Specific forms of data organization

analysis, simulation and synthesis. 4. Assist the task of spatial reasoning, by providing for efficient retrieval of data for complex queries.

5. Create people compatible output in varied forms of printed table , plotted map, picture, scientific graph and the like. In a sense, the spatial information system lies within a larger framework of information requirements of the human world (Figure 1.11). Just as the plumbing system in a house has se parate but functionally related wate r, odour and waste subsystems monitored by the householder with the aid of a se t of plans (or the map in the plumber's head) , so there are physical, informational and guidance systems for phenomena at large. In a spatial information system context, there is the component of observable phenomena, the component of 'knowledge' about those phenomena , and the component of guidance (the planner,

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I, Disciplin es that have d eve lo ped conce pts fo r dealing with space: co gnit'ive scie nce, geo gra phy , lin g uistics, psycho logy . 2. Fie lds that d e ve lo p practical tools a nd instrum e nts for obtainin g o r wo rkin~ with s pa ti a l data: ca rtogra phy, geod e tic science, pho to gr a mme try, remo te se nsin g, s urv ey ing e n gineering . 3. Di sciplin es that provid e fo rmalisms a nd theories fun da m e ntal to our workin g w ith space a nd a utomation : co mputer sci e nce , geo metry , inform a ti cs, a rtifi cial inte lli ge nce, semiol ogy , sta ti stics. 4, Fi e lds makin g s ubstanti a l use o f autom a te d s pa tial info rm a tio n syste ms: arch aeology, civil e ng in ee ring, for estry, geotechnics, la nd sca pe a rchitecture, urb a n and regio na l plannin g. ,"i . F ie lds th a t provid e g uida nce about in fo rma tion : la w , eco no mics . The s tre ngth a nd inte ll ectu a l inte rest of th e s pa tial inform a tio n systems derives fro m the c ross-fe rtiliz ation a mo ng these ma ny fie lds. W hil e some writers use the term geographic info rma ti o n system s to re fe r 10 th e composite fi e ld of study, we pre fer the te rm geomatics as a n Illllbre ll a cove ring all th e fi e lds li s ted a bo ve that are to d ay impo rtant fo r 1IIIdersta ndin g and furth e r d eve loping sp ati a l informatio n systems. Idc as a nd tools from ma ny field s a re, th e n, necessa ry for a full IIl ll icrs tandin g o f sp a tia l info rm a ti o n system s, A s use r require m e nl s and ,',x pcctations in crease for suc h reso urces and too lbo xes, so th e re are IIl lcll cctual pressures o n the th eo re ticia ns and prac titio ne rs of geoma ti cs I I I coo pe rate in inte rdi sciplin a ry research s tudi es a nd de velopm e nt I" (lj ec ls to further imp rove o ur autom a tio n to ols, F o r exa mple , in th e I IS !\ th e N a ti o nal Sci e nce Fo undation is p rov iding about $6 million ove r . 1 li vc yea r p e ri o d for a group o f schol a rs to und e rtake a thirteen-item I l ' ~l' ; lr c h pl a n . This pl a n cove rs muc h theore tica l resea rc h into to pics like "I ': lli :1I stati s tics. me thods of spati a l analysis , da ta base struc tures, \ 1 ~ ll
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man y practically ori e nted fi e lds. P e rha ps 80 per ce nt of d ecision s by sta te and local go ve rnments invo lve a spati a l co mponent e ithe r directly or by implication . G eographe rs a re used to thinking spatiall y, a nd geography h as the same common inte llect ua l root as gcom e tr y. Architects , eith e r building or land sca pe, work w ith s paces. So o ur search for organi zation of m a te ri a l ahout s pali a l illfo rlllaliol1 sys lc lll S li as 10 look 10 many di sciplines

and li c ld s of sllldy . 'I'II l ' S l.' :11',':

21

22

Ceoma tics

based o n ma ny exa mples, mu ch like a n und e rsta nding o f fin e a rt co mes a bout by looking a t a nd st udyin g m any pictures. We m ay acce pt a uto ma ti o n as the way of the las t decade of the twe ntie th century, but we must a lso have a view of non-a utom ate d spati a l inform a tion syste ms , fo r most o f the info rmati o n abo ut and g uid ance of th e wo rld 's physica l sys te m a re not ye t within a computerized e nviro n me nt. Whil e the concepts of data stru cturing, concepts of spati a l re lati o nships, a nd ma tching too ls of drawing and tab ul a tin g ex istin g in the no n-automated world have bee n take n to the automated world, it ha s no t been witho ut cost. So we presume , o n th e one ha nd , tha t the spati a l informati o n sys te m in a computerized e n vironme nt a llows some probl e ms to be handl e d more e fficientl y a nd cos t effectively . On th e ot he r ha nd , we ex pect to be able to und e rtak e pro bl e ms otherwise co nsid e re d in so lubl e. Automation allows faster creation a nd use of la rge quantiti e s o f inform a tion , speeds up ma the m a ti ca l or log ical operations on data , a nd makes possible tasks that are o th e rwi se unman agea bl e. The spatial information system , then , is a compute ri zed e nvironme nt whereby utility progra ms performing specifi c fun ctio ns a re used in an integra ted e nviro nme nt , in which the use r is shi e lde d from the details of computer processing, to achieve so me goal of resea rch , edu cat ion or decision making. The inh e re nt form of s patia l data re prese nta ti o n a nd o rga ni zation must be de signed to support effective ly and e ffici e ntly the kinds of que ry and a na lysis re quire d by many use rs . Th e perfo rm a nce of computer sys te ms is a re flection o f ha rdware techno logy a nd software e nginee ring , but also re fl ec ts th e data structures a nd the qu a lity of a lgo rithm s. We approach th e questi o n of d esign of spa tial inform a tio n sys te m s at th e conceptual leve l. Elaborating on Figure 1.1 as Figure 1.13 , we prese nt a process (taken up further in Chapter 9) demarcated for practica l convenience into four stages , proceedin g from use rs' views of their needs to the computer's physical e ncodin g a t the leve l of hardwa re ( known in computer j a rgo n as the byte leve l) . Using the te rmino logy from the field of informa tics, th e real wo rld is seen from an ex te rn a l view, re prese ntin g the need s, objectives a nd process o f extraction o n the part of different peopl e or organizations. E ach such subset of the rea l world ca n , refe rring to the info rmati o n system too lbox, be th o ught of as a n appli ca tion . Conceptual modelling, which we treat exte nsi ve ly in P art Three, covers the synthesis of the exte rn a l mod e ls. It is he re that we can think of common ground among use rs, in part the unive rse of discourse, in pa rt the unive rse o f m o de lled e ntilies. Using tool s ;1I1<J form a li sill s di sc ussed later in thi s book , Ih <.: ph e noill e na of illll'l'l's i ;lll' idc nlili <.: d , th c ir pcrtincnl c hal ;l(' t ~' ri~tiL' s (k ~ l,Til lL· d . ;1I1l1 how objcct., H: lat l' to c; lch olhcr is

The role of automation

23

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II Llpped o ut (IS well as pOss ibl e . Diffe re nt views a re integ ra te d via a ' "lll lllon la ng uage and structure. Th e logical modelling stage tra nsla tes I" ,' co nce ptu a l o rgan izatio n int o so me thing more practi ca l, pe rh aps " "pl y thought of as putting num e ri ca l values into tables of d a ta, but I\ It;d ing the detail s of sto rage of da ta o n physica l medi a. It is at this leve l tl1.lt ph (; llOlll e nii are orga ni ze d into (I d a tabase as tabl es of d a ta records I li d ("() llIle ctioIlS to other tahles. TIl(; int e rn a l mode lling, which we d o not '''''ll'S,\, IIC, lI s til l' olganiz
Ceomatics

HibJiography

25

24

think about the value of th e information in the database. The contents, certainly reasonably viewed as a commodity, are presumably valuable for one or more purposes, yet such data alone are not enough. Users provide the added value by interpretation and analysis , and by understanding the nature and quality of the data. It is important to focus on the message, the contents of the envelope, not the messenger, the postal system. Consequently, our discussion of data structuring in an automated system

is put in the context of the semantics of the data.

Automated spatial information syste ms, th e n, are many things to many

people . By way of summary, they are: A data depository

A toolbox

A resource

A technology Frames of mind, one or seve ral As a set of software process ing routines in a hardware setting they are a new kind of toolbox for practical problem solving. As a new resource compared with paper based map making, they represent a new technology; and, through their emphasis on spatial data, they stand, for many people , for a different approach to thinking about probl ems and knowledge. Computers provide compact forms of data storage, fast retrieval and mathematical equation solving; can store vast quantities of information, but especially allow link ages to be made among separate and apparently unrelated pieces of data. But computers are not enough . The use rs of information systems must have a good sense of human error and be capable of making wise judgements. The path from the real world to people's views to digital encoding must be trodden with care; otherwise we will not be able to make the return journey, from data to information

to

knowledge.

1.5 BIBLIOGRAPHY The references given at the end o f each chapter as Bibliography are th e principal works used in the preparation of each chapter. M any oth er books ma y have been utilized peripherally. or represe nt writings on recent technical deve lopments. In some pl aces important rderences for historical p\lrp()SL~S arc likewi se not 1ilic indicated hy name . We d() not incIl rpor ,lIc IOllg ddailcd hibli()gr;'1 li sts or in text citations. Thl' rL'('( 'nl 1ll;Ij"r work;.; cilt'd (,"" 1)(' \·'II1.;';IIIICd f()l additillnal likra\url' . 11I . lt'xl ('il ;,III"I·.. h\' n', ;I,S ill illl,,,,IIIt'I"I Y \>""k s ill Il sc d

I~\'n\' r;d,

sraringly, indicate sources for particular illustrati ons or techniqu es . For th e fi rst chapter. we provide a selection of items broad enough in scope to provide materi al on the gen eral n ature of spatial information systems. This includes inh:.oductory tex ts on geographic information sys tems, geom atics , and clo se ly rel ated topics that readers may consult for alternative approache s to the ;.;ame topic of spatial inform ation sys tems. In subsequent chapters the biblio .l!, r. 1111 1111 , 1\,11,\' 11
Geomatics 26

Nagy, George and Sh arad Wagle. 1979. G eog raphi c data processin g. Association fo r Computing Ma chinery Computin g S urveys 11(2): 139-1 81. Raper , Jo nathan (ed.). 1989. Three-dimensional Applications in Geographical Info rmation Sys tem s. Lo nd o n , UK: Ta ylo r and Fra ncis . Ro binove, Charles J . 1986. Principles of Logic and the Use of Digital Geographic Informatio n Systems. U.S. Geological Survey Circular 977. Washington, DC, U SA: US Government Printing Office. Sa met , Hanan. 1990. Applicalio ns of Spatial Data S lruclures: Co mpuler Graphics, Image Processing and G IS. Reading, Massach ussetts, USA : Addison-Wesley. Smith , Terre nce R ., Sudhakar Me no n , Jeffrey L. Star and John E. Estes. 1987 . R eq uirem en ts and principles for th e impl e me ntatio n and co nstructi o n of large sca le geographic information systems. Inlemalional Jou rnal of Geog raphical

Informalion Systems 1(1): 13-3 \. Star, Jeffrey L. and John E. Estes. 1990. Geograp hic Info rmalion Syst ems: An Introduction. Englewood Cliffs , New Je rsey , USA: Prentice H all. Teicholz, Eric and Brian J . L. Berry (eds). 1983. CompUle r Grap hics and Environmental Planning. Englewood Cliffs, New Jersey. USA : Prentice H all. Tomlin, C. Dana. 1990. Geographic Info rmation Systems and Cartographic Modelling. Englewood Cliffs, New Je rsey , USA: Prentice H all. . Tomlinso n, Roger F. 1988. The impact of th e transition from analogue to digital cartographi c represe ntatio n The A merica/! Cartograph er 15(3): 249-262. Unwin, David. 1981. Inlroductory Spalial Analysis. London, UK: Methu e n. White. Marv in . 1987. Di git al map require me nts o f vehicle navi gat ion. Proceedings of the Auto CarlO 8 Conference, Baltimore, Maryland , USA, pp. 552- 561.

2 Nee'ds Purposes and types of spatial problem .lane Forester wants to know how much timber there is in the landscape she can see from a lookout tower. Jean Paul Pompier needs to determine the fastest route to send
lit (his chapter we treat the specific spati ally oriented tas ks that can arisc IV lt II in a nd across different domains of study or analysis or decision 111:lking. We present several approaches, each with a crude cl ass ifi ca tion ," specific spatial questions. In time we expect there will be a more 1I 'IIIled taxonomy of spa tial problems at a high le vel of detail , possibly I 1" :I Icd as a result of co-operative work combining several di sciplines, but IIIl' yo ung field of geomatics is not yet so developed,

I)ROBLEMS TO BE SOLVED; TASKS TO BE

PE RFORMED

\ "l' l l.~iillc approach to desig ning , evaluating, buying and usi ng spatial 111 1I 11 111 ;11 io n systems is to know what particula r , prefe rably well Ii, II l.lIv; l(cd problems, scientific o r practica l, have to be solved . A study II I j ll ' II ~' I ; 1i C
28

Nee ds

Problems and tasks 29

WHAT IS THE HEIGHT OF WATCH TOWERS? WHERE ARE THE WATCH TOWERS? WI-liCH TOWERS ARE CONNECTED BY TRAILS? WHICH PATHS WILL SHOW A JOURNEY TO EACH TOWER? WHICH PATH WILL INVOLVE THE LEAST TRAVEL? HOW MUCH TERRAIN CAN BE SEEN FROM A

PARTICULAR WATCH TOWER?

WHAT KIND OF VEGETATION IS THERE IN THE

FOREST MANAGEMENT REGION?

WHAT IS THE AREA OF EACH TYPE OF VEGETATION? WHAT IS THE AREA OF EACH TREE TYPE FOR

DIFFERENT ZONES OF ELEVATION?

WHAT IS THE AREA OF DIFFERENT TREE TYPES IN 500 FEET BANDS ON EACH SIDE OF THE TRAILS?

geome try a nd exe mplified , a t leas t in Som e pa rts of th e world , by precise bo unCla ri es used fo r de ma rca ting pri vate pro pe rty. Sch OO lchildre n a re o ft e n as ked to look a t m a ps a nd to fi nd a pa rti c ul a r fea ture 'or to id e ntify som e p a rti cula r a ttribute o f a n o bj ect a t a give n locat ion , fo r exampl e, wh a t kind o f fo res t is foun d nea r the E qu a to r Th a t is: WH A T IS IN A PARTI C ULA R PLA C E ? FIND THE P O SITION OF SOME OBJEC T . T hu s , C ity M in Fig ure 2. 2 is pos ition ed at th e inte rsectio n o f Free ways D and L ; o r a lte rn a ti ve ly , at 25 units go in g east fro m the left side of th e m ap , a nd 12 units up . Th e qu es tio n ' Wh a t is in a pa rtic ul a r pl ace? ' may he a nswe red by findin g a ll thin gs within th e circle. Alte rn a ti ve ly, it ma y res ult in a spec ific kind of phe no me no n lik e the tra nspo rtat io n ne two rks , ,IS mig ht be necessa ry, in the s pirit of Fig ure 1.7, in cases o f mainte na nce O J' re pa irs. Thc de te rmin a ti o n o f bo th ca tego ries of info rm a tion , a kin to t he in fo nn a ticia n 's no tio n of da ta base qu e ries , may includc bounds p la ced o n specifi c qu esti o ns. Th ese co nstra ints m ay be o f se ve ra l types:

Figure 2.l So me specific spa ti a l qu esti o ns .

r *

id e nti fica ti o n of appli ca tio n subdo m a in s, th e boxes , r e vea ls th e ty pica l s itu a ti o n in la rge gove rnm e nt o rgani za tion s o f m a ny se parate uscs o f diffe re nt p ieces of data co nta in ed within th e o rga niza ti o ns' fil es , wh e th e r a uto mated o r no t. This approach hig hlig hts th e co nce pt o f use r vie ws o f a d a ta base reso urce. In co ntras t, F igure 2.1 focuses o n th e sp ecific qu esti o ns be ing as ked within a ge ne ral 3rea o f fo rest m a nage me nt. The ques ti o ns may be e asil y t ra nsfe rre d to o th e r do m ains of stud y. Fo r e xa mple , we may wis h to know th e he ig ht of te le ph o ne po les o r o f buildin gs. W e ma y wi sh to meas ure dista nces a lo ng fo o tpath s within a co mplex o f buildin gs in a city o r ascerta in th e so la r shadow reach o n the g round . So o ur a ppro ac h to st ructuring a n inform a ti o n sys tem in th e are a of spa ti a l a naly sis is to ide ntify types of p ro bl e m based o n p a ram e tc rs o f sp ace a nd locati o n. W e will imag in e th e questi o ns via th c mind of a nin e year ()Id in sc hoo \. At a bo ut th is age , th e yo un g mind is "hie t() h;llIdlc cO ll ce pt s li ke th e qu ;tii ties or S P ;I ': l' , S licil
~

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Needs 30

Problems and tasks

31

1. Based on character; that is , by a range of specified properties (values, quatities, combinations), for example, the kind of forest. 2. By count: that is, looking for only a certain number of objects, as in finding"three plants in Figure 2.2. 3. By distance from a specific object; for example, what exists at 100 miles north from Darwin, Australia? 4. Included within zones defined arbitrarily by geometric figures, such as the circle in Figure 2.2, or existing as real phenomena , like a continent or country.

Vi

What do we have on this line, at this point, or in this area?

Some of the many varieties of queries that are spatialJy bound , such as the distance and inclusion categories, are demonstrated in Figure 2.3. The variety exists for both the nature of the spatial unit that is found by the query, and the object that defines or constrains it, as shown in the top-left hox. Students may be asked to make measurements, perhaps on the ground or from maps, for example. the number of paces in walking from home to school, or for the airline distances between cities. That is:

What do we have in this a;ea or on this line?

What do we have in this circle zone ?

MAKE MEASUREMENTS FOR OBJECTS IN SPACE or , indicate how places relate to each other; for instance, who are their Ilcighbours at home or in the classroom? In other words: IDENTIFY SPATIAL RELATIONSHIPS

What do we have in this buf1er zone?

What do we have in this zone?

Spatial measurements involving one or more objects on the earth may Iclate to many properties: the distance apart of administrative centres, llie pattern of cities as seen at night from a spaceship, the size of lakes, Ihe total area of grandfather's several scattered pieces of land, or the disl;1I1ce to the nearest gas pipe or bus route (Figure 2.4a). The category determining spatial relationships also has varied content, such as d I:Iwing cognitive maps , or demonstrating a knowledge of world issues by k llillg friends the names of the countries which border China (Figure

t"

, ,Ill).

( 'hildren likc maze games; they may be asked to solve problems like 11'1 Inc on a beach they could set up an ice cream stand to make the III/',!'csi profit. A tough earth science instructor may even ask the pupils to V \l l' ~ ~ whal is under the ground of their state or county. That is , we have l"I'I('s Ldlillg into two categories of spatial problem solving: Figure 2.3 Some types of spatial queries.

Needs

32

.; :~

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Spacing 01 administrative centres

Size 01 barrier islands

Network distances

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(a)

Iran

Neighbours

Cognitive map

a nd spati al re lations- (a) Figure 2.4 Some types of measure me nts kinds of ph e no mena. (b) Ex amples of measurements for diffe re nt Ex amples of spatial relationships.

The first of these, representative also o f spatial decision making pro ble ms , may include a variety of ph e no me na with diffe re nt spatial form _ de te rminin g th e best location for a hospit al or rest aura nt , finding a path through a maze , creating a path ove r mo unta ino us terrain , deciding which childre n go to which school , layin g o ut the a ttractio ns in an amusement S = school A = farm boundary

==---

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33

p a rk , o r all ocatin g differe nt types of land to parti cul a r fa rm e rs (Figure 2.5) . ( Spatial inference ma y be necessary for domains lik e min e ral prospect ing whe re..- it is impossible to ha ve many direct o bse rv a tio ns, or for inte rpre ting digita l da ta tra nsmitted from sate llites. Fi gure 1. 9 s uggests th e conditio ns th a t might need to be assessed to be a ble to de te rmin e th e e xiste nce o f a n aquife r based on limited info rm a ti o n fo r s urface wate r hydro logy, a nd a fe w bo reholes or tunnels , pe rh aps a lso na tural ca ves , pro vidin g subte rra nea n data . It is not inco nceivable that children could a lso e ngage in more challe ngin g activiti es like making maps th a t re prese nt so me vi ew o f the future, o r maps that represent the combin a ti o n o f specified co nditions. Th a t is: PREDI CTIVE SPATIAL MODELS

shOpS

(b)

Loca tion and character

Thi s ca tegory includes activities like answe rin g 'wh a t if'? ' questions or und e rta kin g simulations of known processes to produ ce different o utcom,e s . Fo r e xampl e, the best places for building mo re houses in a region in th e urba n pe riphery of Washing ton , D C (Figure 2. 6) may bc diffe re nt according to the specific perspectives o f a n e nvironm e ntalist, co nstructio n compa ny or local governm e nt ho us in g a uth o rity . This cate go ry o f acti vity also includes sci e ntific simul a tio ns to unde rstand be tte r th e e xisting o bserved phenom e na o n the ea rth ; fo r exa mple, that of pred ictin g pa tterns of rural land use o r o f mode lling th e deve lo pment o f rive rs in a ge ol ogic time span . Figure 2 .7 provides one summary of th e va ri e ty o f spa ti al situations lh a t a rise. D e te rmining position and cha racte r a re pe rha ps th e most basic geogra phic a nd spatial questions. Th e y ca n, h o we ve r, be ve ry different d e pe nding o n the restrictions or qualificati o ns of th e qu e ry . Measuring , prope rties of spatial entities or rel a tion ships a mo ng se ve ral objects ttnd e rli es much scientific endeavour. Choosing pl aces or other spatial l' ntities by comparison of alternatives re fl ecting diffe re nt opportunities or l'l lll straints is a major type of public or priva te d ecisio n problem. Spatial plcdicti o n , pe rhaps used for inferring the loca tio n o r existe nce of objects, ,I I ide ntifyin g and assessing different sce narios, ca n be p art of many tasks. 1.. 1. LOCATION AND CHARACTER

All oc~lli on and d1" lrtcll.nq pro t)I1) 01

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IYIH' S of "1':11; :11 11IO\lklll.

. 3yOU\

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ril e th e mes of locati o n ,lIld c har:tell'l' , fllnd :lI11 c ntal to Ihe o ther activities l' lesc nl ed ah(lve , rc preselll Iwo di I'i'c rl' 11 I views . (W(l c(lntrasting thought IlI llCCSS.CS. In a praclic:d Sl' nSl·. III\' Y , ': 111 1)(' dinin"l 10 handl c. Fo r 1' :-;; lllIplc , illl :tglllC :I hlindrol
Needs

35

Location and character

34

NATURE OF ACT I VITY

TARGET AREAS FOR RESIDENTIAL DEVELOPMENT:

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Northern Prince George's County, Maryland

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Make measurements

A re a Distance Shape, etc.

Ascertain spatial r el ationship s

Inclusion Direction Ad j ace ncy Conne c ti on

Make deciSion

All oca ti on Locatior: Rou te or pa th Layout

Make prediction

Geometric inference Simu l ation Mode ll ing

_ VE RY H rG H PO TE NTIAL l' OT E NT I AI_.

IID MODr '-
Character 1St ics Names of entities

P O TE N1"IAL

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NO P OfE NT "£AL It

H IGH

~ FIR E

S CHOOL S TA T T O N

N I N TLR S T An : H IGHWAY

J

Figure 2.6 Pred ictive modelling . T his map portrays land in five catego ries of potential value for residential development. A spa ti al informa tio n system was used to create digital ve rsions of maps for many th ematic variables: to remove some land areas from considera tion, fo r example, too close to highways or own ed by gove rnment ; and to und ertake mathematical modelling before conve rting sca led integer values into five categories fo r mapping purposes -

h e li copter: ' Where a m I?' What provides a fr a me of refe rence fo r this person? Equivale ntly, 'wh at is the n ature of the Sahara Desert?' But what determines the boundary of the desert? Or what provides o rie nta ti o n to a n o n-reside nt com ing to the s urface o ut of the Paris ra il Metro syste m into th at city's C hin atow n? Frames of reference , pro ducts of a t least twO thousa nd yea rs, may be perso n a lized, su ch as distance a nd directi o n from where he la nde d for the m a n in the forest , o r external , as in the worldwide syste ma ti c grid of la titude and lo ngitude. Such refe-rences m ay be polar , also ca lled azimuth a l, based o n distance a nd compass directioil from a specifi c point ; o r Cartesian , th at is, havin g two values rc prese ntin g dist
Figure 2.7 Categories of spatial acti viti es_

11tL' inte rsection of the Equator and G ree nwic h Meridian, Or th ey m ay be :Irh itr a ry ', based o n a loca l fram e of reference not ti e d to a ny standard , ' ll e ll as for g ri ds used in city street atl ases. Also, th ey may vary in tim e, .t ~ in the a ncient wo rld 's li at earth compa red with the mode rn world's " It lt ere,

T oday, s pa ti a l inform a ti o il SYS il'III S illval'i:lill y li se a ('artesian frame of <'I'C I'L' II(;(;, lI slI :dl y pl :"I
36

Needs

l.ocation and character

37

Euclidean geometry as a basis for identification of absolute position and for distance measurements. Yet there are alternative methods that may need to be available for some purposes, and there are cognitive and linguistic parameters that are important in the development of information systems , partly for reasons of personalized frames of reference , and partly for designing the style of and tools for the transfer of instructions from person to machine. While we do not yet know much about mental spatial data processing, there are some fundamental matters that can be included in this book. Firstly, there is not a universal acceptance of a Euclidean rectilinear frame of reference. People often use personalized frames based on their most familiar locations, like their home, or they may use also a focused territorial system, like that used by helicopter pilots. Or an azimuthal system may be employed , like the one Hawaiian Islanders use to determine positions based on distance out from or into the middle of an island or around the perimeter. Secondly, landscape objects may be used for reference purposes, as is quite common in Europe , where the same street may have many different names, and where there are many landmarks. In contrast. in the USA, highway or street road numbers are more often used in an urban landscape with a smaller density of unusual objects. Thirdly, the language of space varies considerably - words like up and down, in and on, inside or outside, have different interpretations. And common words for some spatial relationships, like north, have different interpretations depending on circumstances. For example, north may be identified as a line concept, a single line pointing to the compass point north; or as distance from a line running east and west, thc basis for identifying latitude and the concept of north implemented in spatial information systems. Customarily, earth phenomena are located in some kind of reference system. We use the term position to refer to where an existing object is located, as differing from location, which implies solving a problem like where to put something; although in practice these terms are often used as synonyms. Assuming for the moment that phenomena can be clearly demarcated , that is , we can recognize (bounded) entities; then some method of access is needed to get to a part of the entire world, either by marking on a paper map or doing its equivalent in an automated system. Reference, that is, a method of describing an absolute or relative placement of an object or its representation, C
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Figure 2.8 Spatial reference devices. (a) By coordinates. (b) By neighbours. (c) By boxes. (d) By place names. (e) By tiles.

The concept of a spatial reference takes on several practical forms ,( h gllre 2.8): I. Specific Cartesian or azimuthal coordinates. Identification of neighbours; that is, the entities, either line or area Iype, that touch a specific entity. \. ;\ linear locator, such as a mile point on a road. I. I': nclosing rectangles or boxes of minimum size (based on the lowest dl1d highest Cartcsian coordinates in x and J( metrics).

, ;\ pl;lce n
iI,

References to chllnks, liles, or other hlocks of space, either regular or irrcglll;lI' in sli,lpe, such ;IS 1ll;ljl Shl,~·ts or cOlin tries.

l.l·dvJ,JJ/'. 1IIi1i1 1;Ifl'r d disl'll.'isi()1l ()r l.-lill"III<·I)IS I() tlic h;l.sic concepts of :llid sp;lli;J/ ;,,' (CSS 1(I:l d l l l1 lh,, ~v IV \, (" 'lld"dc Ihis sCclioJl wilh

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Needs

Measurements and spatial relation ships

39

38

a reminder that differentiation is needed between identifying some arbitrary portion of the world in order to learn its character, and retrieval of information for an entity. The third and fifth items in the list above serve to refer to space , but not to particular phenomena , whereas the second and fourth imply that spatially defined entities exist. This fourth implies that the geographic name, like Sahara Desert, or a numerical identifier like census tract 1012.02, has an associated spatial definition, or possibly several. And, at times, the same name is used for different landscape features; for example , Geneva for a lake and city, or New York for a city and state.

2.3 MEASUREMENTS AND SPATIAL RELATIONSHIPS Taking a view that entities of the world can be clearly demarcated and positioned in some coordinate frame of reference, different tasks using spatial information may require the specification of spatial properties , such as shape, and tools to provide some concrete measure of those characteristics (Figure 2.9) .

~ of underground oil

River [email protected]

Perimeter and of properties

ama

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Indentation of a coastiir,e

Gradient of an escarpment

~ of political districts

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2.3.1 Spatial properties Taking individual entities first, we may have: ~

1. Length, as for a river or boundary of a lake. 2. Surface area, as in a lake, or a barrier island, or a piece of property. 3. Volume , for example, the amount of earth to be removed to build the English Channel Tunnel. 4. Shape, such as circular, or not , as for a political district, or a straight railway. 5. Irregularity in shape , as for an indented coastline. 6 . Orientation, such as the principal axis of drumlins. 7. The middle of a line or area such as the centreline of roads or the middle of a city. R. Slope as in the case of a mountainside.

Superficially, the measurement of such properties is straightforward, in the sense that if we already have a lake shown on a map or in a database (or on the ground for that matter) we can go around its perimeter. However, we will get different answers as to length depending on the precision of the measuring device, and on the scale at which the da ta were originally encoded . As we explore more fully later, the length of the coastline of England and Wales is much shorter if measured from a generalized map at a scale of 1 : 100 ,000 than it is if based on a walk (by a person, let alone an ant) along the entire shore. Other aspects of variations in measurement include the measuring instrument used, the ~()urce of data , approximations being satisfactory or acceptable , and the ;lmount of error that can be tolerated for a given situation . For example, the association of form and function may simply utilize visual judgements or may require more precise measurements. Moreover, some properties are multi-dimensional. Shapc may be seen :is the degree of conformance to a regular geometric figure like a straight li nc or circle, or it may involve extra properties like indentedness. Measurements may not be exact, or several may be required. If there are different definitions for objects, then several measures may be required, (If a probability concept utilized. For example, outlying counties around tile New York urban area mayor may not be included in the Illctropolitan area definition. That is , there could be a measure of the ,'xlcnt to which they arc attached, referred to as membership degree, li slially indicated hy a percentage of proh;lhilily . 1'~lltitics or the same general 1111'111:11 it' d:l ss , for example lakes or Iligilw
40

Needs

created . The specific stati stics like area or shape can be defin ed for one instance, and summarized o r generalized across a set o f instance s, includin g:

1. Minima, maxima, ranges of properti es. 2. Averages and va riabiliti es for those properti es. Moreover, som e spati al properti es can be defined onl y fo r sets of insta nces, not sin gle units . While ar ea and shape ca n be m eas ured for one lake, th e nature of the pun cturedness of a fo rest requ ires m eas ures for all hol es, even if shape is the basi c property. E xamples of th ese more compl ex co ncepts (Figure 2.10) are:

1. The pattern of di stribution of unconn ec ted feat ures like schools.

2. The layo ut of a housing developm ent , or the separa te land pieces of a farm . clus t ered

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Measurem ents and spatial relationships

41

3. The distance between trafflc li ghts , or th e aggr ega te trave l tim e to reach a city ce ntr e from different places. 4. The numb er of neighbo urs for cou ntri es, or th e punctured ness o f for~t land use.

5, Th e connections in an air traffic sys tem . 6. The predo minant fl ow of migrants across a co untry, 7. The seq uences of phenomen a, as for ro ad char acte r , o r land

lI S\.',

Fo r o ur purposes in building up th e stru cture of particul
, 1.2 Spatial relationships

air routes

1II II,Iit'i t ill th e examp les gi ve n above for properties of si nglc enl il ies :lIld I I', Il l' l' llt i ti es of th e sa me class , arc so m c o th er fundamelll'al (li SlinClioll s, 111',1 III : 111 , we different ia te betwccn properti es that rClJuire Illcas lII'L' IlI l' II I 11 01 " ) ' coord ln atcs, thilt is (gco)mclr ic informil t io n , 1'01' CX~ lIl1pl c 111 (' 1I I 1I 1(', i: 1 coordi n
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migration stream s

ENCLOSURE

CONNECTIONS AND flOWS

landuse sequences

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S E ~ UENCE S

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42

Needs

Categories of spatia l problems

43 routes; orientation is illustra te d in the direct io n of flow o f fivers; a djacen cy is see n in the case of neighbouring countries and layo ut, and co ntainm e nt is invo lved in th e bare places in forests. Some spatial co nce pts m ay be meas ured in both geometric and topological do m a ins. For exa mpl e, the distan ce between cities in the glo bal air ro utes syste m ca n be represented by the mil es trave lled , or the numbe r of legs o f a route, where leg is d efi ned as each pa rt o f a jo urney between sto ps . Prox imity m ay be thoug ht of as a djace ncy or a Eu clid ea n di sta nce. Secondly , id e ntification or measurem e nt o f properties of spat ial re lati o ns hips lik e distance or co nnected ness re quires that we define a spec ia l cl ass o f entity, the object pa ir o r dyad. Th e chart of driving times between cities, with li sts of t he places in both th e rows and co lumn s of the table of informatio n , has a city pair as the basic unit, no t the individu a l city. The driving tim e is a property of the co m b ination o f two places . While the dyad is not necessa ril y a physica l e ntity, it is non e theless an importa nt spati a l unit in many fi e lds of stu dy , s uch as po pul at ion mig ration, social contacts a nd weather prediction: and the physical re prese ntati o n may be very e lusive as in the case of linkin g a wa ter flow via a sinkhole and s rrin g. Thirdly, additi o na l spatial propert ies are encountered if we exte nd to differe nt attribute them es. The co nce pt o f spatial colocation or covariation , th e e xtent to which different phenomena have simliar occurrences in space, involves a co mpa riso n of the distribution o f one ph e nom e no n with a no the r, givi ng ri se to co ncepts like overlap (for example, o f soil type A and wheat production, or sa les territories) , and intersect ion (such as rail ways with roads) o r not inte rsectio ns (as fo r sewe rs with gas pipes), by th e additi o n of a ttri b ut es to the spa ti a l condition s (Figu re 2.ll). These concepts a ppl y, too, to the three-dimensional world, a ltho ugh they a re i \I ustra ted o nly fo r inte rsections , via th e exa mple of a irsp ace. Other e xa mples are th e thre e dimensional cova ri a tio n o f wa te r content and geo logica l st ructures, or ocea nic propert ies of te mpe rature and fish density. In this context it is important to recall the differentiatio n , firs t mad e in Chapter 1, of information s yste ms with a lay e re d architect ure and those o riented to objects. In a layered syste m , where each theme is se pa rate ly e nco ded, all spati a l covariation tas ks require brin gi ng together differe nt layers of info rm a tion ve ry muc h lik e the manual process of overlaying maps on a li ght table. In this process it is us ua lly necessa ry to dcmarcatc new obj ects , for exa mrl e the new rol ygol1s containillg pa rticular combinations o f so ils and land usc, tile : 1I·1':1 o f over/:ljl of Ill: wsp:qKr circulation te rritofies , Ilf t\),' fO ll es III wllll'li :Iirsjl:ll"l ' (· ;111 Il l' II Sl"( 1 hy p<JrtiuilM c: 1l L') :<1I i,·" "I' :li rn:iI't ,

SO il s an d wh ea t product ion

r o ad s and r a ilw a ys

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OVERLAP

INTERSECTIONS

Figure 2.11 Spatial covariatio n .

2.4 CATEGORIES OF SPATIAL PROBLEMS ()lIr struc turing of particul a r qu estion s in the s pace d oma in Can now trc a t

Ille categories defi ned ea rlie r o f where to loca te so met hing . Many public

:lIld private authorities m a ke d ecisions on loca tin g specific phenomena

like traffi c lig hts or railroads, or make determinations abo ut se ts of

pllC llom e na lik e th e intercity rail services to be prov ide d by a nati o na l

I:,i lwa y comp a ny o r where to locate a set of hosp itals. Scientists g rapple

IV ilh ques tio ns o f how hum a n societies are orga ni zed a nd parti c ular topics

111( 1' th e impact of a n e a rthquak e in J a pan o n the coas tline of Peru.

I

A . I Types of spatial problem

t(' ()g r;lphc rs in pa rtic ular have atte mpted to o rganize problems seman .ill y from a spatial pe rs pective. The s p atial concepts in this di SCiplin e , 11111 IltllCrs , rrovid e a valuable additional pe rspectiv e on s pati al infor II l.1 liPl1 syste ms . Particular catego ri es o f spatial organization (Haggett, t' l( ,'1) dre ( I'i g ure 2 . 12); «

ii i

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44

Needs

Categories of spatial problems

45

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• • e :3 -2 • Hierarchies

3. Tile elements that are given, and those that have to be found. 4. The"objectives.

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1. The types of entity involved, and their combinations. 2. The spatial properties involved.

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For example (Figure 2.5), finding a route through a maze is essentially a linear sequence problem - putting together different segments of the maze in an order to guarantee a path from entrance to exit. It
Nodes

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The location problem may begin with a known location of customers, a known transportation system , or il known set of available plots on which to build, and require the determination of which point in space is good 1'01' maximizing sales. Or two unknowns may have to be determined, sueh ;IS the location of hospitals and the hospitals to which reoplc will g(), in Dther words, a combined location-allocation problem. Otherwise, the loc,ltions of supply points and demand points fO'r some mobile resourcc, I'llI' example, bottled mineral water, are known, and the dement being :Iddressed is the allocation of quantities among supply and demand roinis. Specific types of location problem (Figure 2.13) arc:

1935 ~ 0

.

Diffusion

Figure 2,12 Categories of spatial organization.

5. Fields or surfaces,
I. L ocation of some point entity, like retail commercial centres. A llocation of resources frolll points to other points, as in petroleum I'rom oil fields to refineries.

6. Diffusion of disease , ideas or innovations in space. This different approach to spatial relationships reflects processes of natural and social ordering and organi zation. Movements of goods match up places of supply and demand; networks are the channels of communication or flow; possibly, by reasons of greater connecti vity, some places become more important than others or have to deal with greater volumes of water; and entities may have influences extending over shorter or greater distances. Some conditions may be thought of as spatial structure, like morphology and dimensionality; others as movement, including circulation and diffusion; and others may be processes like growth and organization. While this conceptual framework is particularly valuable for dealing with human organization and artefacts , it is also applicahlc to physical and biological phenomena, for example. animal Icnitoriality , hird migration paths, and overland flows or pollu\;IIl". Generally, Oil thc iJ
I.

1

II

I{ oute-finding , like getting from Square in London, England.

Heathrow Airport

to Trafalgar

"
('rcalion of area units, as in redistricting for school areas.

I ,
To delllonstrate potential variations on these themes, we take just one .I"'gory. Path-finding situations may consist of the choice of one route !l111 ()j \cvcr,!I to get through city streets, or may be more complex , as ti " IIIDIlSlr;lted in latcr ch1(' svslcills or IOUIL' .\ Ill;IY i!;IVl' 10 1)(' dl'si!',Jll'd, ;IS ror the provision of 11 11 \' II'ily 1:lil I);IS.\l'lIg('r s('r vin's hy III:IIIV 11:111"11:11 rdilw;IY companies. ,\ IIV "Ill' tVjll' or IO(';lli()1l proJ,I"1I1 111:1\' I,,· ,,·t "1' ill dillercnt ways, lil lll"II)I.11 ;!II('III:IIIVt", :11" 1101 ;11\\';1\", illll l1 lll, 'I" ol,,,i')l1\ Tr:ldillg L'L'llt cs, r

46

Needs

6 666?6 Commercial 6 centre

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47

require an area type unit to be recogni zed as a da ta type. At the sa me tim e, th ough , it requires that th e dyad be a recogni ze d spatial unit.

~

~

Ca tegories of spatia l problems

Layout

Figure 2.13 Some types of locatio n proble m in two space dimensions.

whether ex tensive mode rn superm alls o r loca l co nven ience stores, m ay be seen as points in territory. Ther e are models and too ls to so lve a retailin g distribution of thi s kind by demarcatin g th e m ark et area in which a sto re or commercial centr e dominates competitors, [h at is 10 say, all area entity unknown. H oweve r , it is mu ch easier and ju st as effective 10 Ire
Th e parti cul ar probl ems are further differ enti ated by goa ls and by the spatial prope rti es of th e unknown. G enerall y so m e quantity is sa ti sfi ed o r optimize d, for exampl e minimi zin g the overa ll distance that peopl e mu st travel to hosp itals from their ho mes, o r minimi zing th e costs in deliverin g pizza. U sually , a geometric spa ti al property is specified , but in so me r ro bl ems a topo log ica l element is also prese nt , as in the creatio n of votin g distri cts th at do no t co nt ain encl aves o r do not have exclaves . The cllndition(s) bein g optimi zed usuall y co mbin e ex plicit spa ti al properties li ke mil es of travel , with ot her att ributes such as re ta il sales profit o r ;Ihse nce of bias in voting districts. An interes ting ex ampl e from today'S world of fast -food is th e pizza delivery company in N ew Y o rk that uses a spa ti al information sys tem to transfer a telephone o rd er from th e IH'e ivin g store to ano ther store whose territ o ry covers the ca ll er's address ill o rder to have a more effective service. Som e spatial probl ems invo lve a time element. Perhaps thi s is onl y l illi e of travel , whi ch ca n be represe nted as (I n attribute of a pair of 1, I
'I li ltii l ioll S for different countries.

I 'Ill ' impli cat ion s of clea rly id entifying the spatial tasks in the co ntex t ," " p;11 i;ti in format ion syste ms are few, but very i mportan t. So me types of l'I"' ·'·'s ill f, w ill he difficult , perh aps impossible practically , if th e 1I i1 " l lild li ol1 sys lem docs not treal parti cul ar types of spatial unit. We iJ .1\ (' ;tlrc;l(ly I1H: llliol1 ed th e need lor lil e sp
Needs

49

Exam p les of multi-faceted need s

48

2.5 SOME EXAMPLES O f MULTI -FACETED NEEDS The complexities a nd uncertainties of spatia l pro blem solvin g are best d emonstrated by e xamples . A first example is a scientifi c activity to ascertain the e ffect of patte rns of land use o n nutrient discha rges into a ri ver syste m . A second task is a natura l reso urces ma nagement activity to determine the land uses associate d with wetland habitats. Th e third illustra tio n is an acti vity to discove r th e e xistence o f phe nomen a th a t are not directl y obse rva ble , that is, whe re infere nce is necessary a nd a fourth is a n exampl e of simulation .

2.5.1 Example of flows over landscapes Eco logists at th e Smithsonian lnstitution in th e USA arc inte reste d in the effects of landsca pe characte ri stics o n the flows o f chemical s, p a rticularly nutrie nts, ove r a surface (Fi gure 2. 14). It is clear that wate r containing che mical s will flow over a surface gra vita tionally, will soo ner or late r become ch a nne lled, and may also see p be low the surface, possibly be in g chann e lled into subte rra nea n strea ms. Not onl y does the terrain slo pe affect tlo ws, but surface land use a nd land cover influe nce the loca tio n of

[ anrlscape un It s

El ev ation

(no t show n)

t~ y

/~~ X

ne t wo rk [Io d l'

nu,trie nts that can be picked up by the water. It is important to know not simply how much agricultural la nd (receivin g fe rtilizers , fo r example) th ere is, but especially if it is adjacent to riv e rs. A rilGdel of the level o f nutrients at diffe rent places must , there fo re, recogni ze m a ny spatial eleme nts and types o f spatial p ro blem. The po int sources fo r chemicals may be regul a rly or haph azardly dis tribute d in space ; the slope a nd as pect of the face ts of te rrain must be kn o wn; data are needed for conditions be nea th the surface; water must be tracked th rough a netwo rk ; some wate r bo dies ma y ha ve substa nti a l surface area; a nd the pa tte rn of land use must be kno wn . Objects are see n in three dimension a l space, and co nsist of po lygo ns (of land use type ) , line segments (of the river syste m) , pa ths (of the flow o f nutrients) , e le me nts (of te rrain), and coo rdinates fo r esta blishin g th e position o f specific fea tures or attributes .

2.5.2 Resources inventory (;overnm e nt planne rs ha ve bee n cha rged with the responsibility of :Isce rtaining the ex te nt o f differe nt kinds of la nd use occurring within slirface a reas designa ted by legislation as e nvironment ally se nsitive wetl a nds (Fi gure 2.1S). Specifica lly , a process was required to de termine :Il'l' urately the la nd use or la nd co ver in zo nes o f o ne th o usa nd feet fro m ho undaries o f we tlands (Lai, 1988). Starting with origin a l ma ps and air pho tograph s o f te rrain conditi o ns , lwo types o f wetland we re identified: those that co uld be represe nted at a ;\'og ra phi c scale as having area l e xte nt , and those represented as linear il-: lllIres (Figure 2 .1Sb). Critica l zo nes, or buffe r re gions , dra wn a t the l'I\'sc ribed dista nce from th e single demarca tio ns of the we tlands we re I ',I:t hlished , a nd combined (Figure 2. 1Sf) . The pol ygon zo nes were the n II"I'll as 'pastry cutte rs' o n the land use map, alth o ugh in practice the I ',I I IiClil a r software co uld no t directly de tect the inte rior ho les (Fi gure , I 'i ii) . Aft e r unw,mted pieces we re e liminated (Fi gure 2.1Si), common l'tl llllll;lri cs disso lved , and true we tl a nd areas o mitted (Figure 2. 1Sj) , the 11 1, ' : 1 Illl':ISlircs we re o btained.

Agricullural producl ion

Dr' aina g<'

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-) Predicting the location of miner•• 1 on' d~posits

,,jog i, ts ofl ~' 11 nn· d 1\) illkr lli t' O ~' l"I lrl\'ll("(" (If plienolllcna th at ca nnot I" rlll.!'!" llv IlI)sl' rvl·d willi ill.~IIIIIII (' III ', ( I'I)',III \' .) I()), f,: lyillg po ssibly o n

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50

Needs

Urban or bUl l l - IJD land

203

res Idt'M la] parI<, rac e lraC l<, goi fcours e Agri cu ltural land cropland and pasture Forest rand decIduou s ror est

evergreen rorest mi xed rore st Water

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bays ana estuarte s Vletland

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42

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SlrlD m in es, quarnes, Pits tr ans iti ona l areas

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51

ACREAGE

LAND USE CATEGORY

Th iS set of maps illustrates Ihe sequence of actions applied to a series of map layers 10 produce some statistics . These. given at the right. indicate the amount of lar1d used. by major category. in acres. within critical zones created by legislation These critical areas lie within one thousand feet from water bodies. In this case these Jre wetlands of class PFOI A. the Palustrine forested broad-leaved deciduous temporary. Details of the processing are discussed in Chapter 8.

Examples of multi-faceted needs

~

~

Figure 2.15 La nd use inventory for cri ti cal areas. (a) fnventory of lilnd use. (b) Map of s urface hydrology. (c) Extraction of wetland polygons. (d) CreCltion of lOOO feet zones around areal hydro logy Icatures. (e) Creat ion of IOOO feet zones abou t linea r hydrology 1L:;ltures. (f) COlllbinCltion of two sets of 1000 fe et zo nes. (g) Removal ()f L:ommon boundClri es . (h) Use of 1000 feet zones to cut out pieces ()f thc land use map. (i) Elimination of la nd use for the undetected intnior hole. (j) Removal of we tl and areas from th e land lise map. (This example was L: reated by, and is used with the permission of I'lOicssor Poh-Chin L,i, Ohio State University, CO lumbus , Ohio.)

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Needs

Exa m p les of m ulti-faceted needs

53

52 Vein material

Tightne ss of folds

Elongatio n of oolites

Fe in sphalerite

Mg in calcite

. - - - -----, G

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Data samples f or t en measurable prop er1i es Spacing

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I

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0

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Detormati on

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sta tisti cal and spati a l tools can help in s uch tas ks . Not onl y is it necessary to guess boundari es of the s ubsurfa ce stra ta containin g the min e rals or wa ter , but it m a y be ne cessary to use so m e o bservable attributes like iron conten~or po rosity to stand fo r th e pre ferre d co nce ptu a l criteria lik e susceptibility to d eformation (Klo va n , 1968). In three-dimensio nal space the re is no ex plicit fram e wo rk exce pt that prov ide d by boreh o les a nd tunn e ls . Surfaces must th e refo re be extrapol a te d o r interpolated fro m just a fe w triads of positio na l coordinates. It is important to re alize that an a lytica l o r infere nti a l ac tivite s like the se may have processing require me nts beyond what sp a ti a l information system s ca n to d a y underta ke. T he de vel o pment of software has bee n segme nte d hi sto rically, Sta ti stica l d a ta processin g o pe ra tions implie d b y th e first st ages of the min e ra l prospecting exa mple ( Fig ure 2.16b) , o r the mat he matical mode llin g invo lv e d in the nutri e nt di scharges e xa mpl e, d o no t necessarily ha ve to be done within a spa ti a l info rmati o n syste m , and histo rically we re acco mplished by sepa ra te compute r progra ms . Today, re garding softwa re tools, there is d e ba te as to how mu ch p rocess ing th e re should be within the s pa ti a l in fo rm a tion syste m itse lf, and how mu c h should be d o ne by separa te s pecia l purpose progra ms .

L 5.4 Engineering network simulation In the d o m a in of utility network e ngin ee ring , it is ofte n impo rtant to have facility fo r simulating th e be haviour of infras tru cture e le ments . F o r inst a nce , le t us examin e (but possibly no t too c1osely!l) a se we rage Ill ;ln age mc nt system, Suppose we have a d a ta base integra ting the sl' wc rage pipe d esc ripti o n , includin g fea tures like s ize, locati o n, g ra dient :l lld flow capac ity, toge ther with the topo logy, that is , pipe co nn ections , Wc also m ay need to determin e th e loca tion of filte rin g a nd purificati o n ,,1 :III( S. Ily me an s o f a spati a l info rm a ti o n sys tem , we ca n simul a te the effect o f ;j i:l inla ll in order to see whe th e r the sewe rage syste m can abso rb it a nd wll l' rc (h e o vcltlows mi g ht be lo cated. Th e sa me model can be exec ute d ,I ~ il (h c storm ccntre is go in g upstrea m o r do wnstream re la ti ve to the ',,·wn;lgc. Wh c ll d o wnstrcam , it icitd s to a sort of accumul atio n in which lilt' pipl's 1l1llSl ahso rh Illore ;tlld morc watcf pe rh a ps suggesting pipes ', I"'llill ),',l' ( I:l rgn III (hat dll C(' II OIl . 1'01' tilc lbtl y ro utine use of th e 11I 1Iilil' ;llio n, we (';t ll :li so l"(l lllP:III.' til l' :l r lll ;II ;111<1 ~ il1lltlat e d behavi o urs in " ,.1" , In k'H)W wilen' tit,: Illlldd y w:ll l' l IS 1' H':l ll' d ill o rd e r to cl ea nse th e ,i lI'l' I:l I'l' . III :ldllilillll ,:t Sl'Wl' Ll g" ,'1":111 ', 111 1' IlIlll'I:II,ic 1.' <111 hL' l'stahli shed I" ",' Ill' t1l1k I ltl' W'lI" I II CUi'I \·" i<- Willi II ", IIIII \' ~ <11 tltl' h('s l wl': llhcr I hc

(b)

Targ et area Known ore body

(c)

1 arCjet area de-ter mlned

t

Figure 2.16 Spa ti al predictio n . (a) Data samples for le n 1I1 C; iSllr;. (r) M"p based preliiclioll . (Thi ~ c':-;; 11l1pk is \); N 'li <> 11 111 ;11 <, 1';;11 ill Iii,' P;lj ll'\' I, V KI (lv; 1I1 , I')(,K )

54

Needs

conditions. Moreover, when city developers are designing new allotments (property subdivisions), it is possible to check whether the actual level of sewerage can absorb new dwelling discharges; and, if not, determine where to enlarge or double some pipes. One can also test the necessity of installing more purification plants.

2.6 MAIN CATEGORIES OF TOOLS IN A SPATIAL

INFORMA TlON SYSTEM

A toolbox in the context of physical, documentation and guidance subsystems in an automated spatial information system, much as for a non-automated system , must include resources and capabilities to keep track of data, bring information into the database, produce output , provide security and allow for a range of non-spatially oriented processing. Several classifications have been made at varying levels of detail, and there is no one universally accepted framework. Some criteria for undertaking a classification are: 1. The type of data being processed: words , pictures, numbers, graphs and maps, sound. 2. The stage of processing: input, intermedi a te point, or output of data. 3. The management: control, guidance, documentation , security, book keeping. 4. The type of operation on the information: spatial. non-spatial, combined.

These are not mutually exclusive: number data can be input or output , and many specific operations being performed on the encoded data will involve several types of processing. Some need may be met by a single command, by a single command that invokes a series of operations unknown to the user , or by a series of commands given by the user. How a task is undertaken derends on the level at which the user wi shes to , or is forced to, work . The style of the person-machine interface is not as important for our purposes as a discussion of th e functionality of software. Data acquisition involves the direct capture or oat;1 by electronic imaging, either remote (for example, SPOT satellite im;lgery) or close at hand (for example , by still video camera lIsed on
Main categories of tools

55

of prior processing or human activity. Direct digital encoding via satellite is followed by a step of interpretation. Information about the same phenomena may be held in a database in differ~t ways, or may be put there via a different process. In any event, though, there will be a need for tools to manage the material, to provide security. to allow for verification and validation, to control access, to correct and update, to convert from one form of internal encoding to another, to organize efficiently, to help the user, to identify and describe the contents of the database, and to facilitate import and export of digital data. Tools for output may be many , depending in part on the kind of information in the system, and in part all the mode of use. Analytically oriented systems will most likely provide capabilities for the creation of physical documents - maps. scientific graphs. presentation graphs or other images , tables of numbers or narrative text. Query oriented systems will focus more on the preparation of virtual output in the form of quickly produced maps or graphs , the contents of particular sections of the database, or derived statistics, all made available immediately in response to a user request. The same physical resources and capabilities of thc information system may be used in both cases, such as basic drawing capabilities. More important are the processing functions, as these are the tools by which the spatial analysis is undertaken. Again, depending on factors like Iype of data . which will be discussed in more detail later. we can identify functions involving or not involving spatial properties. Many require mcnts have already been reviewed. There also are categories of processing for just the attributes of phenomena, such as arithmetic ()perations. logical operations and parsing. For example , census data in jlublished tables may require aggregation. A query oriented system is Illore likely 10 emphasize functions of search and retrieval. A product ()!iented system is more likely to emphasize measurements. The one is OJ ienled to bringing together disparate pieces of information via I)!owsing. by cognitive associations or providing capabilities to proceed [mill general to more detailed information. The other emphasizes the "Olllpllting as opposed to information handling capabilities of computers. ( ;cller,dly spcaking. we can distinguish among several major types of "I il"IlLlli()1l (I"igurc 2.17):

'(

I. /\.111(>111<11,'(/ III
Main categories of tools

Needs

56

AUTOMATED MAPPING

-

•

-e-r-p-Io-t -e-t:~

•

Digitized or scanned

Comput

\.sPATIAL BROWSING

57

DiIiLIDap-' caiend;li]

~

~

•

map

<=) -------.. ~

[~J

Paper map

THEMATIC

MAPPING

Rule 1

19] Statistical areas

••• ~ -

+10

s+-

+

+ 20 4 10+ 10+

Figure 2.17 Orientations for spatial information systems. (This diagram is an extension of one devised by Patrice Boursier , Universite de Paris Sud , France.)

3. The variety known as composite mapping, characterized by a map

Map thematic

layers

SPATIAL STATISTICS

SPATIAL ANAlYSIS

· O •

•

Measurement

• •

Slalistics to< neq,bours

~

••

•

~

D".

•

./ ' - I "

Nearest neighbour distance

Binary similarity for neighbours ~ C = 0.75 Bina~ similarity for non-neighbours

A

SPATIAL

QUERY

D =1.17 n

Summa~

•

25

A'

~ 1~: D

99

1)

•

Rule 4

:bioali) •

Rule 2

Rule 3

Symbol map

Census data

MAP OVERLAY MODELLING

SPATIAL REASONING

~

Ic 6

82

1

overlay concept. 4. Analy sis of spatial data , encompassing tasks which deal with the attributes of entities, like the average size of citi es or the degree to which crop yields are related to climate conditions. 'i. ' Creating spatial statistics covering ta sks that require measurements of sratial properties of phenomen a, like th e shape of drumlins or the volume of timber in a forest. () Analy sis of spatial statistics, that is tasks which treat spatial properties as attributes, for example th e co rrelation between hi ghway network connectivity and levels of economic development in the world. 7. Spatial analysis, encompassing tasks , including simulation , which use " variety of tools of spaticll statistics and location based problem solving . ,e.: Spatial browsing, or generally exploring the contents of a database containing spati::d information of some kind. ') Sp,ltial querying , obtaining informatio n from a database in response 10 identification of particular conditions. I II. Spatial rroblem solving , such as geometric reasoning , for example ,Iulucing inclu sions of roints in rolygon s\ or for spatial decision Ill;d( illg incorporating iloth sp;ltial ;\lld logK('al deductive reasoning. Tlll" :O-c dillcrclll oriL:II(;ltiolls rL"ikct 1101 applic'llion dOlllains but general l'I''' T~ ~ill!'. l("(jlli rL' lllclllS 1ll'lt :lillill!'. 1'1 .~II:III:d d :II :I , sp
Needs 58

terminology, much more than we will to subject matter application domains.

2.7 SOME IMPLICATIO NS FOR TH E DESIGN OF SPATIAL

INFORMATION SYSTEMS The foregoing discussions of types of investigations, themes of location

and character, measurements and spatial relationships, categories of

spatial problems, and major categories of tools are intended to provide a

perspective useful for both evaluating and designing spatial information

systems. It might he regarded as trivial to say that information systems

will be better if we know what functionality is required. However, there

are some important practicalities that complicate what otherwise might be

regarded as a simplc process of matching demand and supply:

1. Users do not know clearly what they want to do.

2. Users may think specific software systems can solve their problems when, in fact, they cannot be dealt with via the data structures used. 3. Data and processing requirements are mutually dependent. 4. Some operations may be theoretically possible but practically very

Bibliography

59

preSent, but shape measurement and comparative characteristics on either side of a boundary were infrequent. Area measurement, and attribute and location specified searching were most common for this set of responses, The study clearly showed, too, that there is confusion over termin ology, in part because of differences among disciplines. Moreover, users of spatial information .systems are often not oriented to spatial concepts, let alone knowledgeable about specific measurement tools like spatial autocorrelation. From the perspective of automation, it appears that tools for efficiently handling spatial relationships and spatial problem solving are not well developed in the context of present comprehensive conventional infor mation system toolboxes. Because of the huge number of spatial relationships that arc possible, we may not be able to compute and store some attributes for future usc, for example, all possible distances for a set of points, or to anticipate all possible combinations of spatial entities, recognizing their type. One implication is, of course, that we should not rc:ly on general purpose information systems, but have a number of special purpose systems. After all, we have many specialists such as plumbers, lorry drivers, teachers, geographers, informaticians and creators of spatial information systems.

5. costly. The means of imp lementi ng particular types of function may be confused with what is needed. 6. The commercially available systems must be judged from a market place perspective in a historical context. 7. Choices are made because a neighbour or a sister agency used a

:l .1I BIBLIOGRAPHY ('l'lleral, comprehensive reviews of spatial and locational concepts are provided e/ aI" Cole and King, Haggett, and Unwin,

[,y Ahler

particular type or brand for sharing costs or knowledge, While there are many lists of requirements associated with software procurements on the part of businesses and government organizations, often following extensive formal needs assessments, there are few systematic academic studies that give a profile of the functions expected of spatial information systems. One study is described here as an example of the process to obtain good material. A selection of users of spatial information in the field of natural resources management have provided some indication or their needs by indicating how important certain kinds or activity were for them in their ordinary work (Tsai, Il)S7) A qLlesti()lll1ilire prodLlced patterns of response to sixleen ilclllS, sCided ill ~IX ciil c go l il's rrOl1l very rrequc:nt to not
/\I,lcr, ROil, John Adams and Peter Gould. 1971. Spatial OrganizatIOn I 'Ilglcwood Cliffs, New Jersey, USA: Prentice Hall IkIIY, Joseph K. 1987. Fundamental opcrations in computer-assisted map ;111;ilysis. In/erna/;onal Journal of Geographical Informa/ion Systems 1(2): [[')-1.16 (ILlITl', .Incl ;11lc! Pierre DUlllolard, 1989. Initia/ions aux Pratiques Informatiques ,'// (;':()!!,I'IIIJ/iic 1,(' IJ)!!,icicl INFOGEO, Paris, France: Masson. '''['' .Iolin 1', ;llld Cuchl;lillc A. M King. 1968. Quanti/ative Geography, London, III\. .Iolin Will'y. ~( 1I,III,I'c'1I1l011l1, -'dl'k. I')tn. A clas,ilil';lllllll or soitware components used in 1" '(}I:I;I['''ll' IllrOlllldlioll ,'ystClllS 1111)"'111:1.1, I'cul[ucl ,Ind John O'Caliaghan t " d,),

1)('11.,:// (///" 11//1,1"1//" 11/(///11//

(If" ((lIIIII/i/"1 {'(/s('(1 (;('()!!,I'{/{ihic Information

1\llill"I',I, Nnv ) (}I k, N ,' w )' ''1 '' . 1 I,' \!\: 111IC'llldliOIJ;iI C,cograpilical 11111\111 ('IlIIIIIII"illll ,III ("'(}I'I;II ,lIl l':1I I I,lf " \ "11',111)' ;111" l'IIll'l's,illg,

,\" ' 'o/''/IIS

Needs

60 Goodchild, Michael F. 1987. A spatial analytical perspective o n geographic informati on systems . inlernational Journal of Geographical i nformation

SySlemS 1(4): 327-334. Goodchild, Michae l F. 1987. Towards an enumeration and classification o f G IS function s . Proceedings of the international Geographic information Systems Symposium: The Research Agenda, Arlington , Virginia . Greenbelt , Maryland , USA: NASA , vo\. 2, pp.67-77. Haber, Wolfgang and Jorg Schaller. 1988. Ecosystem Research Be rchtesgade n Spatial relations among landscape elements qua ntified by ecologica l balance method s. Paper presented at th e European ESRl Users Confere nce, Munich , Germ any , 1988. Haggett, Pe te r. 1965. L oco /ional Analysis in Human Geography. London , UK: Edward Arnold. Klovan , J . E. \968 . Se lecti o n of targe t areas by factor analysis. The WeSle rn

Miner (Fe bruary): 44-53. Lai , po hChin . 1988. Maryland D e partment of Na tural Resources Pat uxent Basin geographic information syste m. Repo rt prepared for the Maryland Departme nt of Natural Resources, University of Ma ryland, College Park, Maryla nd, USA. Tsai , Bor-Wen. \987 . A que ry approach for GIS database systems design: the case for national resources analys is and management. MA th es is, Depa rtm e nt of G eography. Unive rsity of Maryla nd , College Park, Maryland, USA. Unwin , Davie\. 198 1. Introductory Spatial Analysis. London, UK: Met huen.

~

3 Semantics Objects, surfaces, data The manager of a geographic data vendor decided to try to sell an electronic atlas of countries of the world, but empbasized the tabular and grapbing aspects ratber than the map displays. The map libraria n created a system to select map sheets tbat contained many features but could not allow for selection of the maps based on tbose features. The politician bougbt a statistics package to help in redistricting but was unable to ascertain if one election precinct was next to another. The geologist tried to infer the houndaries for petroleum deposits in sa lt domes that could not be seen, A Londoner says that Eskimos are to be found just 'a fter ' Scotland, The visitor to a restaurant in 'The South' of the United States was intrigued by tbe curious paper placemat map depicting 'Old Dixie'.

Thl? content of this chapter extends the discussion of se mant ics from !leeds to a more particular leve l, that of the nat ure of spa ti al data. Spatial problems may be evaluated differently if they dea l with phenomena from ~Jlecia l viewpoints represe nted by idiosync rati c trea tments of data and spacc. We distinguish primarily between an entity- or object-oriented vie w and a continuous space perspective. The chapter also reviews the ddTcrcnt kinds of data , not just spa ti al, th at may be part of a spatial ill rormation system.

1.1 THE INFORMATION IN A SPATIAL INFORMATION

SYSTEM

l 'lll' discussion in Chapter 2 of the task s that may need to be performed "il li the aid of a spatial information syst em indirec tly introduced the topic ,>I llil' n,llme or spatial inform;ltion. L et LI S now trea t the topic of the IIW;llIll1g associated with phenol1lena 1ll oF6..._directly. In simple terms , 1III:I,I,.ill(; I Ii;} I I()r S()llle pllrpose we h;lve gO!le through a process of t1" llllil ioll, till' COllCcptll ;d descr iptloll oj" ..-pil enomena , fo r cxa mple , " I ' i" ' i ryilil~ tli(' pmJll'rt i,'S IILII ;Ii"t' IH 'l"("S';, 11 y ,lIld slifli cic lit I()r identify ing a 1011. ,', 1,"('ol'lli l,i()ll , 111(' OI'Sl'IV;IIIOII 01 IIIl" ,. , is\(" 11("( ' or 111()se prope rti es

62

Seman tics

such th at a claim can be made that so mething obse rv ed is a lake; and demarcation, th e delimitation of th e feature on th e gr ound or a representation of it as in a photograph.

Th e information in a spa tial informalion system

'-~--------------------------------

63

3.1.1 Spatial entities We use the term entity to refer to a phenomenon th at ca n no t be subdivided into like units . A ho use is no t divisible into houses, but can be split into rooms. An entity is refere nced by a single id entifi er , perhaps a place name, or just a code number. Even though it may be composed of seve ral pieces, it does have an overall identity , for exa mple , a city or a watershed. The entity may be a mappab le feature like th e lake, but perhaps not readily mappabl e, like the salt dome, beca use it ca nnot be seen o r because it is an arbitrary unit like a statistical unit. It m ay also be a co nceived unit , necessary for argument or imagining, but not yet ha ving an ex istence in the real world. The indivi sibility is based o n th e properties used in the definition, and there is no t a necessary implication of co mpl ete hom oge neity within the spatial extent of the entity because so me characteristics, such as depth of lake , are not included in the criteria use d to defin e and demarcate the entity. In the spati al infor matio n systems field, there is no comparable standard structurin g of ph eno mena as in th e physical world's molecul es , atoms and quarks , to help us in id entifyin g th e scale of pheno mena as impli ed by divisibility. Whil e in everyday language th e terms entity and obj ec t (or thing) are sy no nym ous, we shall follow th e terminology of th e field of inform atics to use entity when spea kin g of th e conceptual organi zat ion of phenomena, lind object when referrin g to th e digital represe ntation . H owever, as the cho ice of term is co ntex t dependent , we will occas io nall y use entit y and obj ec t interchangeably . At tim es, oth er terms like geo-object or fea ture may be encountered. The former term impli es ph enomena pertaining to th e earth . The latter, a mo re confusing term in th e context of spatial info rm ati on systems, has th e connotation of geo-obj ects th at are show n o n maps. Because of cartographic conventions used to ameli orate the difficulties of show ing the large curved ea rth o n small flat pi eces of paper, features o n a map may not be positioned as the y arc on th e eart h , and so me rea l entities may not appea r at all as features on lll ;lP S. The phenomena exist as perceived by people (Figure 1.1) , so th ere may be different definitions for lakes. Hut there Illa y ,tiS() he dirferent demarca ti ons on th e ground or in th e ,1\Jstract as;1 restllt or ' n,tture' heing at odds with th c int ell ec tual de lillitioll s. hll' IIlSI,I Il ("C , til l' krritori,d ex tent o f ;1 hod y or wall: r (:,I n v;lry ;11 dilkll'llt lilllt'S
Figure J,1 G eographic e ntities. This sketch demonslrates .ius t a fe w of the sema nticall y based e le ments in a geograph ic landsca pe. Bound a ri es may be imprecise, and data represe nta ti o n is no t a simple one-to-one match.

,':I rth cru stal or o ther processes, only one exte nt beilJg co mpatibl e with lite definition. Lack of clarity may be represented by a co ncep t of l11l' Jllbership function. For exampl e, determinin g th e extent of low I11t'Ollle neig hbo urhoods in London req uires a doubl e meas ure: low If lUlIllC and neighbourhood . The spatial extent of th e entire area as "l1 :lgIfl CU in th e stated requirement co uld be seen as a probability (pe rcc ntage in practi ce) o f association with a Core .

1. 1, 1 Cllegories of informalion '-

'· \ " IIIII'ifl!'.

t 11;11

Iltl'

l'ilr t h\

s llr!';l , ,.

,1 , ' I1I : IfT; II:t!>\(- pIt C IIOIII l' Jl : I, WI ' ": 111

'''1 ~' P' T itJ(' I'fl(ill \, ~

( 1I g IIII'

; . ,).

1" 'II!;I II I'>

' ·'lfl '. II kl

,'k;" Iy

definahle

and

III', · l'; 't l' )~I>1 I t'S o r iflrOflll il ti o fl

Semantics

64 SPATIAL

PROPERTY

CHARACTER ATIRIBUTE

IDENTIFIER

pOSITION

School

Name

CoordinaleS lor one poinl

to number

Coord inates lor location at surtace

Depth

Borehole

coordin ates lor surtaces

Volume

Aquil er

Nam e

Coordinat eS lor one point in time

Type

??

Volume

Cloud

ENTITIES

Distance 10 anolher SChoo l

Number 01 students

BEHAVIOUR ATIRIBUTE

Educalio n

Size 01 hole

purpose

Water quality

Change in water level Dissipation

Code number

Coordinates lor endpoints

Connection 01 tw o cities

Trallic volume

Use

Railways

Coordin ates lor perimeter

Composition

Number

or name

Total area

Barrier island

Change position

Name

Coordinates lor bOundary

NeighbOurS

Cou ntry

Gro ss national product

W ars with neighbOurs?

Figure 3.2 Example data for spati al e ntities.

1. An id e ntifier ( name , cod e number, or o the r device). 2. Th e position o n the earth' S s urface ( locator) .

3. The characte r o f the e ntity . 4. Th e role, behaviour o r function of the entity. 5. Spatial properties of the entity. An identifier prov id es a m eans to refer to different e ntities, although it is possi bl e that the namin g syste m used is not very precise, consistent o r able to guara ntee uniquen ess. For example, there is more th a n o ne Derek Thompson in the world, a nd more than one Paris , although a ppare ntly o nl y one R obert Laurini . Identifie rs may be geog raphic names, e ither in loca l usage o r offici ally recognized by nati ona l or interna~ional organiza tions. For example, the United States G eo logical Survey has an ex te nsive list of official pl ace na mes for the USA , co mpile d large ly but no t solely on the basis of no menclature appearing on topographical maps , a nd approved by committees at different levels. Entities m ay be given a rbitrary n,tnKS solel y for the purposes of usc by a particular person or organi7.ation. At tilll es, Iltllllcri ca l cmling tll ,ty be used in stead nt', o r in co njunction with, pLttT namc s ()tl a glo\1 ;t1 sca le it is likel y that llot ;tli e ntitie s It ;tVL' p\;lCl' lI ;tnll'S, ;1I1l1 therc ;11 \' type s of

The in forma tion in a spatial information system 65 ~ ------------------------------------------------------

e ntity in a spatial inform a tion system that d o not need na mes, but are referred to by unique numbers . Some phen o mena m ay have di ffere nt na mes at different tim es, like th e Volgograd and Stalin grad in th e USSR, o r the change from Cape Canaveral to Cape Kennedy and back aga in to Cape Cana veral for a promo ntory in Florida. Info rmation abo ut position (discussed more fully in C ha pter 4) is needed for locati ng or delimiting na tural o r ma n-made objects on the ea rth's surface. Usually numerica l coordin ates a re used, but th e re a re ot he r form s of spatial referencing , such as a postal code or a narra tive description. Positio na l inform at io n m ay be used in conjuncti o n with names for identifica ti o n purposes. For exa mple , th e ce ntre of th e city of Pari s, France is loca ted at (approxim a te ly) 48° 50' north la titude , and 2° 20' east lon g itude , but it is unwise to use such information as the so le id entifie r for that settlement. Very few place names give a n y indicati o n of position, a lth o ugh th e re a re some that d o , a nd even then not a lwa ys co rrectly . An e ntity m ay no t chan ge in any regard except for wh ere it is positioned, for exa mple , a rea li g nme nt in a highway . But if that occurs , the n a ny identifier that is based on coordinates will have to be altered. Ch a nge ma y a lso occur via ea rth processes; for exa mple, a point in three dim e nsional space may no longe r be there after a n earthquake or other dis turbance. Positional inform atio n is a lso subje ct to limitati o ns in acc uracy , reflecting principally the instruments used for reco rding purposes. Not only is accuracy limite d if positi o n is meas ured from movin g ve hicles; but it is also a ffected by the mode ls used to establish globa l measure me nts, th a t is the shape of the geoid a nd th e datum used lo r recording distances above and below sea leve l. Beca use s patial properties are conditions that use positi ona l inform a tion ill so me way, we treat th e m as a sepa rate category, although the data for hasic characteristics like perim e te r of area ca n be, a nd so m e tim es are, (ll.~tomarily conta in ed in th e informatio n system in the sa me mann er (t
Semantics 66

Non-spatial attributes

67

~~----------------------------------------------------

3.1.3 Metainformation

but for which data are collected and assembled in an information system. Some descriptions of entities may refer to behaviour in time in contrast to static conditions, as with the property of erosion of a barrier island. Some characteristics may represent functions of an entity, for example the activities undertaken in a building. Descriptions of form, the spatial characteristics, may be insufficient but, most likely, necessary for many phenomena. Whatever the kind of attribute, a particular instance of an entity category may be distinguishable from other instances by the particular condition. For example, particular occurrences of the entities labelled cities can be differentiated on the basis of size (number of inhabitants). Instances may be differentiated solely on the basis of a non-spatial attribute value or for a combination of characteristics. The entities may also be characterized by spatial type, as discussed quite fully in later parts of this chapter. For now, we mention simply point, line or area type of expression for phenomena. Entities of similar nature may be conveniently grouped. For example, highways, railroads and canals can be grouped as transportation features. The grouping may be done on the basis of spatial form. For instance, the category of transportation features could be only those of linear form, the railways, etc., but not the stations, conceived as point elements; or the cat egorization could ignore spatial form entirely. Figure 3.3 identifies the commonly encountered terminology, although we must point out that there is no standard nomenclature.

In an information system, indeed in any collection of data, the entitles carry a variety of properties and names, but there is also material about the criteria and procedures used for definitions and demarcations. Sometimes this documentary material is referred to as metadata or meta information. Without going into details at this point, we see the ancillary and complementary information including such data as:

1. Definitions of entities. 2. Definitions of attributes. 3. Explanations for measurements of attributes or coding practices. 4. Explanations for false colour encodings used in maps based on remotely sensed image data. 5. Accounts of rules used for delimiting entities spatially. 6. Guidance as to data sources, quality and date. 7. Explanations for missing values or inappropriateness of a measure. K. Any other information to provide clear explanations of the data. Indeed, in a spatial database there may be spatial data for some ;lllcillary information. Think of maps showing the variation from place to place of the accuracy of the representation of entities. Furthermore, some material can itself be recorded as attributes: for instance, a column of (\;tta could carry the name of source documents or of who encoded the ITl formation. Textual data could be stored separately, but associated with Ill\; numerical tabular information. We return to the topic of meta Illformation in section 12.6.1.

Primitive spalial (pOSSibly non·spalial) unil as perceived (exiSling) or conceived (projeCled), for example a school

ENTITY

Physical represenlalion of an entily OBJECT

I.'l NONSPATIAL ATTRIBUTES

A particular occurrence of a given type, for example Roosevelt School INSTANCE point, line, area, elc.

SPATIAL TYPE

Descriptive characteristicS of an entity, lor example, type 01 school ATTRIBUTE Category comprising similar attributes, lor example graces of students ATTRIBUTE CLASS Particular measure of an atlribute for an instance of an entity, for example, ATTRIBUTE VALUE

number of students, or type of school Category of entities with some similarity, for example all educalional

ENTITY CLASS

establishments A combinalion 01 several entilies making up a complox unit

FEATURE

FigUH' ., ..\ I )l'lillilioll~ for v;i1l'I',onn; of IlIfOI'll1;i11011.

l' lopcrties of static conditions, function or behaviour may be numerous. I'v cn though only a few are needed for definition or demarcation, a ,', Hllprchensive inventory may produce hundreds of attributes for "I I('odi ng in an information system. Accordingly, attributes can be l ll:lrdclcrized according to purpose or value, possibly by creating attribute d asses: for examplc, those characteristics needed to define wetlands can I' I ~ I' )('('ially markcd, Groupings of allrihul~s or entities are done for , 'HIV('INCIlCC of workillg wilh data, hili 11I
Sem antics

68 Phenomenon

AERONAUTICAL CHART INFORMATION 1

\

10POGRAPHIC AL DAIA

Feature group

AERONAUTICAL DATA

\

r

I

CULTURAL HYDROGRAPHY

~

HIGHWAY RAIL \

AIRPORT \

Feature

COMMUNICATIONS

class

TERRAIN

\ AIRSPACE

i-l-l

POPULATED

PLACE

PROCEDURES

'

Entity

ZONE

\

\

Attribute

r

Value

TYPE

TWO-LANE

OIVIDED

(a)

r

CULTURE ML

. _. - - - - - - -' - - - - - - - - - - - - - - - - - - RAIL·ID RAIL_CLASS RAIL_NAME RAIL_LABEL FRO M_ NOD E TO_NODE X· COORDINATE Y_COORD INATE

~

relerence numbers lor rail segments single track, double track, eleclr~ied, etc name lor par1 icular railw ay e.g. "under construclion", "railroad yard" 10 lor beginning 01 rail segment ID lor end 01 rail segment X coordinate lor a po,nl on segment Y coordinate lor a point

ROAD

HYDROGRAPHY RIVER

(b) Figure 3.4 Example of classifyin g informa ti o n on ae ro nauti ca l charts. (a) Cl ass ifica tion , (b) Some specific items.

phenome na shown on Unite d States aeronautical charts. All initial distinction is m ade betwee n topogr a phical and aeron <\lIti cal in accn rd with the data handling rea liti e s of the orga ni i',ation Pl'l ldllcin (!, tile maps , essentially re prescnting two differe nt 110ws or matcrial. Th e IOI)(lgr;tpllieal and aeronautical C
Non-sp atial attributes 69 ~ --------------------------------------------------------

th e data flow within the organi za tion for the air travel info rmation a nd th e custo mary classification of topogra phical m a p d ata, partly itself reflecting map sy mbology a nd production processes. Within fea ture classes a re the diffe re nt e ntities that , for the purpose of map-m a king, are meaningful and are di vis ibl e no further, for exa mpl e , highways, So a t the next leve l, highways are described by attributes like Road _class, Roa d_n a m e, Ro a d_ID , R oad_type . Road_number , some of which are identifiers for re<:o rd keeping, like 10 ; or official name s, like Pe nnsylva ni a Turnpike . For each attribute the re a re attribute values, the spe cific quality o r quantity assigned to an a ttribute, for exa mple , Nation al , for cl ass of road; or 2,500 metres for road le ng th. D efi nitions a re affecte d by cultural diffe rences, for example, th e phys ica l entities of city and tow n are de fined differently in m a ny countries. Definition s may change in time, as in the U S A with the co nce pt of poverty threshold_ A single term m ay be used for di ffe re nt ;Ittributes, In Fran ce , for insta nce, th e word fo r landown e r propri e taire, has two m e anin gs (polysemes): th e persons wh o have signed th e property docum e nt to a no ta ry (holders of property documents) or th e person s payin g property taxes. The quantita tive or qualitati ve prope rties m ay result from simple m e as urem e nts, m ay be produce d by m a ny or complex mathem at ical ,)pc'rations, or may be nal rative desc riptions. Some characteristics may rcllect a sta te of p ossession , as with th e prese nce or absence of fo rest; or l
"- Semantics 70

display techniques with the basic nature of the attribute data. Attributes may also be classified or characterized with respect to:

1. Specific values or ranges of values that are correct for a given attribute. 2. [f an attribute is original or derived. 3. If it is necessary for tbe definition of the entity, such as a n identifier. Specification of different values that are possible provides a means of "SCALAR" RATIO INTERVAL

"CATEGORICAL ., ORDINAL NOMINAL PrQR~[\ias

Presence/absence Difference in degree or qualilY Counting Equality of calegory

~ CLASSIFICATION

ExamRI~;;

Name of city Type of landuse Name of highway

Sequence Relative position Greater, lesser or equality

Differences Arbitrary zero

Ratio Real zero

Addition subtraction

Multiplication Division

MEASURING

ORDERING

Large city Wettest soil Primary highway

Bearing in degrees

Number of people Number of air passengers Distance between two places

~

3

E 2

",M

SChoolS Highways Land use

Type of school Class of higl1ways Housing type

Figure 3.5 L evels of measurement. Altributes

Number 01 sludenlS Volume of traffic Density 01 housing

the attributes

Non-spatial attributes

71

checking a nd validating the encoded information. For example, only fiv e types of land use may be recognized , or lakes must have an area greater than zero, or people may be no older than 120 years. In a database, one or more a ttributes may have no information recorded , or if they do, they may have been derived from other attributes contained there rather than being indepe ndent measures or qualities. For example, population de nsity is derived from th e number of people and area of census zones. At a given mom e nt, or even permanently, the values for density may not actually exist. Good documentation about th e attributes and entities may be undertaken by specially marking those properti es used for definitions. Many techniques are availab le for creating derived measures, and many different operations may be involved . Some types of operation a re: 1. Value reclassification from rea l numbers , that is , interval or ratio scales, to ordered categori es . 2. Value substitutions, such as averages, particular values like a maximum. 3. Value creation through combinations of values for different attributes via arithm e tic operations, perhaps using advanced statistical techniques. 4. Value creation through Boolean logic combinations; via trigonometry or interrolation. 5. Value assignment, substituting values from one variable for another. fl. Those which involve a spatial property as well as non -spati a l attributes. 7. Those which result from simulations, or produced by 'what if?' modelling. Thu s, we distinguish betwee n measured properties and values which dre produced for simplification, because some prope rties are inh ere ntly measurable, or because new scales are needed for mode lling purposes. I'he example of Figure 2.16, showing the prediction of the location of a 1;1 rget area for min e ral discovery, uses da ta for several measurable v;l riables , such as iron content in rocks , to stand for unmeasurabl e qua liti es like rock deformation. The exa mpl e of Figure 2.6 uses a pro n ·dule of comhining different original attriqutes to estimate the potentiality 1m resid e ntial developm en t by means of J- new sca le of values. Because our primary purpose is to e mphasize spatial data, we shall not '.v.,It:Ill
\... Semantics

73

Spatial characteristics of entities

72 POINTS

LINES

AREAS

Rivers

Wetlands

BLOCKS

3.3 SPATIAL CHARACTERISTICS OF ENTITIES Entities may also be classified according to spatlal pro pe rties . Accord ingly , the e ntities are ofte n thought of as basic spatial units , the fundamental, practical, non-divisible el e me nt ma tching a perceived e ntity a nd holding th e e ncoded data. These units may he seen from three perspectives:

1. From that of the numbe r of spatial dimen sio ns. 2. Fro m that of th e type of spa tia l prope rt y. 3. From that of how th ey may be co mbin ed for a particular purpose.

•

•

• Boreholes

Aquif er

H

H

3.3.1 Dimensionality of entities Suppose we see a pa rt of the earth' s surface o n which are scatte red

phenomena; we ca n think of the entities as po ints, lines, areas, volum es

o r combinations of those. Assuming we can recognize and dem a rca te

separate entities , we have different spatial dimensions (Figure 3.6). In

principle , the po int , lin e, a rea a nd volume descriptions refer to th e ze ro,

one, two and three dimensions of geo met ri c figures , with respecti ve

pro pe rties of nO possible measures, le ngth , le ngth an d width , length , n width, a nd de pth . The idea of ze ro , o ne or two dimensions is perso made, fo r in rea lity eve rything is three-dime nsio nal. As a practi ca l matte r, a dot on a map has size. O n th e other ha nd, the dimensionless concept of a po int for a city will , upo n e nlarging a scale of vie wing , as for the map re prese ntation, beco me a n area object , and at tim es a volum e e ntity as in a pbysical three-dim e nsio na l model. The well kn ow n ada ge of ' th e furth e r away yo u are the less you see' is as impo rta nt fo r spatial inform a tio n syste ms as for m any o th e r topics. Thi s scale of o bse rvation or m appi ng will affect what mi ght be re prese nted in a d ata base. For exa mple, so me e ntiti es m ay disappear altogether beca use a minimum size of unit is establi shed for a particular sca le. This concept of resolution , the minimum addressa ble spatial unit , for exa mple a line no less than 0.005 of an in ch , o r a square of 2.5 hectares, is related to scale but is not the same. Point entities , defined for a given scal e of o bservati o n, will appear in a database as se parate lin es of information in a tahle, with two ite ms of info rm ation for position , o ne, or possihl y IWO , il ems for ide nlili cat io n , a nd o ther attributes of inte rest. Each poinl is see n
H

Sanc hez

Hisloric sites

Streets

Land parcets

BU ildings

River names

Country names

Building names

Elevat ion layers

Subsurt ace depths

.

Barce lona

Madri! • Lisbon Granada Place names

3 16

X 475

X

X 296

400 300

Spot heights

Contou r lines

\ Figure 3.6 Types of spat ia l unit s.

\.. Semantics

74

not important. As we shall discuss late r , points may represent phenomena of higher dimensions , or may simply be locations for the placement of names on maps. Point spatial units are especially important in spatial sa mpling , for example where to drill boreholes. In utility and tran sporta tion systems inventory and analysis, s uch zero-dimensional units record the location of telephone poles or sewe r manholes , or are used in recording data for individual house s with discrete addresses. In geology and surveying, point positions on the earth's surface, known as control points, are important zero-dimensional entities.

Line entities (also referred to as polylines, arcs or edges) in a simple

view of phenomena like pipes a nd railroads, are separately occurring

phenomena, unrelated to each other in any way. So we think of the Mississippi River, the M I motorway , or an air link from Madrid to Miami. But these may be parts of a set of linear elements, all the rivers of the Mississippi system, all motorways in Britain, all air routes of th e world. The entities may be directly observable or may be conceptual only, as is the case of air routes, and the most interesting combination of a sinkhole and spring in a karst (limestone) landscape. They are not necess arily only in a horizontal plane, for example a bore hole or a water le vel in a mine. Linear features are important in transportation studies, hydrology, utilities manage me nt and geology, and are prominent features on many types of map.

Some spatial properties defined for linear features are:

I. Length of entity: for instance, le ngth of canals. 2. Sinuosity, for example, the Lower Mississippi River bend s. 1. Orientation , such as compass direction of mineral veins. Area entities (sometimes refe rre d to as polygons, regions or zones) may be identified for natur a l or man-made phenomena , assuming in most cases a world of the earth's surface without height. The area spatial units may be natural entities lik e lakes, islands, territory with a particular soil type, or the top of a building. The units may be artefacts used for statistical reporting lik e census zones or delivering mail like postal zones, or discretizations (the creation of pieces or segments) of continuous space like climate regions. Boundaries may be unclear, multi-attribute, changing in time , variable according to definition, and may not be directly observable. Area units are important in socio-economic studies, analysis of terrain conditions, land use and natural resources in ve ntory, and recording of real estate.

Particular spatial properti es associated wilh arl:a cnlities :Ire:

1. Areal extent: for exa mple , the size or I:tkcs. 2. Perirneter length: f()r l'xdnlplc, thl' ex!l'1l1 of a sIHlrl"iIll<:.

Spatial characteristics of entities

75

3. Being isolated, or connected to others: for example, the separate elements of the Hawaiian Islands, or the contiguity for countries of the Middle East, or the historical case of the West Berlin exclave of West Germany. 4. Being punctured or indented: for example, the District of Columbia ' hole' within the United States, or the sma ll Spanish village, L1ivia, completely s urrounded by France , or, in North America, the Chesapeake Bay with islands and a very irregular shoreline. S. Overlapping: for exarnple areas of circulation of different newspapers , or non-overlapping , for example school districts. Volume entities (solids or blocks or polyhedra in some contexts) bring the third dimension of height or depth into the picture , giving us recognition of buildings or earth features like volcanoes or chasms. The phenomena may have clearly marked boundaries, or they may not be directly observable with current measuring in s truments, like aquifers or mineral zo nes below the surface. The volumes may be three-dimensional equivalents of statistical artefacts, for example, the faces or polyllc
Area for each two-dimensional cornponent.

I. Len g th of perimeters. I. Be ing punctured or indented.

\ . Ilaving isolated or connected pieces.

I I. I'rofil es and cross-sections.

\ I. Ll C('ometric elements

1:111111 Iii" V(I.'wI'Oinl or 1'111111:1111("111 :1\ :, "I I ~'·"III I.·IIY, Illl' oillccl poinl is 111·,llt.' d I)), \.: vcly ("IOSSIII !1. 1)1 1"llIlill:llill" II I li" I.· ~ (l:i)!.\IJ"\: 1.7d). ( "II ~ \" "III.'lIll y . pOil' lt.~ ;111 ' ~;Iid II) I" , IV IIIII,"1 '.1 /, ' TI,,' IIOlioll Ill' ;1 lillc 111111 '· ... h01l1 0111 "i',II;lii/:llillll (II "I li lil l" 'H'I',II,III'"1 1""111'1',·,, IWI) tli.~lill\ ' l 11'1111"', :illd Ih,' ""1)(" ', floll 111111"'.1 ' 111'1( 111
~

Semantics

Spatial characteristics of entities

77

76 Vertex (Point) Vertex

0'

Vertex

(a)

Line 1

Vertex

Vertex

Links (Edges)

(c)

(b)

TWO DIMENSIONAL CASE Shell Solid

(Block)

and o utside (Figure 3 .7d). In this type of e ntity so me faces touch a nd so me hav e co mmo n vertices. Fo r th e same set of points, three dimensional figures may be regular o r irregul a r in appeara nce (Figures 3.7e and f). Th e o ute r bounda ry of th e complex figure, a pol y hedro n, is known as the s he ll , and a ny se t o f lin e segments is a loop. Solids may be puncture d, h avi ng one o r more ho les, and m ay be co nn ected . Gove rnm ents, acade mic ian s, a nd oth e rs , besides geo ma ticia ns, have prod uced classifications of spa tia l obj ects which o ften use differe nt terminology from that use d in the fields of geo metry o r computer graphics , a field very impo rtant to th e practi cal creation of spa tial inform at io n systems. On e rece nt effort in the USA (N at io nal Co mmittee for Digital Cartographic Data Standards, 1988) has produced a schem e proposed as a sta ndard for two-dime nsional ca rtograp hic entiti es. In this sch e me ze ro - , on e-, a nd two -dimens io nal cartogra phic obje cts a re class ified acco rding to three circumstan ces of spa ti a l property : I . Geometry-only ope ration s (as generally invo lved in map drawing). 2. Topology-only ope ratio ns (as might be require d or sufflc ient for some

kind s of spatial anal ys is, like accessibility of node s in netwo rk s) . .1. Geo metry a nd topol ogy.

' (d)

(e)

Hole

(f)

THREE DIMENSIONAL CASE

Figure 3.7 Ge om etri ca l units. (a) Linear units. (b) Co nce pt of a line. (e) Units for complex objects . (d) Solid object. (e) Simplest solid. (f) Irregul a rly shaped so lid with hole.

to the o th e r (Figure 3.7b). The idea of growth in o ne direction leads us to say that lines are o ne-dim e nsional , a nd to di stin guish between extent and dime nsionality. The latte r concept holds whether th e lin es betwee n points

,

are straight or curved. Conside ring sets of fundam e nt a l geometric entities (Fi gure 3.7c), we recognize a node (usually call ed a vertex in geom et ry and engineering) as a point that terminates a line or is the po int at which lines cross. Therefore it has a prope rty o f co nn ectedn ess, bcing rel ,)tcd to the lines. These lines a re called links, some tim es arcs~ a nd arc refe rred to as edges in geometry , if both e nds termin ate in a nod c. Ev e ry link has I wo nod es. A set of three links is a minimum ne eded to houlld ;1 two-dimellsiollal surface, or area. F or thc third dimension tll c s(llid is IlLlcl e lip of ;1 sc t or areas or faccs (facets), whi ch cO \1lhin c 10 PWt\II l"\: ; \11 o \l.jn·1 willi ;Itl itl si dl"

In ca rtogra phy, ze ro-dim e nsio nal e ntiti es m ay be perce ived rea l Ili1 e nom e na, an entity point ; points with text , a label point; points for ':I rryin g at tributes of areas, a n area point; and topological junc ti o ns or L'lld po ints, a node , when d ea ling with ne tworks of linea r features (Figure U~). A se t of coordinates specifies th e loca ti o n of any type of point. !\ lin e or o ne-d imen sio nal object may be a strai ght line conn ec ting two -IH)iIl IS, a line segment; or non-straight , a string (ca lled polyline in

X

Point

A zero·dimensional object which specdies tocalion

•

Enlity POlf't

Identif ies an entity

CJ

Label POlili

Poinl with text

0

Area pOint

Carnes allributes 01 an area

•

Node

Top()lo~ical

_/

Mo scow

jUllcllon or end·point

"'i~llI"(" ,U! I'oilll e:lI"I ())! Llpliir () I )j l: l" I ~ . (llased ()II 111 :l lcrial ill till: II 'c·" "111l (" ldaliol, "f Iii.. l iS N:ltilillitl ( 'llIlIllIilllT f" r l)igil ;tI ( 'a rl o I '.t:q, lii ,· IhLI St:llldilld , .; " l :t ll'l I'ltldl ':IiI ·" . 1' , N :lt,,, " ;" (·lIlIlIlJiIIIT lor I )'rI1 ltl ( ·; 11 ")I ~ I :'plii,· 1),,1 1, ,"'; 1:11111 111 d'. 1

Semantics

79

Spatia l charac teristics of entities

78

//~

/ Line segment

(Line)

Link

Directed (oriented) link

Arc

~~~~

Ring of string s (Clo sed polyline)

C hain (Arc)

Ring of arcs

Simple polygon

(Line)

(Edge)

String (polyline)

Interior area

Network (oriented) chain

G

Ring of links (ClOsed polyline)

Complete chain

Ring of chains (Oriented polyline)

Figure 3.9 Line ca rto gra phic o bj e cts . There is no stand a rd te rmin ology fo r the na mes o f cart o graphic entities; so me sy non yms are indica te d. (Base d o n mate ri al in the recomme ndati o ns o f th e US Nationa l Committee for Digital Ca rto gra phic Data Sta ndards, as la te r publi shed as Na tio nal Co mmittee fo r Di gital Ca rtographic Data St,lnda rd s, \988.)

computer graphi cs) , bein g defin ed as a locu s of points o r a se t of lin e segments (Figure 3.9). Th e locu s of po ints, an are , ma y be defin ed by a spline curve or po lynomial m athem ati ca l fun ctio n, or o th erwi se. If the topological propert y of co nnectedness is co nsid ered , th e basic unit is termed a link ( part of a set in a network), which may be directed (usuall y by specifying ori gin and destinatio n nodes) or a chain if sev eral lin e segments are involved . If th e propert y of neighbours fo r the line features is involved , then we define an area chain; if both conn ecti on an d adjacency properties are involved then the sct of link s produ ces a complete chain . T o demonstrate the vari ation s in terminology, nol e thai th e word arc is used in sev eral w ays: as in Ihi s propo~cd s(;lnd ard, in SOlll e software sys tems as a sy nonym for comp le le chain , ()r , ill I :r
(only one boundary)

Complex

polygon

(several boundaries)

•

Pixel

•

Grid cell (pixel array)

Figure 3.10 Are a cartographi c o bj ec ts . T he bo und ary is no t of interest fo r th e ' area' unil. A co m plex po lygo n may a lso have di sco nn ected pie ce s. (Based on ma teri al in th e reco mm e nda li o ns of the US Nati o na l Committee fo r Digita l Cartog ra phi c D a ta Standards , as la te r pu bli she d as Na tion a l Committe e for D ig ital Ca rt og raphi c D a ta Sta ndards, 1988.)

spati al unit is called a ring . Rings may bc crea ted from strin gs, arcs , links , l)r chain s (Fi gure 3.9). T wo- dimension al entities (Figure 3.10) are bo unded sp ati al units which ma yo r m ay no t include th e bo und ar y for ca rr ying info rm atio n - th e ilo undary bein g necessary for dcmarca ti o n purposes. If the boundary is ex clud ed , th e term intedor area is used ; if th e boun dar y is inclu ded then \ \l' have a polygon co nsistin g of the in te ri o r area and an o uter ring. A !,,, Iygo n ma y be simple if it has no interior rin gs, or co mpl ex if i t has at k :l st one interior ring. (In so m e co ntexts th e te rm s closed pol yline and 1l( llygo n are sy nonymo us w ith rin g.) Three-dimension al obj ects, no t covered by thi s pro pose d stand ar d ,1 : 1 ,, ~ ili ca ti on , with t he ad diti o n of t he verti ca l dim ensio n ca n be trea ted ',i lllil :lrl y to two-d imen sio nal obj ects. Thus we can recognize th e primitive ,' I<- Ill enl s o f lin e segm ents and area segm ents, or faces, and wh et her or 11,, 1 I() pology propertie s arc in vo lve d. I k pen d in g Oil circllllls l; (h .~ III \: ~ .II) , I)(, ~,s i l" v 11i(' 11\( ) ~ t eomplex <JnL! 1" 111',C's lcd (I f :III V killd (II 11" ' 111 :11", 11 1" 1' ( " Vdl!111 :1\'l: l llllJI Il en ls 10 know " I I I,, ' ,' \I SIt-I" ',' (:11111 , (II 1.'(1111 "(' , III ,' 1'''', 11 1(1 11 ) " I ,1I1 ~ I' : W V I ,,·, tri ckd t()

Semantics

Combinations of entity t yp es

81

80 military aircraft - the boundary condition is important, not the character of the airspace itse lf. But at the same time, th e civilian aircraft may need to recognize zones within which particular radio fr e quencies can be used. In some fields the three-dim e nsional obj ects are discre te, for exampl e , separate architectural structures. In oth ers, th e y are joine d, as in th e example of geologica l strata. The character of entities and space is often expressed in d a tabases by the use of discrete primitive recording units as demonstrated above. How we ll the data re triev a l part of th e information syste m wo rks d e pend s on the particula r circumstances of the uses of the data. It is in this regard that we now look at co mbinations of spa tial entities and alternative ways of recording spatial variations.

3.4 COMBINATIONS OF ENTITY TYPES Spa tially oriented processing as d escribe d in C hapte r 2 can involve Illany types of entit y at one time , o r for one probl e m on many occa s iolls Cartographic representa tions can d ea l with many types al o ne lim e .

3.4.1 Combinations of spatial units Spatial unit s may be combined ir; diffe rent ways:

Figure 3.11 The example of aeronautica l charts. Thi s chart is included as an e xample of source doc ume nts with a hi gh den sity of data, and many diffe rent types of spatia l entity. This example is a portion of a n ori ginal multicolo ur aeronautical charI of th e area of Cheyenne, W yo ming, USA. The example shows th e ele vat ion zo nes, airport characteristics, airspace , communicati o n fa cilities. urban areas, and o ther cultural a nd aer o nautical informali
I. . Spatial proble ms can be described acco rding to which spatial units are involve d, and particular tasks may require severa l type s of sp atial unit. ...:. ) Complex entities may be created by combining units of diffe rent spatial dimensions and type . .l. Spatial units of one type are combined to pro duce a new type, a compound entity. . ~ One type of spatial e ntity may be ~re!lsform e d to anoth e r type . ') Some spatial entities have du a ls, that is , a matching entity of diffe re nt dim e nsiona lity. 1,(lcalioll problem s or spatial qu e ri es may require m a ny combin a tions of 1111i1 S. Taking only pa irwise or hinary cnmhinations for poi nt, lin e , or a rea "llIilies, Ih e re arc ninc poss ihilities (hgllrc .1.1 2). Il may a ppear a t first , il!ill lilal lh e rc arc oilly s ix: poilll point- point-lill c, point-area, IIII<' liill', lillc- ;Irc
I'

~

Semantics

82

census tracts (point within area) , or the areas conta1I11I1g talented and gifted students (areas containing points). A problem may be posed differently conceptually in terms of apparent causal d epe ndence or, as a practical matter, the processing sequence for a spatial problem of points and areas can give different results.

qU.~

~

0

0 0

~ 0

0

o

Public buildings

B

Schools

)'"

c·: _<: u

~

Is on? Is near to? LINE-LINE

LINE · POINT

o

•

Case 01 lung cancer

~

poSlal zone

Is • inside zone? LINE -AREA

0

Passenger rail

o

Shoreline

Marm

M

Is near to?

~

POINT - AREA

POINT -LINE

POINT . POINT

/'"

River

~

Trai

Town

Does line seNe town?

AREA· POINT

Does postal zone encompass schools?

Is river inside zone? Do river and zone boundary coincide?

Do trails cross rivers?

AREA-LINE

Do zones encompass railways?

AREA-AREA

Do zones touch? Do zones overlap?

Figure 3.12 Hinary c(l lllhilialiolls (It" spali:i\ 11111" .

Combinations of entity types

83

Each combination has assoc iated operations th a t may require specific tools in an information system, as illustrated. For exampl e , point-point re lations can involve distance measures or comparisons of coordinate values . Point-area rela tions are inclusion queries or problems like finding the extent of visibility from a he licopter. Line-line re lations include such elem e nts as coincidence of fea tures, or d e termination of inte rsections. Secondly, entities may be defined as combinations of more than one spatial unit (Figure 3.13a). If these co mbination s consist of different types of entity , we have complex objects. A parcel of land with a house, wa ter utility pipe, frontage onto a road and a garde n may for so me purposes be treated as a spatial e ntity, call ed an improv ed lot. A collection of parking spaces a nd adjace nt kerbs, a parking lot , is anoth e r exa mple. If th e combination consists of diffe re nt instances of one type, for example links in a transportation network , we hav e compound objects, also, of course, a type of complex obje ct. In many spatial problems, the unit of interes t is a combination of two places, or spatial dyad (Figure 3 .13b). Most often this is a pairing of point objects, like citi es a t the beginning and end of air routes, or points re presenting area c ntiti es as in the study of strea ms of mi g ration betwee n countri e s. Howeve r , more gc ne rally, it is a linking in space of particular e ntities as impli e d in Ill e r~s uring di sta nces , or finding directio n from an origin to a d e stination . IJin ea r and a real spati a l entities may also be either co ntinuous or discontinuous: that is, th e re may be break s in a set of pipes , or ga ps in a land parcel system. (n th e transporta tion case we have th e unit of the net or network, or graph , a set of links (Figure 3.13c). Thi s unit has pro pe rties in addition to those of an individual link, for example the poss ibility of identifying p:llhs, and th e definition o f network co nnecte dn ess: Point objects may be ((l illbined to produce lattices of points (Fi gure 3.13d) ; these can be lliought o f as networks or arrange ments of po ints in diffe rent patte rns, like squares or tria ngles. Area entities are combined to produce fl-sscllations (Figure 3. Be), se ts of area l objects of either regular or II I( 'glli a r form that cover all space in a -defined area~ Facets are combined II) produce polyhedra (Figure 3.13f): In summary, we produce new 1' 111 i I ic s at a higher leve l than the elements from which they are '( IIISlillltcd .

1.4 .J. Suustitutions of sp.lli.ll units

111I11I1 v. pdllinti :1I IVI)(" '; 01" 1IIIil II HI\I II(" ', 111> -. 11111(( '11 lor ollil"l"s, involving 11111("1:1 InliH"lioll 011111 '1('; 1.'.1' III dlllll ' II·.II)II : IIII v. ,'()illis olliliVs 111:1\1 slalld

"--

Se mantics

85

Combinations o f entity types

84 Darwin /" ,

Oslo

/

L.onOOn ___ ---

/~

~ -h~dam \\'''~

"Moscow

--.

/

'~"'" ~ranklurl

Paris

/'" , /

'-....

I '\\.---""'"

\

\ ___ ~\i J-

o/~

\

ot o'r.-/\

o \. ..... I ./

. Zunch

sidewalk

DYADS

(a)

COMPLEX OBJECT

NETWORK

~

TESSELLATION

(f)

(C)

m

~

I

~

Trees in orchard

0

0 0

( d)

LATTICE

-

I

(e)

,

'-.~y,

0, !Adelaide

\.

Canberra (b)

,

..........

I

Sydney

0 I Melbourne

(c)

Figure 3.14 Substitutions and du als for spati al objects. (a) Centroids for polygons. (b) Prox im al areas for points. (c) Points and polygons for a graph.

:b~VJ'SERVE Bouodary 01 pnvale property

= house 101 wilh ilT4>ffivemenls

.... _Brisbane ,

/

I

.....

Rome

parking spaces

--I

'

0

~

e\~~~ I , __ _

0

,I

-I

P rth

(a)

Madrid

(b)

.....

0

CONNECTED POLYHEDRA

Figure 3.13 Complex and com pound spa tial objects. (a) Complex obj ects. (b) Dyads. (c) Network. (d) L att ice and mesh. (e) Tessellation. (f) Connected polyh ed ra.

The rev erse, an in crease in dimensionality, sometimes o ccurs . There are ways to create proximal areas from data for only po ints (Figure 3.14b). That is, territory about specific points is demarcatcd so that all possibl e location s defined by the areas are associated with the nea rest spec ific point. Such a tec hnique has been used for de m(lfcatillg sLllislic;d units representing th e postal areas based o n zip codcs or th e LJllilL'd St.a tes Posta l Service, because few postal areas are delimited. The post;d units are see n by th e Postal Service more from th e po int of view of streets followed by walkin g or drivin g mail carriers, not fo r obtai ning sta t iSI ics ror zones . Simil arly, in th e N etherl ands, the postal zones are the set of street s m aking up a daily workload of a single postperson. I n some cases , point and area spatial units are duals for each other (Figure 3.14c) , that is, pairs of m atching entities. In o ther word s, there ex ists a set of polygons for a set of points, as in the exa mple of the postal I.o nes created from points for post offices . For transportati on networks, line el em ents, a set of polygons and m atching po ints can be defined as dual s. Imagine European countries with capital cities, m apped as points with in polygo ns, The citi es can be connected by links, and po lygons pmdu ce d for those points. ",

:L4.:~

for area units, although in practice it is usually a point lik e a m athematical centroid o r an arbitrary loca tion within or without the spatial extent of the area that represcnts the two-dime nsional ohject , rather th an the boundary line (Figure ~. 14<1) . III
Mixed uses

hlllr(ldy, spiltial prohlem .~() Ivillg ;lIld qucryillg ma y use informati o n for ""vn;1I ty pes oj sp,lIial e illily a( Olle lillll' ( h gllll' 3.1 .')). Indeed , we ean I' T,~g lli ·I,C v; lI ego ri,'s or I<'('; lti'lIl pnllll'·'IIS ill Ihi s w
~

Semantics

Continuous variation over and in space

87

86

Watersh ed catchment area and statisilcs

Gradient, aspect, and spo t hel9hts recorded

for the binary connections among counties o f the United States. A s discussed, the re lationships may be metric, topological or orde red. They may include m a ny insta nces of one class of e ntity, or mixtures of severa l types; and they may be represe nted in ways that are indirect and not obvious. Som e may be encode d fully in a database as a complex o bject; so me software syste m s may not allow this, recognizing only primitive elements of point , line, or a rea .

3.5 CONTINUOUS VARIATION OVER AND IN SPACE A vie w of phenomena as entities is not the only conceptual mode l that can be applied to spatial infor mation. Basic English expressions contrast phenomena that occur on the earth's surface or in the a tmosphe re from phenom e na that a re distributed over space. For exa mple, Finland h as many la kes, and coniferous fore st covers much of its land. Th e continuity of space is also represe nted by concepts of gradients and mathelll
(a)

0

Buildings

• III ~

r? /

/

Fire hydranl

3.5.1 A field view of spatial variations

Fire slation SIreel Main slreel Service area

(b)

Figure 3.15 Examples of th e usc of m a ny types of spatial unit. (a) T e rrain analysis. (b) Fire protecti o n services.

a study area can be meas ured by a summ a ry spati a l statistic requiring initial measure m e nts between nea rest ne ighbour points. The flow of chemically laden water ove r a landscape involves point-to-point prope rties of gradient and flow through channels (Figure 3.15a). Th e de termin
D isc ussion s in section s 3.3 and 3 .4 h ave dealt with th e entity-oriented view. Whil e that tre atment of spatial e ntities did include re ferenc e to the . possibility o f continuity , as in the case of a ne t wo rk or latt ice, and impli e d ily a m e ntion of contour m aps, th e discussion was focused on spa tial units :IS separate entitie s. In the field view of space (Goodchild , 1987) we expli c itly recognize continuity as the principal e le ment. That is, a ny positi o n in two- or three-dime nsio nal coordinate space ha s a value o f an ;il trihute. Even thoug h in prac tice there is a limit to the numbe r of poss ible cases, in th eo ry we h ave: {a :x-, y , z, t} where x, y , and z a re th e ,'()llventioJlal symbols for pos ition, t is the time e lement, and a is a singl e 1)1 set of attributes. Th e va lue of a/ is, in principle, e mpirically 1"I1I tfirJ1lable. That is , if a persoll is dropped from a plane to a location o n III(' ,'; Irlh , the COJlditioJl oJ one or more ph e nomen a at that e xact place is , ,j ),-;U 'v; Ii) Ie . III C()Jltl;ISt, tile ('ntity-oril'lIh'(1 vinv 1i;1.' Ihe cOllditioll that a is not ,' vL'r ywlwre
"- Se m antics

88

19 18

•

(705)

•

. 25

./9 (691)

(562)

i4

,7

(5'5)

(747)

(417)

3~

•

SaIT{lUng points (wealher sialions)

18

Temperature in degrees Celsius. C

(562)

.21 (696)

•

Elevation in mel res

DATA -- { C. X. Y. Elevation}

22

(634)

. 28 (592)

(295)

(a)

+

r 150

Space polential. SP. 01 population in persons per mile SP isolines at 20 person inlerval up to 250. There aller al only 400.600, 800

DATA--

SP ; L P I D where P is lhe number 01 persons at each stale ce ntroid +, and 0 is lhe distance lrom a centroid to another. The PI D quantity is determined lor each centroid relative to an origin centroid , and all vatues in

the set are summed. Thai number, SP,

is the aggregale inlluence 01 the populalion

qu anlily at a distance on the centroid 01

origin: il is that number, e.g. 95, thai is

shOwn on Ihe map. Isoline s are th en

created to give the sense 01 a continuous sul1ace 01 population iniluence.

I

SP, X, Y

I

(b)

Figure 3.16 Continuous space. (a) Discrete (point) o bservation s of te mperature . (b) Aggregatio n over space using a space po te nti al index,

bo unding the point , lin e , area and vo lum e s patial unit s . Natural , physical, or biologica l phenomena arc ofte n clcarl y con tinuously vari a bl e - temperature, b a rometric pressure. o ccan salini( y , so il we tness , slo pe of terrain la nd o the r a ttrihutes ( Fi g urc j. 16a) . At timcs discontinuities ma y occur , as in cliffs (lr wc
Continuou s variatio n over and in spa ce

89

p a rtic ul a r scale, or can be me asured by de riv ed statisti cs th a t re present a fi e ld vi ew, for example , va riations in po pul a tion density, income level s, agg rega te accessibility . Fi gure 3 . 16 b has exa mples fo r a spatial statisti ca l a rte fact, a co ntinu o us surface o f po pulation pote nti a l, o r the integra tio n o ve r area. As a prac tica l matter, th o ugh, the continu o usly varying attributes a re ofte n pe rce ive d as discre te e le ments:

1. As sa mple points, lines or areas. 2. A s iso lines and ba nds between th e iso line s. 3 . As connecte d pi eces. Sampling , no t a lways sci e ntific, creates a reduced number of loca tion s for ca rrying the attribute da ta , so that x, y , and z a re a subse t o f all possibl e tri ads of latitude. lo ngitude , a nd he ight (Figure 3. 17a) . A set of geod e ti c co ntrol po ints , o r spot he ights is a comm o n technique for ca rryin g data for e le va ti o n of terra in a bove some datum line, that is a n acce pted horizo nt a l base leve l re la ted to som e sea le ve l definiti o n . Th e po ints may be irregularly di stribute d as in the case of s urvcy triangulation based contro l po ints, or regul arly dis tributed as in the casc of el c vati o ll s inte rpol a te d from a g rid s upe rimpose d o n e xisting topog ra phi c tl lll:tP ~ Lin es o r a re as are at limes use d as sampling units, as in land scapc profil cs or pl a nt s pecies surveys using quadra ts, res pectively. T he contouring of te rrain eleva tio n or atmosph e ri c te mperaturc usc s the spatial unit o f lin e segm e nts o r a rcs of equal a ttribute value (Fi g urc j. 17 b). More gene rally kno w n as isolines, thi s unit partitions space into band s or zo nes , within whi ch so me assum p ti o ns a re made a bo ut the nature o f th e variation in th e phenom e na o ve r space - pe rh a ps directly propo rtio na l to dista nce be tween isolines , o r perhaps resulting from a hi ghe r order math e ma tica l equati o n . At times th e re ma y be no info rmation pro vide d as to the va ri a bility betwee n isolin e s. The produc tio n o f segments, o r piecewise models, is a common prac tice 1m some a rea phe nomena , esp eciall y soils, la nd use , land co ve r , a nd ()(h c r te rra in conditions; but a lso for'-s.o me socia l phe no me na. In this plOcess . a re al units a re d e limited some how so that bound a ries can be tlr ,lw n for polygonal e ntiti es, producin g la nd use zones , a nd th e like. Th e :t!tributc information is carried by th e a re a e lements, a nd th e bounda ri es I ~' flccl v'lr yin g ass limptions of change betwe c n adj ace nt units. A sha rp 1>1(·: tl< or di sco ntinuity is gcn c rall y assulllcd , with(lut any possibilit y o f 1>l'Ov idin g
' 90

Continuous variation Over and

In

space

91 (18, 73)

(96,91 )

I

Y

-- SOL_ I

o0 -

50 (96, 91)

~

X

IX

Sampling location based on coordinate values, selected randomly

Point samples

Traverses

Uniform plan e - travel equal in all directions

Highway reduces travel cost in the easl·west direction

Quadrats

(a)

400 - 12 -11

4.

1000 100

- 7

8

9

-8

3 - ·5

Point form

. Isalines

Bands

(b)

Figure 3.17 Practicalities in dealing with continuous variations, (a) Some spatial sample types, (h) Continuous field of temperature,

networks. The difference from th e ca se of the complete surface is that x , y , and z have null valu es in many places: that is, they are not defined off the graph Otherwise, there can be continuity of values along the links of a graph; sediment load and vehicle densities are two attributes of this killd for hydrology ,1Ild transportation network s resp ectively, The discretii.ation can similarly usc point or line unit sampling , or segment into particular line elements characte rized by homogeneous or other assumed characteristics sufficiently different from neighbours to justify dividing into new units; for example, to recognize variation s in pave ment conditions.

3.5.2 Isotropicity A particular aspe ct of the continuity of phenomena is the differentiation between isotropic a nd anisotropic (Figure :I Ik) An isotropic condition exists in geometry wh e n conditions do not vary wilh direction, Thill is, the measure of di stance docs not vary with directioll I,Oill ,I p()illl - ,I Illile is a mile is a mile, In contr"s! , hCC;IIISl' S(}1l1 l' plll:Il
Travel costs increase proportionally with distance

ISOTROPIC

Travel costs increase exponentially with distance

ANISOTROPIC

Figure 3.18 Anisotropieity,

the value of space does differ depending on direction , being associated with presence on or off the network. Most clearly associated with networks, this anisotropic condition is , een, also, in the difference be tween kilometres distance and temporal distance. In geography it is often repres.eOjed by the measurement of the distances between places by differe nt measures of accessibility, usually cxrressed and displayed by isoline maps implying continuous variation wilh directional components , as demonstrated by Figure 3,18.

:1.'l3 Discrete and continuous views 'vVl' (';111 IIOW CllL'goJ'i /,(' pll('1I0111<'11:1 illl() liv(' fllild;lIlIl'lllnl types of unit , fll\J()wilil-: (;()()dcilild (1'liO ) , All' l t ,,1'1 (,, '1', :11" dill l'rL' lIli;ltcd from the "IlIIII('('\vd di:;u c tc jll'T( 'S 'li 111111 " III II 11\ . 11 1" 11' ;,1 III:J\,, ' IIjl d IvsS('/!;,lioll

~

Sem anlics

Spatial and non-spatial properlies togelh er

93

92 supe rimposed on co ntinuous spaces ; lin es are diffe re ntiated fro m line segme nts. Points a re clearl y de fin e d in onl y o ne se nse. The set o f non o ve rlapping units re presenting discre tization o f space is often created in a practical se nse by producing sets of polygo n units from di gitized line data; the se t o f te rrain elevatio n va lues for a ll possible locatio ns is often crea te d by sp atia l interpolatio n from a lattice o f a small number of po int locati o ns. A fi e ld vie w of rea lity seems to offe r advantages in dealing with s patial

a nd attribute d a ta e rror. That is, it is straightfo rwa rd to ima gin e a n error

b a nd for th e values for an attribute meas ured at a pa rti cular point

loca tion ; but it is difficult to establish e rro r based on d iscre ti zations. On

the othe r ha nd, treatin g complex and co mpound obj ects is not consiste nt

with the continuou s surface view o f the wo rld ; and peo pl e tend to think

of o bj ects not gradie nts. At the sa me time . as discussed more full y la ter

(initi a lly in sectio n 4.5.3), inform a tio n syste ms ca n be more effi cie nt for so me purposes if re ality is mode lle d as a continuo us map rathe r than as a se t of po ints, lin es and po lygo ns; but the re a re proble ms in e ncodin g a very la rge number of points in space as implied by th e fi e ld view.

3.6 SPATIAL AND NON-SPATIAL PROPERTIES TOGETHER Even though at times resea rch and practice ma y be able to u se data onl y for spa tia l fo rm or locatio n of obj ects, much analysis . mo delling a nd queryin g is based on attributes tha t a re non-spati a l. That is. we a re inte reste d in the ch a racte r of space, spa ti a l cova ria tio n , location- alloca tion pnlbl e ms, etc . Several types o f a nalysis ca n be re cognized o n the basis o f spa tial o r no n-spatial pro pe rties . types o f spatial unit , a nd nee d to cre ate ne w spati a l units (Figure 3.19): l. An alysis of the attributes of a sin gle cl ass of entities . 2. Ana lysi s of the a ttributes of pa irs o f places. 3. An a lysis usin g bo th attributes a nd spatial pro pe rties for a sin gle class.

4. A nalysis with both for pa irs o f places. 5. Anal ysi s th at involves mo re th an o ne type of spa ti a l e ntity, fo r attributes only. 6. An a lysis o f attributes a nd spatial pro pe rties for diffe re nt e nlily types . 7. An a lysis which crea tes a new cl ass of e ntity. The first two types of anal ys is consisl or enquiri es for o nly Jlon -spalial pro pe rties. Th e e nquiri es may be re lri e v,d fronl l
ATTRI BUTE S ONL Y

ATTRIBUTE S A ND SPATI AL PROPERTIE S

SINGLE CL ASS OF ENTITIE S

Parce l s ow ned by Pi er re L econt e

Uses o f l and ow ned by Pi err e Leco nt e near to a coas tlin e

PA IRS OF PL AC ES

Numb er of a ir passenge r s r or se lec t ed c it y pair s

Corr el a ti on o f mi gr at i on s trea m volum es w ith di s t anc e betw ee n zones

Correlati on of cr op y i elds with dat a fr om w ea th er sta t i on s

Anal ys i S o r hea lth condit io ns f or neig hb ouri ng co un t r ies

MORE TH A N ONE SPA TI AL ENTIT Y CLASS CREATI ON OF NE W CLASS OF ENTI TY

(n ot app li ca bl e J

NOi se polluti on bands on e i t h er s id e of highwa ys

Figure 3.19 Attri bute and spa ti al data. (Afte r Goodchild. 1987.)

o wned by Pie rre Leco nte , o r anal ysis o f a ll records co mparing o r correlating two attributes. The differe nce be tween the two categori es is the need in the second to re late two spa ti a l units in o rde r to create the dyadi c unit. (A s su ch , we do not co nsid e r creating a dyad to be the sa me as the se venth catego ry , to be discu ssed below .) In practice, the quer y a nd a na lysis needs a re met by info rm a tion syste ms and sta tisti ca l a nalysi s so ftw a re pack ages or routin es. Anal ysis may invol ve bo th spatial a nd no n-spatial attributes. An ,Issessme nt o f political gerrym a ndering , (-Qsiginall y, de libe rate and bi ased hound a ry drawin g to fa vour one po liti cal party o ve r another) might require data for vo ting splits fo r diffe re nt politica l pa rties and fo r the sha pe of the votin g districts. The e mpirieaLstudy of distributio ns of drug le lated violent crim es could be ne fit from findin g the mea n geographic n' ll!re o f !ow income apartm e nt dwellin gs. An ex ampl e fo r units which ;lle dyads is th e correlatioll o f Ihe volume o f mi g rants fo r th e pa irs of pi;ICCS with the di slallce or Ih e llli g l;]tiOIl Stl': ;llll. At liJlle s 11l0lL' tklll OIl L' Iypl.' ()I' sl' ;l li;d ullit 111;] Y ilL' re quired, espec ia ll y Ii' d Y; ld s ; 111.' illl.'illd n i. !-'tH' ;111 I: ,\ ;IIIJI'I, ' wltlltltlt " V"ds, th L' nop yield s ill ~ ' '' "ll'k ; 11 (, ;1'. II!;I\, Ill' n llll 'I",,'" 111111 1<'11I1" ' lI ltll'I< ' ;111<1. IHl'l'ipilalioll

~

94

Semantics

cond iti o ns at wea th e r sta tion s (po int objects) . For m ost s pati a l inte rac ti o n s tudies o r modellin g th e attributes of th e p a irs of places, fo r exa mpl e, th e numbe r of mi g ra nts between specifi ed origins a nd destinations of mi gra nts, a nd th e nature of th e o ri g in s or destinations se parate ly a re req uire d . An assess me nt of sp atial a utoco rre la ti o n co uld use a s ta ti s ti c compa rin g the a ttributes o f po lygons which a rc ne ig hbo urs w ith those which a re no t ne ighbo urs. Thi s statistic req uires access to the a rea type unit and to the lin e e ntit y (t he boundary for ad jace nt pol ygo ns) . Tra ffi c studi es m ay req uire th e co mbin at io n o f p o int a nd lin e e nt iti es: for insta nce, traffi c li ghts are features associa te d with th e lin es (s tree ts) a nd po ints (inte rsec tio ns) . Some analyses require crea tio n of new e ntiti es, e ith e r un a ry o r bin a ry. If o ri gin a l data co nsist o nl y of po in ts in sp ace, th e n investigations of p a irs of places requires crea tion o f dya d s. If d ata ex is t fo r points o nl y , it may be necessa ry to c reate po lygo ns to w hich th e att ributes of th e points apply, as in the use of Thiessen polygo ns to pro du ce pos ta l code a reas fr o m the loca ti o n o f post offi ces. Wh e n buffer zo nes a re created fo r o ri gin a l po int or lin e e ntiti es, a new polygon unit is formed. Also , new in s ta nces of o ne o r more sp at ia l unit types m ay be crea ted in so me processes, eve n if a new class of e ntiti es is not c rea ted. For exa mpl e , pl acing distance based buffer zo nes aro und polygo ns prod uces m o re polygons. If the spa ti a l data a re co ntain ed in a la yered database, the n ove rl ays produce n ew a reas, lin es and points. N o r a re the s pati a l linkages the o nl y relati o ns hips th at m ay be in vo lved. E ntiti es o r a rbitra rily discretized pi eces of co ntinu o us spa ce m ay al so be re lated in functional/behavioural and ow ne rs hip se nses. The firs t of these includ es rea l world situ at io ns lik e s trea m flow , the ph ysica l co nn ect io n between e ntiti es o r th e process re lat io ns hips of c ha nges in the loca ti o n rath er th a n attributes of e ntiti es , s uc h as the e ncroac hm e nt of a barrier isl a nd o n a lagoo n . Own e rs hip re la tio ns hips refe r to grouping o f e le m e nta l units; for insta nce, co unti es, into hig he r leve l units, lik e s tates o r co untries. Studies in vo lvin g spatial relationships a re, th ere fo re, not inh e re ntl y simple. They m ay re quire computa ti o ns to prod uce va lu es for s pa ti a l m e tri ca l prope rti es, o r sea rc hing to estab li sh topo logica l a ttributes. Whil e not a ll o p e rat ions will be needed in a give n case, it ca n ha rre n th at so m e proce dures m ay not be possible at all with a particular so ftware rack age because it is not well ma tche d to a p a rti c ular co nce ptual Illod e l rC4uired fo r anal ysis. The exceeding ly complex spal ia l rca lil y rnu st Ile co n ve rled to a fix ed an d some wh a t limite d form to rcsidc ill all illforillati o ll syste ill . As demonstrated here alld in previolJs ch;lpt c rs, the rt' , 11 \' IllIIII CH)l IS a lte rn at ives , eac h witll its (lW Il IllL'l'it s ;llId lillli(;lli o ll s.

Mechanics of spatia l data organi7. dtion

95

3.7 AN INTRODUCTION TO THE MECH ANI CS OF

SPATIAL DATA ORGANI ZATION

M a ps have bee n used for ove r two mill e nni a by m a ny individuals , cultures. o rga ni zatio ns, profess io ns ilnd disciplin es to represe nt knowledge a nd perceptions of space a nd its c ha racte r. The spa tial for m of buildings 'is' eas ie r to convey in pictures than with words. H o weve r , so metim es for co nve ni e nce, o r so me tim es beca use of ava il ab le instruments, tab les of da ta a re ofte n th e reposi tory of sp a ti a l data. A tabl e is no t practically co nsiste nt wit h a field view , b ut obj ects o r discre tizations of co ntinu o us space, for Som e p urposes, ca n be effec tive ly represe nte d in tables of num bers o r other sy mbo ls. T he tCl b le by itself has no planned connotatio n of spat ia l locat io n . whereas a m ap by definition co ntain s th e position o f ph e no me na. T ha t is , in a table. the o rderi ng of rows a nd co lumn s is not n ece~sa rily impo rta nl , b ut it is for a map .

3.7.1 Tables and matrices

Tables, o ne , two o r multi way, lin ea rl y o rga nized information, C
III all cases, o nl y at tribute d,(ld arc usod, and the ord e r of rows is

IlililllPOrl
IlIellided in Ih e lailic a I'll' I' they
/(11' pos iti o ll or lo p()l ()gy. ( '()1I1Pil' X ;lIld CO lllPOlilld olli"( 'ls 111 :1\' I'l' illl'liid ed ill suc h a mat ri x lI l,I',: 1I1i :;:l li(l ll , 1(11 ('X :lillpk , li l(' I ()W,~ , '(Jilid h (· p:lllIs Ihr()l l ,l~ il ;1 ne two rk , /1111 fur S(l ll l<' sp:lli:d ('IlIili \:,\ II I~ II leil \, 11 ',v / 1I1 leI 111 111" (l IIIL' wil ;lt "11 /"1 " 111 Ivp,' " i f:dl il' . I " i'l " 'I:dh' , lit ,· . 11'lull. "'"1/h , ('(1111 :11111 11 1',0 11" o r

~

Sema ntics

Mechan ics of spatial data organiza tion

97

96

y

SPATI AL UNIT

NUMBER OF

SPATI AL UNIT

AREA

PEOPLE

LENGTH

(mil es)

\ Road segme nt

43

\ """ "gm,", 2

"

Argentina

Brazil

Road segment 3

Chile

}g

TRAFFIC VOLUME

SUR FACE TYPE

~

• c·

}5,000

conc r ete

18,500

Asphalt

17}50

Asphalt

A· 8

•

I ----'-T ).

Ven ez ue l a

Roa d segment 567

0.6

9.B50

6

\:3

Concrete

( b)

(a)

I:

Po lygon ID

Lin e 10

DESTI NAT ION

New York

London Dakar RIO de JaneJro Cape Town

ORIGIN

3500

New York London

3500

DaKar

3BOO

RIO de JaneJr o Cape Town

7900

3800 2700

oBOO 3 100

2700 5BOO

} IOO

6BOO

4100

Point 10

7900

A B

6800

C

~IOO

0

X 37 67 4.8 1.4

Y

2.S 2.0 5.2

6.5

(2.0, 47) (S.O, 1.9) (10. 2.0) (3.1,2.9)

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( c)

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Figure 3.20 Data matrices. (a) Po int o r area e ntiti es. (b) Line or dyad objects. (c) Dyad ic o rigin-destination matri x.

Figure 3.21 Tables for coordinates. (a) For point entities. (b) For line objects. (c) For po lygo ns.

m o re attributes for object pairs , a rrayed in both ro wS and co lumns, should be ide ntified. In this m atrix one row represe nts conditio ns for places perce ived as o rigins, one column re lates to d estinations, a nd one cell has so me property re la tin g to th e pa ir of places (Figure 3.20c) . Topol ogical properties of co ntiguity a nd connectivity are nea tly repre sented in this form. as demonstrated in secti o n 5.5 , and mathema tical operations appropriate to matrices can be used to derive o ther pieces of

tion conte nt. Aeronautical cha rt s (see Figure 3.11) a ppear to win hand s down for the hig hest density of inform a tion prese nte d in ma p form. Th e table and ma p can. of course, be used fo r containing inform atio n selected or po rtrayed according to o th e r views. Maps of o nl y topological info rmation ca n be prepa re d from data embod ie d in topog raphica l maps, as discussed in more detail in section 5.1 a nd illustrated there in Figure , 5. 1 for a rail tran sit netw ork. Th e reve rse process m ay produce several ' maps' from one se t of topo logical info rmation , because the metric inform at io n has been thrown away. M aps of topo logical spa ti a l properties
info rmation.

3.7.2 Maps M aps have been used as devices for revealing details of shape of e ntities as well as position a nd variations in attributes. A field vicw can be well acco mm oda te d, although some discre ti zat ion is often pract isc d for portray al of the continuol,'s surface , albe it not always errcctivcly. Maps can show clearly objects, simpl e or complex. unitary ()I' compound . but aga in certain cartographical tech n iq lie s arc em ploycd 1'01' erred ivc Il l'SS. Sollie! i Illes th e imagination orC
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99

98 A B

c (a) Icctoooooo polygon model, grid-cell data structure

o (b) polygon model, vector data

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(c)

line model, vector data structure

(a) An example of a Figure 3.22 Data model and data structure. (b) An example of a polygon model with a grid-cell data structure (c) An example of a polygon model with a vector data structure.

vector data structure for a line model.

3.7.3 Data models The map model has limitations and may indeed constrain the perception of space. The map, predominantly oriented to a static, two-dimensional view of the world, is only one intluence on the conception of spatial objects and variations. In a database context of computer based numerical information, the mechanistic organization of data into tables or other forms can be, and generally is, quite different from the map representation. We use the term data model to refer to the mechanistic organization reflective of the logical organization of data, an organization which has a semantic basis. The example of tables for holding coordinates represents a Euclidean geometry view of the world. From a spatial data user point of view, the terms data model, data structure and file structure are often encountered because there are no standard terms across different fields. This is true especially for data model and structure, essentially synonyms for organization, but used differently. The expression data modelling is used for a comprehensive set of conceptual tools for organizing data. These are of general utility and can be formally defined, as examined in Part Two of this book for spatial contexts. They include notions of tessellations and poly lines. The organization itself may rely on a variety of spatial concepts, like spatial interaction or fractal geometry, that facilitate our understanding of spatial reality. The data structure is a more detailed practical description of spatial phenomena, being concerned not so much with entities like polygons or tessellations, but with tools like run-length ene()din).', for savill).', storage space for representing phellOlllell
used as effective devices for revealing data models, the data structure itself focuses on numerical data organization into lists, matrices or diagrams that are aids in addressing the computing issues of storage requirements and performance. The file structure is the other aspect of internal representation within the computer, the particulars of location of data in hardware devices. Data structure and file structure are not usually of concern to the data user, but if an information system has a data structure of a certain type it may foreclose some kinds of activity for the user. Using Figure 3.22, we first contrast two structures for a polygon data model. Phenomena are seen as polygons, yet on the one hand they are created from squares, and on the other from chains of line elements. Both structures allow operations like finding the area of polygons. Otherwise, comparing Figures 3.22b and c, a vector structure is used for both, but on the one hand data are available only for the boundaries of polygons, and on the other for only the interiors.

3.8 PERSONAL SPATIAL SEMANTICS Whether the purpose is to find a better way through a city or til understand better the political conflicts in the Middle East, peopic need to acquire spatial information. The facts may be landmarks or turning points useful for giving directions; they may be images of how places relate to each other. We do not discuss here the process by which people capture information, but we do point to some topics of significance for 'designing and developing spatial information systems: I. The personal spatial reference systems that people seem to use. 2. Spatial language. 3. Perception of objects and relationships among them. Psychologists seem to have established that a child's ability to deal with spatial relationships develops early, perhaps by the age of six or seven years. The ability to work with geometrical concepts seems to come later, although once that facility is developed it appears to be very dominant in sOllle societies. For instance, distances are measured metrically, in kilometres or minutes, andj much attention is given to the precise location oi boundaries oi real pr(~)erty, Cognitive lIIap.~ ;lppe;lr to di.stort sh;lpes but do preserve the relative l()vaiIOILS ;11\(1 propl'l"lics oi ;ldjacl'llcy ;11\(1 cOllnl'ctivity, although the ,'XlVIII I() whicil ;IIlY SP;III;t! Ililorlll:tlioll i.s ;1(il'qll;lIdy ILIIHlled depends on llil' ;1111011111 ;1I1l1 IVI)(' ()I" 111;11 IlIf()I'II1;lli,)II. :\ kll()\.vkd".l' 01" pLlu:.s is oitell

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revealed by drawings from mental maps, although the conclusions about that knowledge are affected by the ability to draw it on paper (Figure 3.23). I t appears , in general, that cognitive maps emphasize topology rather than geometry; are azimuthal rather than Cartesian ; have non linear scales for distances; and are object oriented rather than focused on continuous variations. From this we might conclude that Euclidean geometry frameworks are not necessarily optimal for spatial information systems users , but that people feel more comfortable visualizing relationships among point, line, area or volume objects in different spaces. How much precision is needed in locating objects is more a matter of the purposes in having data in the first place (for example, inventory of real estate), but, for many purposes, topological graphs or highly generalized metric maps may be sufficient for visual operations by users. Also, users may wish to have several maps , at different scales, or one map with a variable scale. Teachers of geography and other spatial fields ' have long used these techniques ill pre.~cllting concepts to students. Certainly , (IHHlgll , dfurls .~Il()lild Ill' 11I;\(I<' 1101 10 IlIl'selll sp;Il;;" fac\.> in misk;td;lll', Li~lli()lIs , ;\., II;I.~ 1);I[llh'II(,
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GOODE Figure 3.24 AlielJl ~lliv(/ views til' !he world. Rectangul<:lr and nonree!;\llglI!;II' projeC!iOl1S will preserve only .'Ollle of the basic proper! ies or 'rill' sh;rpe , :lrC;I, disl;lllcc ;\lld direClioll. (Reprinted wilil Ihe PCIIlIISSIOIl oj Tile ( ';lIlogr:1I1iJic 1lllil , I iniversily or WiSC'OIISill, IVLldisOIl, .'i,'" III, ' r<"lcIC'II<'<" I{,,"illso ll , 1()')() , cill'd in

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direction-preserving map transformations to show political units of the world (Figure 3.24), leading to misperceptions about the relative sizes of countries or continents, in particular, the large size of Greenland, or the shortest way from Oslo to Tokyo. Cultural factors may condition users to particular forms or ranges of language for dealing with objects in space. The Hawaiian Islands residents use a radial system for referring to the location of places. The words 'up' and 'down' may mean different things in varying places, bearing no particular reference to north and south as 'top' and ' bottom', based again on a dominant culture of making maps with the North Pole at the top. The concept of direction embodied in northwest is not simply defined , for it may mean objects in a straight line from an existing point , or it may mean everything above and to the left of a single line running northeast to southwest, or the coordinate system qua drant above and to the left of the vertical and horizontal axes. Some fie lds that deal with spatial phenomena place much emphasis on visual impressions of landscapes - landsca pe a rchitecture. civic design and urban planning. Trave llers through city streets or river rafters on the Colorado River encounter varying scen es, judged personally in terms of a sequence of images, along paths which themselves have many spatial properties . Building styles may clash ; some fe atures may not be seen; others may have unclear boundaries. Space, the world in which we live and experience objects and their form and character, may be seen at different levels: physical, activity based, perceptual and symbolic. In so me specific spatial problem solving tasks , the ' real' entities are th e symbols of a landscape or the mental painting of a series of views of the walls of th e Grand Canyon. A better design of interfaces be twee n pe rson and machine for spa tial information systems requires that we know more about personal re ference systems, how spatial data are processed cognitive ly , and if language has any consistencies across cultures or other subgroups of peo ple.

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Figure 3.25 Differe nt trea tme nts of time . (a) Pheno mena observed at diffe re nt time pe riods. (b) Time of Occurrence of specific events. (c) Modifiable spa tial units.

3.9 SOME OTHER ASPECTS OF SPATIAL INFORMATION While this chapter has emphasized the logical organization of phenom e na in space, le t us refer also to so me othcr matters rc\;ltcd to the characterization of spatial information. They include :

1. The quality of dala. includin g idcntification ;IIHllllc ;l s urcnl,'lil orl'llOr.

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3.9.1 Data quality Error can arise for attrib utes , topological properties and positioning information. Measuring instruments, physical or conceptual, can be biased, leading to in accuracy, that is, the property of not producing the real value. Precision of locational or att ribute measurem ents refers to the fineness of meas urem en t: distances can be measured to the nearest kilometre, metre, centimetre or millimetre, for instan ce. W e hope that we do not disp lace objects from a true location , but this so metimes happe ns. R ecently , for exa mpl e, th e United States Geodetic Office publi shed position offsets for places in th e USA that res ulted from the use of a new geodetic d at um. The record ed locat ion may be not only an in accurate x or y coordinate, but may cause topo logical relation ships to be erron eous. Fo r instance, a point feature (a school , say) may be placed on the wrong side of a line feature (the street) on which it is loca ted, when using only the coordinate information . Such situations may arise from bad original data for either the po int o r lin e feature , but can arise from certain computational procedures in spat ial information systems. The view of spa tial ph enomena can affect the treatment of erro r. In particular, an erro r term can eas ily be visuali zed for the co ntinuo us surface vie w by adding, perhaps via a no rmal statistical distribution concept, a measure for the range of possible values for the positional items, x, y, and z, as ap propri ate. At least in th eo ry the true value is empirically verifiable. Howeve r, fo r th e obj ect-oriented view , error is more co mpli ca ted, as it has to be defined for complex objects, se ts of lin e segments, co ntour lines, etc.

3.9.2 The time element Today commercial spatial informatio n systems do not directly address the matter of time (Figure 3.25). However , time may be included in several ways:

1. As a basis for record ing eve nts or attributes . 2. As an attribute of an entity with unchangin g spatial properties. 3. As a framework for observing changes in the spa tial entities. In the first case, ph enomena or charact eri sti cs arc ohserved at va ryin g points of time, perhaps at regular interval s, as for m()st official slatistical ce nsuses (Figurc :1.25 a). Maps Gin he produccd ;It dillercnt point s ill lim c for a single Ihl' lll e. or for seve ral ph CIIOIIIl'lI ;l : dillcrc ill layers ill ;1

database can represe nt observed co nditi o ns for different time s, for exa mple sa tellite im agery every twenty-eig ht days, or po pul at ion di st ribution at decennial census intervals, or eve n surface conditions of differe nt parts of the earth in differe nt geological time periods. For the seco nd case, d ata may co nsist of the date at which events occurred. Thus, a se t of data can have a changing number of in stances of an entity . For exa mple, as new highway acc ide nts occur, there are additional data r eco rds for these events, and the data table co ntain s an item for the date of Occurrence (Figure 3.25b ) . Time can also appear as an attribute as a duration meas ure. T ab ulation s of dem ographic condition s o ften includ e data fo r population grow th ove r a period of time. For the third item , the focus is o n a changing object. Physical features of the ea rth 's surface and subsurface change ove r time. The con st ruction of new hi ghways changes topological relationships amo ng places. The area of parcels of land changes over time as they are subdivided. Time- space interaction s lik e th ese may necess itate use of specia l data o rganizati o n meth ods, o r representation in different versions in a database. For example, fo r the subdivisio n, attributes ma y be pa sse d from parent plot to new lot ; new boundaries are created hut so me arc 1he sa me as before; and the date of crea tion of the new lo t is an importanl attl~ibute fo r tax purposes. At th e le ast , the qu est io n of time di scret izal ion has to be addressed fo r most phenomena; con tinuo us tim e reco rd s mc very cos tly for mos t purposes . Definitions may va ry at different points in time, if o nly for attributes of cntities whi ch themselves do not change over tim e. Unemployment or pove rty rate definitions are cl assic exa mpl es; th e definition of metropo litan \ areas varies natio nally in tim e, as in the USA, as well as cross-n ati o nally . . The recording units th emse lves Ill ay change in tim e. This Occ urrence of 1I111diliabie areal units is es pecially troublesome for social scienti sts using , sucial, cco nomic and dem ographi c data from natio nal censuses for spatial IIllil s in urban areas . The difficu lty may be overcome by the crea tio n of Cs"lv;d)ic al a lal er 1" ""1 III lilli" 1'\11 Ih, ' ",,, l,-d i ll ;II (" I'(lilll , III "P;I( ',· , Tilll \" d,dll,I'.<'., :11 '" ilion'

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Semantics

difficult to handle via an object orientation, because new objects have to be created , as in the ca se of modifiable cen sus units mentioned above, and spatial relationships may change .

Other aspects of spatial information

107

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3.9.3 Intensional and extensional data As we examine more fully later in this book , spatial phenomen a arc . sometim es thought of in extensional terms but are represented in databases in intensional form , or vice versa. In th e second of th ese, the entity is represented by only a few data compared with what could be used. For exa mple , a river is approximated by a set of numerical parameters instead of by a very large number of vertices. For the intensional state thcn , we deal with a small amount of data explicitly , and store the rules , procedure or method for obtaining other information. For example, a straight line can be constructed from an ordered series of points or may be created via a mathematical formul a (Figure 3.26). In contrast, in a database in which inform ation is presented by word s and numbers , as is often the ca se for non-spatial ph enom ena , for example, a bank transaction or a paragraph of tex t, data are almost always given in an extensional way. III this regard the concept of scale , as embodied in ideas of resolution and precision, has different interpreta tion s. For spatial data the precision of reco rding the positional coordinates may vary depending on purpose, but for data lik e thc amount of an invoice in pounds sterling or the balance of a bank account in Swiss francs every digit is important and cannot be generalized.

3.9.4 General discussion In a pra ctical sense it may be difficult to ascertain the semantics of situation s because of the difference between re al and artificial obj ec ts , and because different users view a single context differently. The material on images (photographs, film s, digital satellite data) is hard to deal with. Aerial photograph s contain shadows that mask detail; maps produced from satellite data may use false colours. Measurements from pictures of landscapes have to re cognize and possibly adjust for camera angle, elevation and di stance. [n addition, ditkrent users have different perspectives, so that th e data required for a p;lrticuiar purr)()se, for example transportation planning, v
<:~

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EXTENS IONAL

A and B and an equation
INTENSIONAL

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resort to other means such as num erical measurements, 1ll;lfht' III:llil ·:t1 cquations , or annotated pictures, drawings or photograph s, NlIIlIL'1 il':11 representations of spati al data are characteri zed by many (,llildiIIOII', : l:ntities are multi-faceted ; entities may be represented by dilkrt'lll ,.P:III :t1 unit types at different scales ; they may have inex act boundaries; tlh:y III:IY difler depending on what data are available; th ey may be reprcsl'll k d Il v rules rather th an by empirical d ata ; they may be modelled as Cllnfill\lllll.'i ,"pace; they may have attributes correlated in space and changing ill lilllC ; ;Inti they may be accessed via both properti es and spatial relationships. It is unlikely that all ways in which spatial entities may be reprcsenfcd PI linked in a study can be stored in a digital or analogue database, Bill Iltnc arc good rea sons , semantic and procedural (as discussed ill ('iI;IPlcrs 4-6), for storing so me fundamental spatial relationships . ( :('IIl'rally there is benefit from storing topology properties, becausc thCSL' III;IV rq)l"cscllt user views as match ed by data organization. In cOlltrast, dl"(:IIIl'C hClwcCIl point places, even though there /llay be mallY, is :1 .1"1 i v,'d ;I!frihutc, not ;1 conceptual appro;lcil. But storage and rc prescnl:I " Il
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Semantics

instances of spatial datil can be related, but, as for map design , to recognize the importance of need or purpose. It is here that we recall earlier comments about data and information. Not only do our varying views of the real world lead to a need for specific data and information; but information itself is a commodity with varying usefulness to individuals or organizations . The issues before the designers of spatial information systems are challenging. How should complex reality be effectively and practically dealt with ? How should uncertainty and ignorance about some phenomena be handled? How should different basic spatial recording units be d efined? We have addressed the conceptual issue of how to organize material at the semantic level. Subsequent chapters develop the the me of data organization and prese nt techniq ues for conceptualizing th e complex real world. Formal descriptions arc needed because the meaning of many terms is not clear, and ambiguity ex ists when some fea ture is not explicitly recorded . Spatial relationships should be able to be derive d in a consistent manner even if not explicitly recorded. Perhaps most importantly , we will be able to reason about and from the formal descriptions of spatial relationships.

Bibliography

109

H su, Po-Siu. [990. An an
J5(1): 2 1-2K. Snyder, John P. 19K7. Map Projections - A Working Manllill. liS (;",,1 "11) , Id Survey Bulletin P-1 395. Washin g ton. DC, USA: US GOVCIIlIIICIII /'llitllIl)' OfAce.

3.10 BIBLIOGRAPY For more det ails on c
or

Tomli n , C. Dana . 1990. Geographic Information Sys/ern.1 1/1It! Modelling. E nglewood ClifFs, Ne w Jerse y. USA: Pre nticc Hall.

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4 Geometries Position, representation, dimensions Theodor Topol landed in the wrong part of the d esert and could not find where he was. Hea ther Rose tried to explain to her twelve year old pupils that Greenland was not bigge r than South Amel·ica. Ha n Norse measured the dista nce from his home to the ski slopes by pedometer. Lee Kim prepared to dig a hole in a fOI'est at a location for which his fri end provided the la titude and longitude. Bette Mounds tried to recreate the map of la nd p,u'cels of the seventeenth ce ntury from knowing th e na mes of neighbouring landowners. John Hudson used a large computer monitor t(l visualize a new skyscraper in New York . Art Santander created visua l models of landscapes using d ragons. Doriana Corona thought she could easily measure the perimelel' of Sicily and its isla nds. Ben Almondbread rc-invented dust , sJlowflakes and trees while I'edro Barrador wonder ed how to sweep dust.

In Pa rt Two of the boo k we turn our atte nti o n to some basi c rrin c iplc s lo r o rga ni za tio n of spa ti a l d a ta and represe nt a tio n o f spatial e ntiti es. To a large exte nt drawing o n d iffe re nt parts of th e fie ld of geo me tr y , th e math e ma ti ca l study of the pro pe rti es a nd re la tio ns of lin es , s urfaces an d .s olids, we p rov ide a fo rmal fo und a ti o n for many prac ti ca l aspects o f .spc,tial info rm a ti o n syste ms desi g n o r e va lu a tion. M atte rs o f pos itionin g , re pre sentation , a nd s pa ti a l relati o ns hip s a re d e alt with in C hap te rs 4 , 5 :,nd 6 , respecti ve ly. C ha pte r 7 prese nts procedure s for m a ni p ul a tin g sll
114

Geometries

Different geometries

115

4.1 DIFFERENT GEOMETRIES

n= w-

In the context of spatial information systems there are needs for dealing with many aspects of space: 1. 2. 3. 4. 5. 6. 7.

H-

Two or three dimensions. Planar or non-planar situations. Continuous or discrete referencing. Isotropic or anisotropic conditions. Forms and distances that change over time. Measurements of qualitative or quantitative properties of space. Smooth or rough objects.

Firstly, though, a few words to set the stage. Spatial indicates pertaining to space, the void in which material elements exist. Geometry is that branch of mathematics which deals with spatial quantities and the shape of spatial forms. Shape (or configuration) refers to the structure of those forms. Form is the mode of arrangement relaled to function, or the appropriateness and effectiveness of purpose. For example, the hexagonal pattern of the bee's honeycomb results from the application of both a principle of economy of subdividing space and a principle of stability of structure achieved by the hexagon shape. Mathematicians have developed many systems and tools for referring to spatial objects, for describing the form of objects , for representing entities and for measuring distances. There are geometries for clearly demarcated entities like buildings, geometries for phenomena that do not have smooth boundaries; geometries for plane or curved surfaces; and geometries for the relationships among entities. Descriptiveg.eoJ:netry ..is the measurement of properties of objects in space or relationships between objects, that enables the true lengths, angles, lines of intersection and other elements to be determined by graphical means. Topological geometry relates to the relationships of components of form. Fractal geometry deals with the fragmentation and non-smoothness of form and the dimensionality of objects. Computational geometry refers to those concepts, principles, and tools used in the numeric, in contrast to the algebraic practice of geometry. Geometry provides many tools, and formalisms for helping us work with spatial concepts. Consider first that material objects have length, width, and height, and can be placed in a framework, a spatial reference system, that allows measurements of those three fundamental properties (Figure 4.1a), although for many purposes people live in ,Ind visualize Ihe world as a 'Flatland' (Ahhott , I<)S2). (icncrally, Ihc Ihrec principal ;IXCS are orthogonal (at riglJt ang.lcs) , hll! ill SOllie ;lppli,';dioIIS 01 sl
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IIIClhods , as might oecurjn lll"niplIl;lIill,I', allrihllte data, entities may he viewed III ohlique co,()'rdill:ill' ,'P:I(',· (!-i",lIrl' ··l.lh), reflecting some ('I'i(il-Iln: 111,,1 ;Ihslra,'! Illl'aSlin's, III(' (·<Jlliv:d'·III.' or lile r, y ,Ind z sp,llial ('"ordill ,II" 11,01 d" l,ll y (II ' SIII(H)lhly d(,III;lr ':11('''. The w:ll<'r I('\'d "I ;1 ~ l':t Villll ". , I' " " " }, 11'.1' I" III\.' II ~ (' oi':l1I ;IVCI:l g"

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or maximum water level. or an expression of the probao ility of occurrence of a boundary value, with its spatial manifestation . The lin e rep rese nting the edge of a (wide) highway may be appropri ate for a viewing position from a long distance, but a walk along this 'fine line ' reveals that it is very irregular at this scale of observation (Figure 4.lc) . The field view of spatial objects has this concept of indeterminancy of boundaries, and the geometry of fractals provides some w ays to repre se nt the unevenness of objects. E arth scientists may not always be able to use th e rigid planar assumptions of solid geometry , and so use coordinate systems with curvature. Of co urse the earth, as a globe, has sphericity, necessitating measures over a curved surface for true indications, for instance, of distance and area. Map-making, analogue or digital. automated or manual , has long been practised with the aid of conversions from curved to plan e surfaces. Map projections were devised to alJow representati o n of the spherical world on flat paper. Materi al objects in the one, two, or three spatial dimen sions have different properties. Shape implies the use of absolute coordinate values as a basis for measuring length , perimeter and subseq uent variation from a regular figure like a circle; but many spatial relationships utilize topological properties of connectivity, contai nment and contiguity. Spatial information systems are often structured o n the basis of exp licit topology , whil e some kinds of representations , for example, map model s, show those properti es implicitly - the human eyc is very good at see ing rel ationships between places and phenomena shown cartographically. Al so , recall from Chapter 3 the discuss ion of field and continuous perspectives, and th e di stinction between isotropic and anisotropic. Th ese situations differ in the way they deal with relationships among objects. Taking just the di stance from one entity to another, perhaps the simplest concept of spatial relationships (but also quite complex, as discussed in section 4.4) , meas urements may be co nstra ined to networks , may arise for all possible points in continuous space, and may be different depending on th e dimensionality of an object.

4.2 POSITIONING OBJECTS IN SPATIAL REFERENCING SYSTEMS Positioning objects, fundamental activities ollields ;IS differellt ;IS geodesy or land surveying and thematic ca rtograph y, rckrrill)', to llie re ;iI world and map model domains, re spectively , invol ve l"ollsi,kr;llioIiS of:

Positioning objects

117

l. The geometric character of the reference system. 2. Measurement metrics. 3. Types of reference system: Cartesian or polar. 4. Nature of the origin. 5. Discrete or continuous reference s. For most purposes, people relate to the length and width dimensions, although occasionally, for some people, the height dimension, associated with mountain climbing and skyscraper elevators can be painflilly obvious. Today many spatial inform ation systems have data for only the {x , y, a}, not the z dimension, reflecting origins in natural resource s mapping and management, th e emulation of map overla y modellillg tec hniqu es, and, originally, difficulties in programming comrutcrs to work with data for three dim ensions. Often the z variable is treatcd as
4.2.1 Continuous space referencing Within the two- or three-dimensional vo id, referencing 1I1:I Y il,' ( ': IIII " .I !III or polar (Figure 4. 2a). In the Cartesian case , the positioll IS l"il"11 I, v ,I linear displacement value, as a pair or tripl et of valucs. III lile p0!;11 " :1''' ' there is again a doublet or tripl et, but one measurc is di st: I1\("l" 1"11>111 ,III origin, and the others are angular measures , azimuths or ile
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Geometries

118 Rectangular

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(Figure 4.2c) . often use latitud e and lo ngitud e to refer to m ap shccts. M o reove r , origins may not be singul ar , ma y no t be st at io nary in spa ce, l :ill ' lil'lciJlI l'. :1 jlosition ill rl';t1i1 y. say 11)(' ( 0 . 0) 1;llilIHic- lull giIIHk IHlilll , 11>11'11 , 11".1 Willi il.~ wilcl'c;ill(lIlh Oil ;t 1I !: '1 I . h ';I IIII'l 's III:I V

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Geometries

120

@) @) (a)

Two air traffic control

reference systems

(c)

offset distances via mileposts

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( d)

Figure 4.3 Different o ri gins for spa ti a l refere nc ing systems . (a ) A zimuthal ma ps . (b) Mov ing origin. (c) E xte rnal a nd local refe re nce sys te ms (d) O ffse t di sta nces .

refer to the location o f o bj ects o r eve nts on hi ghw8 ys as an offset distance from a s pecific origin , p e rhaps th e beginnin g of a hi g hwa y, an inte rsection, o r a nation a l bo undary (Fig ure 4.3d). T he refere ncing may no t be consiste nt with ide ntification of o bj ects. For exa mple, a pi ece o f highw ay with pa rtic ular prope rti es like th e number of la nes or s urfa ce conditio n ca n be d e m a rcated by ho w far fr o m a n origin th e co mbin ed attribute o f lanes and surface ch a nges . H o weve r, a topo logica lly hase d d e finition o f a o ne-dime nsio nal obj ec t w ill not recogni ze that segmentati o n into pieces b ased on the a ttributes . But, alas , s pa ti a l objects mo ve . For o ne e xaillpl e. wc citc th e pos ilioll

Positioning objects

121

o f la nd masses of th e E a rth , perh a ps Pangea o r G o ndw a n a la nd , on e billi o n years ago compare d with now . Th e contine nt a l position s a re quite diffe re nt du e to plate tec to nics . In th e m o dern world the dis ta nce betwee n North Am e rica and Europe increases every year by a fe w ce ntime tres . Amo ng o ther exam p les, Scandin avia's la nd mass is rising slightly , a nd for mu ch o f the wo rld th e re is a stead y se a le ve l rise. Anoth er pro blem is th e exis te nce of earthqu a kes, impl ying th e m o ve ment of pi eces of la nd , and th e refore , of la ndmarks. The qu estion ari ses fo r s uch events , as to wh e th er we con sid e r the la ndmark coo rdin a tes as fi xed , o r mov e d. In othe r wo rds, do we ha ve to chan ge the coo rdin a tes in a spatial inform a tio n syste m ? Partic ul a r e ntiti es ma y have th e ir coordinates o btain e d f ro m a vari e ty o f sources. G round sur veys and m e ns uration ha ve tradition a ll y been use d to establish a co ntrol syste m , or, if it mee ts cert a in condition s of accuracy a nd precision, a geodetic reference system. The contro l system is ge ne ra ll y rega rded a s a se t of physi ca l objects , ca ll e d mo num e nts, put into the ground, and th e d a ta describing their p ositio ns, whi ch in comhin a ti o n, form a bas is for est ablis hing th e position o f e ntities on th e e arth's surl 'lee. The geod e tic system p rov ides a uni ve rsal ba sis fo r what wo ulu otherwi se be separa te m easures. T he net wo rk of cont ro l points is , ge ll e r;lIl y, na tionally b ased for th e la titude and longitude dimension s, although Ilati o nal sys te ms may be ti e d into th e whole glo be. Some tim es the Ill o num e nts can be L1sed by travellers to ascertain th eir position s. For in s tan ce, the prin c ipa l g lobal m e ridi a n (ze ro d eg rees) is cl ea rly mark e d at Greenwich Obser va to ry in L o ndo n , and Swed e n ma inta in s a se t of 1llonume nts marking th e locatio n o f the polar c ircle . Moreove r , the offici a ll y accepted horizontal datum, the s t::tnd a rd global rr;llll cwork fo r esta blishin g la titude and lo ngitud e, m ay chan ge in time. \0 h )r e xample , the recent cha nge in th e U S A from th e North Am e rican J)a tum o f 1927 to a new on e, N o rth Am e ri ca n D a tum o f 1983 , ca use d .shilts o f Lip to 100 m e tres in so m e coordin a tes as a res ult o f goin g to a IlC W sph e ro id to represe nt the gl o be fr o m th e e llipsoid used in 1927 whi ch W: IS d es ig ne d to fit onl y t he shape o f the conte rmino us Unite d States. " .iL'vations or de pths arc establish e d re lative to a n accepte d sea level. So nH.' tillles diflcre nl vertica l hases have to ) )e used. Fo r e xampl e, th e re is :1 il e ig h t dille rell ee betwe en se a levels Oil I he A tl'lIlti c a nd M e dite rra ne an '''HIl es or l:r'lllce . Po sitillll ;1i (\ ,ILI III :IY bC('lbLlilll'd ill Iii ,' li t' ld hy s urvey methods, by l' I)( )Jdilldl e ,l.!,eo lll c ll Y CO lllllllLlli o ll :O- ,II i>y ',;ilt'llik .S III VL'y illg . Tr,lditionall y , ',I lillt' 1':11'111 flo ;llllll'S, l's pl'l' i:tll y 1III1j1l' 1IV IHllllld ;lril 's, wcre lo ca te d hy

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metes and bounds description s. Features were positioned relative to others by distance and bearing , even if at times the base points for measurement were not too fixed, lik e trees. Today, space-borne transmitters, known as the global positioning system, are increasingly used

~L

to pinpoint places on the earth. Reference to position may not be based sole ly on the precision implied by a continuously scaled coordinate ax is, but can also be discrete. Refe rencing may also not necess ar ily involve numerical measures, but can use numerals or words for relativ e positioning. For exa mple , a house could be positioned as being in one county as opposed to another , or referred to by a (supposedly) unique postal add ress, as opposed to being positioned as 56,789 kilometres from the inte rsection of the Equator and

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4.2.2 Referencing for discrete entities Discrete referencing , the mea ns of estab li shin g the location of separate e ntities, other than by coordinates in cont inuou s space, has several possibilities (Muehrcke, 1<)86):

1. Narrative descriptions. 2. Street addresses. 3. Blocks of earth space. In many parts of the eastern United States, and in other countries, legal specifications of property have hi storically been given as a narrative deSCription of boundaries (Figure 4.4a). Descriptions would identify landsc;l\)c objects, and possibly give some indication of distance. IZelcrence is made to supposedly permanent objects like Farmer Green's barn, the large oak tree or the bridge over th e River Exe. Descriptions are as precise as the objects used as landm arks or boundary paths , and as practical only to the extent that these features still ex ist , and are re cognizable, and have not moved over th e years. Mod e rn eq uivalents (Figure 4.4b) use coordinate geometry based direction and distance me asures , avoiding statements like ' follow th e C rook ed Rive r for five mil es'. Street addresses provide an almost unive rsa lly adopted mechanism of locating discrete properties , such as a dwelling or a plot of land. As wilh the offset measures for the /continuous lin car addressing casc, localioll is with reference to some selectcd, local origill . ParticllLlr propntlcs call then be matched to linc ohjccts, hy indicatin g ;Iddless r;"' I)',l'S :lIHI odd-evcn numhers iissociiilcd willi 11i1rlluilar stll'vl sq! IlH'llls (I'Jgllrl'

Figure 4.4 Boundary descriptions and street addresses. (a) Descrip tion by reference to landmarks. (b) Description by compass direction and distance. (c) Street addresses.

4.4c), or even to zones. The United States Census Bureau DIME syste m , developed for the 1970 census , was the first large sca le practical application of using street address referenCing with a topologica l based identification of objects. Street numbers for left and right sides of streets , were inco rporated into the basic set of data for the line segme nts, as well . ;IS th e indica tion of the area units (city blocks) on e ither side of the lines. Stree t addressing systems are, though , not without problems. Difficulties call arise for numbering streets bordered on one side by a river, and when prorer ties are s ubdivided. While data structuring using the topological one-dimensional element IS quite co mmoll today, street address based referencing has its own Illilctical utility because many data, particularly administrative or taxa tion I vpc , arc recorded by street address. /While there may be so me ~liorteoll1illgs in precision of location relative to latitude and longitude, :111<.1 inconsistencies or du[:>lications in namin g st reets or assigning 1lIlIllllcrs, th e stree l ilddrc(;;lng SYStCl1l is wldcly ilcccpted througho ut the Willi\! . 1'lil'1I01lIL'llii Illil',' ;IISI) Ill' n·re·lIl',1 10 11V Ilwir I\IL';ltillll ill picce s of th e ,· :I1I11·s ,~\Ill:w,· (lVrllilps 1I;lIur;1I '·1111111 '" , Ilk,' :11 1 ISl:lIld . or :Irl c filcts lik e 1"ISlill /. ()II\' ~ . Wliili- 111('1 (' :11(' 111.111\ 1,,,,·,. ·11,1. - , ' II,, ' IIi! for ("I('i1lillg :lrcilJ

124

Geometries

units for referencing , worldwide the most common a re: zones of postal services , a dministrativ e units , and statistical reporting units for census purposes . As such, these irregular tiles, exa mples of the s patial discretization mentioned in Chapter 3, are useful for accumulating aggregates by p "rticular attributes , a nd for retrieval purposes using names or other identifie rs for those units. although they may lack positional precISIOn. A regular tiling syste m, that is a s ubdivision of space into areas e qual in shape and s ize, may also be used for reference to places on the earth. Most street atlases provide ta bles of street names and a number or code le tter for th e row and column in which the street can be found. Some countries , for example , Britain and Sweden, have at tim es compiled statistical inform a tion for grid cell units , tied into a geograp hi c reference system by being produced by s ubdivi sion of large kilometre square blocks. Referencing makes use of code numbers combining a row and column identifier ; for exa mple , the British Ordnance Survey headquarte rs in Southampton has th e ad dress of SU 387148, based on the set of latitud e-longitude blocks making up the nation a l grid. Spatial information system d a taba ses created from existing map s hee ts may use the particular sheets for refe re nce purposes or for s implifying manage ment of a large database. The USA provid es two interes tin g exam ples of the practical use of discrete re fe rence units. Part of the co untry has rea l estate ti e d into a planar public land survey system , that is based on rea l geographic coordinates (Figure 4.5a). Lands we re subdivided in the eighteenth ce ntur y by initial reference to a principal m e ridian, and a selected parallel as a baselinc for east to west meas ure ments . Square block s of six miles per s id e were identi fied (Figu re 4.5b), lab e lled as townships, a nd the n hi e rarchically subdivided into thirty-six sections. four quarte r sections, a nci even to sixteen te n acre units within each 160 acre quarter section. Spatial information systems using this basis of land re cording arc in ex istence, and have special features of refe re ncing to the geographic coo rdinates, and making ad justme nts for the fact that the geo me trica lly square description s are in fact not square because the length of a degree of longitude gets smaller with increasing north latitude. In some parts of the USA a checke rboard patter n of public and private oWllership exists within thi s framework (Figure 4.5c). The postal code, one of few universal referencc systems useful for spatial data, is ostensibly 'associa te d with "rc,Ii unit s. In n ;;t1ity. tile hasic unit is th e postal delivery routc , a CO llcCpt ;Issoc i;lt cd with s trcet s . lillc
POSitioning objects

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126

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postcode bre aking point Postcode

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co nd itio ns. Most usuall y th ese a re th e coordinates for th e e nd s of lin es o r th e furth est extents of lin es or po lygo ns in orthogonal x - a nd y -axes, as s ho wn in Figure 4.7. Th e ex te nts esta blished by th e minimum a nd max imum coordinates establi sh th e co rn e rs of boxe s kn ow n as minimum bounding rectangles. Thi s mea ns th at e ntiti es will ofte n be re prese nt e d by two kinds of data - the spe ci fics of th e ir location a nd/o r fo rm , " nd til e lr positi o na l extent. By way o f s umm a ry , we use the te rm locator to re fe r to a ny d ev ice lo r indi ca tin g re lati ve or a bso lu te pos iti o n in som e s pati a l re fe re nce syste m . Loca to rs may be Cartes ia n o r po la r coo rdin ates, distan ces fr o m s pec ifi e d bu t a rbitra ry o rig ins meas ure d a lo ng lin es, through space, in o ne o r mo re dim e nsio ns; they m ay be d isc re te ad dresses , or special loca ti o n a l codes k now n as P e ano and Hilbe rt keys (secti o n 4.7).

-1.3 GLOBAl REFERENCE SYSTEMS /

Th e do ma in of three-dim e nsio na l space is ve ry impo rta nt for the (b)

lievelopme nt o f glo h"l d a tab ases such as is prese ntl y conte mpl ated a nd for e nv iro nm e nta l m a nage me nt a nd rese arch into globa l w: ll'lnin g ;lIld it s impa ct. A di scuss iOIl or SO lll l' "s pecls of g lo bal refere nce 'Y:-'\l: ill S w ill all ow us IH2~ () Id y to 1l1 (' Il\i o ll .~O lllL' inll:rcs tin g practi ca l (k vcl o PIll L' llt s hut a lso to' iIIU Slr;llL' S() lll V ()llilL' 11I1l(\;llllL'lltal ma tters to do willl (' lIr vc d .~ lIrl ; I (, ( ·S. di.~c ll sse d

Figure 4.6 Postal codes. (a) Pos tal codes concept. (b) A port ion of th e postal codes of the city of E indhoven, Thc NClherla nds. (From Van Est and Scheurwater, 19R3 with permis~ioll.)

128

Geo metries

4.3.1 Global referencing The bas ic co ncept of globa l referencing (Figure 4 .2d) co nsists o f using a set of two coordin a te values fo r the pos iti o n on the rea l (c urve d , but fl a t) surface. Latitud e is gene rally m easure d north and so uth from the equ a to r by positive and nega tive valu es as a y-ax is; lon gitud e is measured east o r we st b y positive and nega tive numbe rs o f x fro m an arbitra ry prime meridi a n , usua lly the Gree nw ic h Meridi a n b ut no t necessarily . Indee d, in the 1880s the re was much po litical activit y befo re agreem e nt was reached to use th e line passing thro ugh Lo ndo n , a nd several co untri es tod ay have the ir o wn prim e me ri d ia ns for ma p pin g purposes . C urrently th e re ex ists in space a se t o f satellite based e lectroni c signa l ge ne ra to rs th at ca n be used to ve ry precisel y a nd accurately pinpo int a location o n the carth 's surface. This global positioning system (G PS) develo ped b y the United Sta te s gove rnme nt. allows acc ura te spatial refe re ncin g in continu o us space to a p rec isio n o f just a fcw metres , or a bo ut th e width of a n a ve ragc cit y st ree t , or eve n be tte r a s c lectro ni c e quipment improves . It is poss ible to give every hecta re of th e ea rth a unique address, a ltho ugh it still re ma in s quite im practi ca l to coll ect d a ta fo r the a tt rib utes of such spa ti a l units . Th e re a re several di scre te referen c in g systcms for the wo rld. A grati cule o f g rid lines provides a coordin a te meas urem e nt fra me but can a lso be see n as esta bli shin g a tiling syste m . Howeve r, th e b locks a re no t squ are becau se of th e c urvature o f th e e arth. At th e po le s the wid th is co ll a psed to a po int - the tile s are the n triangles. R e fe re nce may be mad e to blo cks of space re prese ntin g co untries, a n irregul a r tiling; m ap sheets or pol ygo ns c re ated by two bo undin g lin es of longitud e a nd la titud e, almost pe rfec tly reg ul a r tilings ; or b y the use of specia l regular geom e tri cal fo rm s suit ab le fo r partitio nin g curved surfaces, as we de mo nstrate in secti o n 4. 3 .3.

4.3.2 Map projections World m aps in a n atlas prov ide a spati a l refe re ncing syste m , but the sam e pi eces of the ea rth ca n loo k differe n t acco rdin g to thc m a p pro jecti o n used fo r the m ap ( Rob inson et a l. 1984). A map pro jecti o n is ,I d cv ice fo r producing all o r pa rt of a sph e re o n a flat shu; !. S()me rroj cc ti o ns prese rve dista nce o r true 'direct io n , so m e maillt i lill c() rrect~ h a pcs .•lIld others prese rve the pro pe rt y of a re,\I si ze . M ilp projectiolls C lil he studi e d from the po int of view () f: 1. Th e form o f th e slIrl';I('l' use d I'() r tlIV PlOll T ti o ll

Global reference sys tems

129

2 . The particul ar view in g origin a nd th e standard po ints and lines used . 3. Spa ti a l prope rti es, pre served o r di sto rted. 4 . The numb e r of po ints use d to tra nsform fro m a sphere to a fl a t surface . 5. Th e fo rmul ae used in ma ppin g from a sphe re to a non curved surface. In m a nne r o f con structio n projecti o ns have been class ifie d first of aJl based o n th e fo rm of th e pie ce of pa pe r th eo retica ll y w ra ppe d around o r o the rwise touchin g th e globe, prin cipa ll y the three types of cyl inde r , co ne, a nd disk (Figure 4.8). Tn the third of th ese d evelo pable surfaces, ass uming to uching, th e chose n g ra ticule and eart h fe atures a re m apped from th e sph e re to a disk ta nge ntial to th e globe a t one po int , possibly a pole as in the illustration (Figure 4.8a), o r a n y o the r point suita ble as the o ri gin. Th e cy linde r o bj ect m ay be w ra pped a ro und the ea rth in diffe re nt loca ti o ns, but to uc hes o nly along a c ircl e (Figure 4 .8b ); while the co ne is ta nge nti a l to th e surface a long th e pat h o f one circl e (Fig ure 4.8c). Tn a ll ca.,es the re is a sin gle line o f po int s (or a sin g le po int in th e ca se of th e azimuth a l s urface) for whic h coo rdinat e va lu es a re corrcc t to sca le , as s uggeste d by th e small circles in the figure. Th e pro jectio n surfaces ma y a lso cut th e ea rth surfa ce ; s uch secant projections will have two principa l axes , as s how n fo r the coni c pro ject io n, the reby re ducing a ve rage distortions co mpared to the ta nge nti a l form. T he secant type is mo re common for co ni c rath e r th a n cylindrical o r azimuthal proj ecti o ns. The de ve lo pab le surface may a lso be oriented in d iffe rent wa ys , per ha ps bein g a l'igne d with p a ra ll e ls or meridian s, o r possibly be in g o blique to th e m , as shown. Cy lindrical pro je cti o ns that are ori e nte d to meridi a ns rath e r th a n pa ra ll e ls are kn o wn as tra nsve rse projectio ns. Diffe rent proj ecti o n origins m ay be used. Illustra ted only fo r pro j ecting to a disk (Fi gure 4 .8d), we ca n im agin e a base o f in fi nity, re presente d by pa rallel pro jecti o n lines, or a n o ri gin at the po le o ppos ite to the ce ntre of the azimutha l p ro je ction, o r loca te d at th e ce ntre o f the g lobe . The particul a r po in t of o rig in will cause the g ra ti cul e o f equ a l degree in c re ments fo r lat itude to be re prese nted by diffe re nt spacin g, as show n . Fo r these a zimu t ha l proje cti o ns (tho se with a co mm o n ce ntre ) , pa ra llels may bc space d accordin g to di Ifere nt co nd itions. (Me ridi a ns a re always radi a ting straig~-t/ lin es.) II the line ar scale is unifo rm the projection is c quidistant ; ir c qll a l ;lrl' ;1 is pr e.~c r vcd th e n th e spacin g be tw cc n succcss ive p a r a llcl.~ or ('li!1iI1 d(,·.I',IlTS iJlLTt'lll c n t will be sli g htl y Icss. Th e urthogr ;lplli ( h ; l.~ ;1 .~illlil : 11 pa llt'lll , lilll Ih l' di s l;\llCC betwee n sli l-ccss ivl' liJ\l'.~ diJ\liJli slw\ JlI (II(' I :1I'1"1 \, Ih :111 II I Ih,' ";1',(' (I I' 111l' l'<Jllid ;lrl'; 1 1YP('

Geometries

130

Global reference systems

rf~ ~

Pla na r projections may be furth e r classified as to the pa rti cul a r properties prese rved. No t a ll properties ca n re m a in undisturbed in a si ngle pro jecti o n. Some guarantee o ne factor; o thers are compromises. The ex iste nce of man y p a rti cul ar projections is no t surprising , g ive n th e four different properties , distance, direction , shape , and area , th a t may be preserved, and their ma n y uses. E ngineers, meteorol ogists a nd nav iga to rs are particularly int erested in distances and comp ass directions; eco logis ts , geologists and geographers often need area preserv in g m aps. Tangential

&~.'" The globe contad s th e cone along one (tangential) or

~'C.

Only one point of the globe

touches the disk. Spacing of

parallels depends on the

origin for the projection .

-
(a)

&

Oblique polar

Oblique

Secant

PROJECTION ONTO A DISK

(b)

PROJECTION ONTO A CONE

Regular

Transverse

Cylinder .A- touche s along a line

-til Equal spacing lor gralicule points. Spacing of meridians depends on the origin for the projection

. (c)

polar

Equal spacing lor the graticule poinls. Spacing of parallels depends on the origin for the projection

Oblique

PROJECTION ONTO A CYLINDER

•

131

No dl6:tortion

OrthoQraphic

I

T

I

~-..-~

Stereographic

(d)

(e)

Distortion increases away from the poinl(s) of no distortion

Figure 4.8 M ap projections. (a) Cy linder developable surface. (b) Disk for azimuthal projection s. (c) Co ne deve lo pab le surface. (d) Origin s for projection to a pl a ne. (e) Concep t o f distortion.

Choices are not always easy; compro mi ses may be necessary , a nd mi slead in g impress ions may result. Recently, professional societies of ca rtograp he rs a nd geog raphers in th e USA adop ted a formal resoluti on urg ing a n e nd to the popular use of rectang ul a r maps like Mercator, Gall, Miller a nd Peters, prefe rring other projections in o rder to show th e ear th as being round , to minimize seve re size distortions of large areas or th e earth (Figure 3.23) , and to avo id represe nting most distances and direc t routes incorrectly.

4.3.3 Some examples of global systems The mo re refe rence points used for the transformation (Figure 4.8e) t he closer is the re presentation to th e rea l s urface. The representa ti o n is mo re accurate th e closer to th e refe re nce points, so that the selectio n of particu la r (standard ) parallels a nd me ridi a ns for certain kind s of proj ectio n is important. Some transfo rm ati o ns project onto a single shee t of paper with a rectilinear graticule ; o the r projections allow curved lin es. In a ny eve nt , only a few po ints are true; ot he rs are interpolated, being defi c ie nt in so me regard as to sca le. In practice thi s means th a t features , IIC rep rese nted hy some selected known points and rules (the tll ;tth e l11ati cal equations) to es tabli sh the coordinates for other points. In ;1 s p;t !ial rcierence systcllvscnsc. tll crciu re. we h;t vc inten sion a l data, a / p()in! !o which we rcturn l;t!n, III ()Idl'l I() jl ['()vide SOIl)l' SI:IIl(i;lr
132

Geometries

Global reference systems

133

poles and it divides the carth into sixty longitudinal six-degree zones, covering the earth from north pole to south pole and having about 670 kilometres width at the Equator (Figure 4.9a). A zone six degrees wide is usually divided into 10 kilometre sections for reference purposes. Points are positioned on a northing scale measured continuously from zero at the equator; and with reference to a zone central meridian with an easting of 500,000 metres. Large countries such as the USA have many zones but these provide advantages of the reduction in distortions that the relatively narrow bonds afford longitudinally. Distortion is absent along two meridians, slightly west and cast of the central one, which has a scale set at 0.996 of those lines. However, the developed surface of several transverse zones will have gaps, causing difficulties in working across zones. Although not global in coverage, the USA uses a system of planar rectangular coordinates, the State Plane Coordinate System, to reduce distortions in mapping an area of continental proportions. Based on either the Lambert Conic or Transverse Mercator projection (Figurc 4.9b), it is similar in intent to the Universal Transverse Mercator but has a differently arranged graticule and measurements in fect not metres. Originally devised for permanent recording of the location of original land use survey monuments, it is now used extensivcly for state mapping purposes. It has 120 zones and a distortion error of no more than I foot in 10,000. Both projections used for this system are conformal, that is, they preserve the shape of (small) areas; and provide for an easy interchange of coordinates between grid and geographic reference systems. Other countries have similar national grids but do not usually need as many zones as the USA or other large countries. Dcvcloping a standard discrete georeferencing system is a little more challcnging because of the properties of curved spaces. While irregular shaped units can be used , and often are (for example, countries), there is interest in a system of regularly-shaped tiles spanning the whole globe. A set of cubes, or octets, may be useful for architects and structural engineers, and, as we see later, for numerous applications in spatial information systems , but this regular geometric figure , an example of a regular tessellation, is not so good for partitioning the surface of a sphere. Geometric structure is apparent in the three-dimensional world stacking of oranges in a tetrahedral cluster, patches of black and whitc on footballs or beach balls, or faceted geodesic domes. The faceted sphcrical surface of a football (not American!) or baskctb,tli consists or mostly hexagonal patches, but also pcntagonal p,ltchcs set ,It predctermincd positions. It can be shown mathematically that oilly liexagolls, or thcir equivalent, tri,lng\cs, will Ilot work ir cllrvnl lilll'S ;Irl' Ilot ;J1lowul ill tlil'

Zones of the Universal Transverse

Mercator (UTM) graticule in the USA

(a)

Zones of the USA State

Plane Coordinate System

(Sixly north-south strips, each six degrees wide)

(b )

(Based on secant conic prOjections, with diHerent Contact parallels)

r:

N

80 0 0

60 40

0

20 0

.. '

......

Im'T I 1 20

10S

•••••••••

1000 kilometre o

square 200,000 m N I

.'

~

I

"umN

,I

,.---,--- ....,

I

I

(I

I I

Equator

I

0

I \ \

300,000 m E

(c)

woo 108 102

500,000 mE

800,000

mE

Portion of graticule with

metre coordinates

FigUJ'e 4,9 Reference systems for global extents. (a) Universal Transverse Mercator. (b) USA State Plane Coordinate System. (c) Detail for the UTM graticule.

graticulc or wireframc model. A set of triangles initiated from an octahedron Illodel for thc/ carth /t;IS rl'cl'ntly been proposed (Dutton, II.JR9; (;oo
I. ILlsi,' ttJlih .)

;tilliost nJilit! III

The fundamental elemen t of distan ce

Geometries

134

135

4.4 THE FUNDAMENTAL ElEMENT OF DISTANCE Di sta nce, or the measurement of le ngth, as enco unte re d in dete rmining relative positions o n the ground or attributing size propert ies to spatial objects, is a building block so important in spatial information systems that it deserves some extra treatmen t. There are many as pects to distance : Metr ic for measurement

Type of geometry

Plan ar or spherical

Simple or composite

Singular or accumulated

[sotropic or anisotropic

Dista nce in a graph

Number of dimensions

Preci sion and acc uracy

Distance (D) = A + 8 + C+D+E+F

Da = Db + Dc

Da=VDb+ Dc

~

~r D

I

)<

/

r

/

T ··

J;7

, ) i/ O J --~.,. ~.'!:

--- f

1"

.,.

,.-4

J

I

(a)

I

,1

D c

~ -,

A Street' t--

£b

U

r----x

Yl

I

X2

1/ 12

B

'

_1 _____

,

c

I

1

V

-

I

~

£t3p-

'--

(b)

lIanhallan

Euclidean Slope faclor

q,

Network

Ground dislance

Figure 4.10 A quate rnary triangular mesh for a globe. The Level 4 syste m of triangles viewed orthogonally from over the Pacific Ocean. (Reprinted from Goodchild and Shiren with the permission of the National Ce nter for Geographic Info rmation and Analysis, Santa Barbara, California, USA.) Correct Rio de Janeiro

3. A set of units for which tru e adjacencies for neighbouring elements are

(d)

~scaIe

(f)

preserve d. Howeve r, the shape and size preserving properties cannot he Illc l wi [hout compromise at one time in /a ny proj ection fWIll a curved to pl
I

(,.. ",/

Ground

Spherical

Incorrecl scale

Map

projection

FiglllT 4,11 Ililll' ll' lIl 1">lI""ph ,d ill '.(: III' T (. 1) ' · IIl'li ,il': 1I1. (I,) T:lxi,':d •. (,') N('IWIllk . (.II ( ; 1'>111101 (\1'.1.111, I ( \ ' ) SI·dll'l i, ';d. (1') 1'1"1,', '111"' .

136

Geometries

Di st ances in planar C arte si a n coordina te syste ms use the data for coordinate pa irs, o ften co mpu tin g th e stra ight line distance as the le ngth of a diagonal of a right angled trian g le (Figure 4 .11 a). Howeve r, there are o the r possi bilitie s, o f which the most intriguingl y na m e d is the Manhattan , taxicab o r c ity block distance (Figure 4.11b). Thi s computes distance by summing the le ngths of the sides o f th e triangle. Algebra ically, it is based on absolute distances rather than on the square d distances for the stra ight lin e be twee n two points based o n the Pythagorean theorem. Th e distance measures may be m a de over a s urface or ma y be computed for th e a ss um e d real cha nn e ls o f move m e nt. Real network distances or driving routes (Figure 4 . 11c) a re a lmost a lways greate r than the dire ct s tra ight line distance. A continuou s fi e ld model for s patial e ntiti es facilitates o bta ining direct po int to point distances , but will underestimate th e reality by ignorin g channels for and barriers to move me nt. Discrepancies between direct distance and 'real ' dist a nce will tend to be higher if ne tworks for mo ve ment are sparse or irregula r , but e ffective distances should also co ns ider the re la tive costs of move m e nt on netwo rks (o f diffe rent kind s) as opposed to simple ove rland journeys. D e ter minati o n o f accurate non-planar distances req uires e leva tion coordinates so that m eas ures may recognize gradients of the earth's surface . On-gro und di stances will also be underestimate d if the te rrain conditions are no t consid e red (Figure 4 .1ld). And , as noted ea rli e r , false impressions or data will be obtained for global dista nces if w ron g map pro jec tions are used (Figure 4.1 J e a nd f). In the human wo rld th e effort to mo ve over th e earth's surface is often tho u ght o f in te rm s of cost, e ith er in time o r mon ey. Transportation planners , urba n geographers , a nd others have d e vel o ped m a ny m eas ures of spatial impedance , and cartographers have devised various teChniques to re prese nt the re lati ve positions o f pl aces in diffe rent m e tri cs. Ind eed unless a s ingle co mposite arithmetic m eas ure is designed for combining diffe rent metrics , it is diffi c ult to de a l with the inh e re nt conflict of pl ace ment of points according to different m eas ures. Sometimes dis ta nce is dea lt with as proximity zones, ge ne rally for simplicity assuming an azimuthal base (Figure 4.12a). For ex ampl e, dista nce ba nd s ma y be c reated for seve ral market centres to show the re lative ease for a farmer to get to e ac h. A comparison of values for each di sta nce zone a t a given po in t m ay be achieved by overlaying the differe nt se ts of cirel es. Accumulations o f distan.ce from o ne point to reich
The fundamental element of distance

137

Distance A 10 B = 60

Distance A 10 C = 40

(a)

Proximity bands

on an isotropic

surface

(b)

•

Estuary without bridge crossing

(c)

Distances aHected by anisotropic conditions

Travel time isolines from central point in an urban area

(d)

Highest cost

0

Lowest cost

Minimum cost path over terrain divided into travel cost zones.

Figure 4.12 Pro ximity a nd accessibility. (a) Proximity zo nes. (h) Travel time isolines. (c) Distance affected by barriers. (d) Terra in 11Iinillluill cost p at h distance.

:ll'l'llllIlIlate d mileages throu gh highway netwo rks a t different points in lilill' ,
Geometries

138 6

2000 miles 1000 miles

y

500 miles ~

5

4

3

2

circles centred at Baltimore; based on a togarithmic scated azimuthat projection.

1 _I

•

/

'f

t

58 cities with highest volume of air passenger Iralfic, 1975

Unit Taper in air ) \ s t Ira vel cost as I distance increases I

Rectangular arithmetic scaled system for referring to places by XY coordinales.

I

o -11-----,------,--- o

2

3

4

x

~-------f

Distance

Figure 4.13 Logarithmic azimuthal representation of relative dis tances from Baltimore, Maryland. The symbols indicate urban areas with substantial air traffic volumes with Baltimore, Maryland. The Cartesian grid is overlaid on a map base of an azimuthal logarithmic scaled graticule centred on Baltimore. The rcctangular grid is used to obtain Cartesian coordinates for mapping purposes. (Based on material originally appearing in Thompson and Murphy, 1975, cited in Chapter 16.)

penetratable barriers, movement stops completely, as at a road construction barrier. or at railroad tracks, or, in some cases, at national frontiers. Permeable barriers allow some movement, perhaps representing a real condition of limits to the number of vehicles able to pass a particular point or journey ,on a selected route, or via an extra cost added to the numbers representing accessibility. They lherdore arc similar to aggregations of several weighted segments, each associatcd with ,I chuilk of terrain of different spatial impedance (hgul'e '1.12d).

Coordinates and splines: line representation

139

Anisotropic conditions require different techniques for expressions of the relative distances separating places. For example, the declining cost of per unit travel rates as distance increases, well known in transportation studies as long haul economies, may require a logarithmic or non-linear scaling for distances, as shown for a map of major air travel destinations lor originations in Baltimore (Figure 4.13). The map uses a logarithmic ,Ii'jmuthal grid centred on Baltimore-Washington International Airport. For many purposes, it is desirable to indicate the separation of places ;IS cost, effort, time or personal travel, or other measures. Indeed, several scales for relative distance may be needed; mental maps often use sOlllething like a logarithmic scaling. An equally difficult problem, Iilough, is when we deal with more than one origin for measurements. Tile polar map is fine for Baltimore, one place, but, imagine a map ~1t()willg relative distances among a set of places. Special mapping I('l'itniqucs are necessary, as shown by the sets of isolines in Figure 4.14 1(" distances from three places at one time. It can be shown Illalilelllatically that some distance representations cannot be achieved in 111:1\) I()rm; we must then resort to dyadic matrices as shown. Tltl' underlying geometry and the unevenness of the world distort 111,' :I SlI rcs or distance. Accessibility is inherently a multi-faceted concept. I )\ ~ I:lllces measured from maps, or digital data obtained from planar IC'f Il "\('lltations without corrections, can be longer than in reality, d"lll'lldillg on the map projection underlying the map. Sometimes, as in II,. ,'x;lIlIpk of walking to school, the temporal distance is most [I I'I'I(lllri;llc. While a Euclidean base is normally assumed for measuring .I"I:III(,·S, ;11 times other geometries may be more appropriate. A ~ LIIIIt;lrl;IIl' or city block distance is based on absolute distances, not the ' 1 '1 Ii (Ii lite sum of the squares for a right angled triangle. Sometimes r/ ",( : II\( ' L' III;IY he conveniently measured topologically, not using coor dill ,ll n :\1 ;111; and sometimes we have complicated multi-faceted relative dl',I,II\('(", wirlt which to deal.

I t, (

()OIWINATES AND SPLINES: THE REPRESENTATION

()I

I I N IS

N II I tlld \' ,11(' tilnL' IIlICVCSlill,l'. (j1l('sli(lIIS ;till)1I1 distance, but there are 1'1 ,11 1,. ,iI 1"."II('S ;ti)()I1( (ite i'q)J"('SCllf:lli(l11 (Ii' (111(" ,Il\(l two-dimensional I".t 1,, 1 • 1,'II\('llh SI1('1t :IS:

II,, ' ,'III()lIllr (II i,.(,rt,·"tlii':lli,1(1 III II,,' I. \l1 \"'\'III:lli(1I1 or lilll'S. I II!' 111.1111<'11 11, III,: tI , I, I( III (Ii rII,' IJ I'I I I, " III ~ II ItI" I, l\ IiII (..,

Geometries

140

(a)

---.

---

Roa:!s Direcl dis lan ce

250 _-- . -

-

,...

Isochrones (se lecled) Irorn 3 origins

2

FROM TO

I

1-:,

Direct distance in kilometres

,:250

, 230',

..

(c)

~

~----

1"

Travel time in hours

V

--~

lD

/370 /

60~

3 435 370

3 ,/

'-

? 2"

1

~ :--

2

2

/

504

""

230

320

/

'fJ

250

/

FROM lD

/

2 504

2

"-J

Relati ve

Caricature via ten straight

segmellls

Rough approx imation via live segments

Even more general - using a straight line

\. Th e nee d for minimizing the amount of data. I. Th e amount of regularity in the entity. The accuracy and precision of repres entati o n. .')illlply put, an irregular line j o ining fix ed and known e nd points, can he : 1 '~ III1ICd 10 be straight , as co mpl ete as the o ri ginal data , o r represente d h \ sO lll e co ndition betwee n these two ex tremes (Figure 4.15). The l l" dlJ("1 ion in the numbe r of pieces of info rmation needed to be ab le to ,lI:l W Ih e lin e can be effected in different ways, systematic or ot herwise. ()I sl'v LTal aspects of generalization, w e are here concern ed with ',IIII I(llhing ,m d line thinning.

,I ,;: . I I inl;' simplification 8:50

Ab so lute position

o

Points tor a Iree'orm caricalure

6:09

,/

/

Two joined Bezier curve segmenls

3

3

8 :50

O rigin a l lor m

141

Figure 4.15 Representation of a line.

250

FROM

2'

/

""

(d)

3

2

320-----'

" 435"

2

3

?

Ro ad distance

6 Travel lime in hours

3

l'

(e)

(b)

Coordinates and splines: line representation

position

Figure 4.14 Multi-origin accessibility. (a) Geodetic positions of three places . (b) Isochrones ' for trave l time from three places. (c) Direct planar distances. (d) Road trave l distances. (e) Tra vel lim es .

( , "II1 I ~ f': lph c rs orten use procedures of line simplification that systemat ic .ill l' I'l illlill;rlc ccrlain points from th ose o ri gin all y recorded o r cap tured 1.11 .1 lil1 l.': lf' icalure. Ideall y, a represe ntat io n sho uld both prese rve the 11 1:1111 1"11 '111(' 111 or shape , and recogni ze topological properties. That is to .11 1111 :1 sci o r houndary lin es, as ill th e c;rsc of the countri es of So uth \ "11 111: 1 .,;flown in l :iglll"c/ l.ll(', it is illlp()lt;lnl to usc the topological lilli' II,IIIS ( llnl' ill[CrSl'cti'Z;'l ls) :IS illvi(Il:II(' IHIIII[s. Thcll thc undul ations 1111 1 1I111 1' IIi:lli()11S ill liI(' sil;l(li' (Ii" [llt ' lill t' Ilt'lw,','11 Ihe [o[1ologieal end 111 11 111' , III :IV 1)(" :till'rl:ti (kpl'llIlilll'. ,III 1111 \1' 111111'11 01 Ill,' origin," shape i s Il tlJHIII,ill l 1(11 :1 1',iv('11 plllp(l ~ " , M !I(' II 'H ' I '. II II IV ,,"1 ,1 II It Ill' ("lllllrOr(
Geo m e tries

142

Coordinates and splines: line representation

143 1

Second example Successive three point moving windows (assuming no point is removed)

2J d (al

~_

~

First

(a)

Floating

, , ,

./

'<.',

>W<

point

./

,

./

),

./

./

./

./

B

',/

AneIlor

point ,

./

Threshold

\...... (tolerance) distance

.

'

~~ A

B eliminated Final form for line AB

o A Successive tests for

perpendicular distances

(c)

0 I

7*°I

6

5

2

4

3

Coding used for eight direction system

Minimum

tangle

largest

2 4 4 12 1 4 7 44 5 76 6 6

22 I

Figure 4.17 Cha in e ncoding fo r lines. (a) O riginal lines (fo r exa mple, portio ns of conto urs). (b) Straight segme nts fo r eight directio n e ncoding. (c) Nume ric codes.

Minimum distance

"'"I ~

(b )

0 I

o

(b)

Figure 4.16 Two diffe rent procedures for line ge ne ra tio n. (a) Three point movin g windo w (a fter Jenks via McMas te r, 1987). (b) The Douglas- Peucke r algo rithm (Douglas and Pe ucke r). that the Florid a peninsul a is not remo ved) , even if th ey are no t necessa ry for a given specific analytica l purpose . The simplest p rocedures f o r line gener alizatio n co nsist o f retainin g only a fr actio n of poi nts at syst em ati c int c r va ls: fo r cxa mple, cvc ry hfth point i s k ept and all ot hers disca rd ed. Thi s proce dllrc docs 110 1 guarant ce shape prese r va ti o n. O ut o f th c ve r y large 11l1l1l lle r o r p roce dllre s thai Iry

1(1 p rese r ve shape, especiall y th e maj o r turning po ints, we illustrate just I W(l to de mo nstrate ce rt ain bas ic pr in ci ples. l i irst of all , fo r an alrea dy se lected line feat ure, we can simplify by I "I IHlv in g po in ts th at are t oo close to neighbo urs o r have a small angle o f ',(' para tio n be twee n vecto rs ( M cM aste r , 1987). E ith er o f these co nditio ns 1II:IV hc exa min ed se parately , o r th ey m ay be co m bined in o ne proce dure ( I i)' ll re 4. 16a). In this process a three po int mov in g w ind ow ca n be ,Ipp l il'd to ,I line f ro m a to po logica l (o r ar bitrary) o ri gin . Th c first thrce \ l' l l il'l.' s L'lu ght in the windo w are exa mined wi th rega rd to two crit c ri a 11' lg1 1i C 11.1 6a). O ne is the di st ance of a ve rtex f ro m th e previo us ve rtex Ii l iJ ;1I di stance is too short then th e seco nd po int ca n be removed . Th e "'; \, 111)(1 cr il eri " relates to the exi st ence of turnin g po ints. If the angle 111 1Wl'l' ll " pro jected line joining th e first tw o po ints and th e lin e fro m .1 '( (liid III lhird is sm clll , then th e third po int ca n be di sca rd ed . T his ( >J II "(' , "S ('o lilinu es fo r eve ry succee din g se t of three po ints in the w ind.ow. 1',· tll ;'1 lS th e mos t we ll known proce dure (Do uglas and Peucker , 1973) , 11·.•·11 ill seve r;" co mm ercial soft wa re syste ms, uses a slightl y differen t .1 1' 1'1 11;1\ 11 10 rCl
Geometries

144

Coordinates and splines: line representation

or o ther regular divisions of a circl e, as in a compass rose , line features can be indicated by a se ri es of num erica l codes showi ng th at general direction of th e line. Assuming eight directions , a se ri es of 0 values would represent a line going from so uth to no rth , and a mixture of the values two and four would represent a sta ircase goin g towards the bottom-right

•

co rner of the map. M o re precise cod in g is possible with more directions. Actu al coordinates are needed for on ly one point , ei th er the beginning of a polyline or an arbitrary point for a closed lin e; and spec ial techniques ca n be used for junctions. However , it is more difficult to co mpare or to apply alteratio ns to lines genera li zed by this incremental type of encodin g th an it is to Jin es represented by coord in ates.

•

1

0.00.0000.0

:o •

...

•

• Original points (,I)

0----..-' •,•

•

•o 8

I

"

'l, %~ 00

--'-"e

,, (b) _ _

Path of curve Junction of two curves

o (e)

000 •

Points with

interpolated

values

Figure 4,19 Interpolati o n of points o n a lin e. (a) Original data. (b) Cubic polynomial fitted to a success io n of four points. (e) Draw ing a curve using very short straight segmen ts be tw een points interpo lated.

With the sa me object ive of red ucing the number of pieces of informatio n, lin es may be rep rese nt ed as a sin gle curve , or as a comb in atio n of smooth curves. Essenti all y, a smooth curve is fitted, accord ing to certain cr iteri a, to a shape for an irregul ar polyline o r polygon represented by a set of specific points, ge nera ll y passing through all or possibly just a sel ectio n of th e specified points (Figure 4.18a). In th e case of a lin e feature, th e curve may cross interm ed iate segmen ts on its route f rom one end to th e other. In the case of a polygon, separa te curve pieces genera ll y are fitt ed to the distinct lin e segments, such as th e fiv e for th e po lygon in Figure 4.18b. Once this is clone, th e coo rdin ates for interm ed iate points can be es lim ated by interp o lation , as shown in Figure 4.19. Of course , if only poinl data arc availclble, then th e sa me process can be used to fit a curve itS op posed to genera li zi ng o ne.

Impo rtant cr iteria th at may be es tab lished for an id ea l curv e arc: I . II sho uld pass thro ugh specific end (o r mid) points. I I sho uld be tangenti al to o nl y o ne point o n an individu allinc scgmc nt, or cross at on ly o ne point. \, II sho uld have no discontinuity where sepa rate curves join whcn 11Llking up a se ri es to represent a co mpl ex polyline or irreg ul ar po lygo n . I. II should have co mpac t mathematical rep rese ntation. I .ill e smoot hin g may use statistical ave rag in g techniqu es o r math e 111 :llic:iI cqu iv,li ellts to the French cur ve, the flexible drafting too l used by 1" :Iphi c :Irlisl s and cngin eers. This spline , made of thin flexib le pl astic , 1.1 11 ( wilh praelicc) be made to fit specifi ed key points. If th e line is " ' I', e
Po l y l ine smoot hi ng

•

•

4.5.2 Smoothed lines

(a )

• ,,

• •

145

( I») I".lIY 'l
Figure 4.18 Line smoo thin g. (a) For a polylinl'. (h) hll'

<J

pol y)!' \I\.

Geometries

146 y

y

y Y = a

a

Y

+

bX

+

a

bX

+

+

cX2 + dX 3

bX

x =

y

x

x

Lin ear model

X

Quadratic model

Cubic model

Cc )

Cb)

(a)

Y

=

x(t )

A

IX2

+ (1 - t )x J

tY2

+ (1 -

=

a

Y (t) = e

/ / / /

,,

/

+ bt +

el 2

+ft + gt

2

+ dt 3

+

hI "

II el e va ti o n , z, is a lso needed :

/ /

,,

z (t )

/ /

,,

,,

///

,

rotat e 0 axes 90

X

=p +

qt

+ rt 2 +

S{ 3

whe re th e pa ra me tric va ri a bl e varies be twee n 0 a nd I , As willi 11t" X'

Three Y values for a given X

Cd)

t )YI

If we se t t = 0, th e n we ge t X I , YI ; and fo r t = 1 we o btain X2, Y2. If we in c rease t fro m 0 thro ugh 1 the n we ge nera te all po ints fo r th e line segment j o ining th e po int X I , YI to X2 , 12. Th e som ewh a t mo re co mp lex case o f pa ra me trica lly crea tin g short arc le ngth s for creating a circl e is ShOW ll as Fig ure 4.2 1b. The in c re me ntal c ha nges ca n be dete rmin ed by equ a ti o ns in this case, but no t every Cu rve has a simpl e ge nera ting algorithm . Th e re is, th o ug h . d famil y of curves, includin g B -s plin e, B ezier a nd H e rmitia n splin e th a t have a basic ma th e ma tica l fo rm o f

Y'

,,

147

conn ecting two p o ints (Figure 4.21a), we can use th e ex pressio ns:

CX2

Y= +

Coordinates and splines: line representation

Y

No multiple Y values

Ce)

(X I Y I )

Y 2

Figure 4.20 Polyno mi al fun ction s, (a) Linear. (b) Q uadrati c. (c) C ubic, (d) Problem of seve ral y va lues, (e) Ro tati o n o f axes. Y

coe f(-i eie nts o f inte rce pt a nd s lo pe by least squ a res sta ti stica l procedures. A curve with o nly four num e rical coeffi ci e nts ass ures a p e rfec t fit. Th e po lyno mi al fun ctio ns are al so used fo r ge ne ra lizin g tre nd surfaces fo r a ttri b ute da ta o r fo r mod e lling surfaces fo r a rtificial o bj ects , H o wever , the re is a major diffic ulty with this fun cti o na l form a pproach - fo r each x th e re sho uld be o nl y o ne Y coordin a te , mea ning th a t within the ra nge o n th e x-axi s the re can be o nl y o ne y va lue. In th a t ma ny lin es do no t meet this conditio n (Fig ure 4.20d). a lte rn a ti ve procedures a re necessa ry. While tra nsp osing th e x a nd y, o r ro ta tin g the axes ean produ ce sa tisfac to ry resu'lts in most eascs ( Fi gure 4.2()e), th e prc ferred alte rna ti ve is to use a parametric form o f re prese llt a ti o ll , Th a t is, in s tea d o f using a si ngle equation to uesc rihe the cur ve we Gin use two equatl o ll s th a t a re exp lic it in a param e ter rck rn:d 10 hy l. h ir in s l;lI !Cc, for ; 1 lin c

Xo Yo

I

o(t (

-,-

I

--r--; XI

X

Y

x

2

l X • ( I - t )X I 7 I Y) • ( I

X

R cos A

Y

R s i ll /\ +Y O

+

Xo

clX

X 1+ (Y I - Yo lcJA

y,,

dY

Y I' (X 2 -

Y

I

Xo ldA

w Ilt'" d ,,,,ltf ,I I " ', II IrlttWI I'l...l l !l oll ,1I" t' l " j l ' !, lI lt lll 1'... 11' ,1 ' 0

( ti l

I·',n" ...· " .21 TIll' I ', I 'I I M' "

1' ; \1 ; 111" " '" ' 1"111/

1111111:111'11 ; 11 III ILIIIIIII'I I) II

Y ) 2

x", x I .

l lVl ,.../

loll

(~

I I

( 1I 11 111.

1'/IU j

II " (

II I

I, ' ,

( l ,' il ',IIII '

,1.. ' 11,

Geometries

148

-'r~

"

(a)

"

t,

~

Curveis close 10 all ooints

I

149

functional form, here the cub ic order is preferred for the balance it pro vides between computati o nal complexity and user fl ex ibility, In th ese equations a, b, C, d, e, g, h, p, q. rand s are special coefficients depending on the coordin ates x I, Y I, Z I , X 2, h, Z2, x 3, YJ, Z3, X4, Y4 and Z4 of th e four points: VJ, V 2 , V 3, and V 4 , Starting from points we ca n write:

/e

~J

Coordinates and splines : lin e representation

I

\..l UI vt' tJCl .::>.::> 't;::.;>

Ihrouah some

(b)

'-----

•

x( r) = A(I)X I

y(t)

=

+ B(t)X2 + C(t)xJ + D(r)x4

E(r)Y I + F(t)h + C(t»)'J + H(r)Y4

If elevation is also needed: xndicates

pulling force

P2

,'~------- --- -

~ I

z(r) = P(t)ZI

P4

,

I

I

Tangent directions of two endpoints are fixed bu1 length of tangent va ries.

I

I

------____

(c )

~

P1

I

--'P3

_____________________

(d)

I

+ Q(t)Z2 + R(t)z J + S( t) Z4

in which the A(l) to S( t) are special cubic po lyn o mi al functions called blended functions . which combine the influences of the se veral end and control points (Figure 4,22), With different influences fro m the end and interm ed i ate points, many composite paths may be deter mined. Generally , the curve-fitting process is gu ided by severed co ndition s: I . The number of points of influence on th e path of the curvc,

2. The number of points th e curve passes through ,

.1. Tangents at the origin and at the extremity,

,~

P4

-90

+45

P1 ( e)

(e )

4

2.

•

3

(()

P4

,

-90 IL

P1

_ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _~

s

JOin pOint

7

(9)

L I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

~ _ _~

Figure 4.22 Spline and B€zier curves, (3) Spline for six points , (b) Spline for seve n points, (c) Concept of control points for spline curves, (d) Fixed directions of t
If co ntinuiti es are desired at th e loca tion where the different pieces or

th e curve co me together.

The contro l m ay be exerci sed by prior specification s, or done ill'lcltively vis uall y, The curvature for a particular line segm ent is ilillucnced by the control points and the conditi o ns to be met where pieces j( lin toge ther. In any event , using cubic rol y nomi als for reasons already ',I :iled , as Figure 4,21 illu strated, the smoothness of a specifi ed sequence ld' curves is estab lished by se ttin g the tangents of two curves at a common Illl illl to be eq ual. in mathematics ca ll ed first order continuity (Figure 1 . ~2 f), The parti cul ar shape of the cubi c polynomials varies as different POill!.' cxert more influ ence, II is important to be se nsitive to the ex istence of several types of '."IIIIl' Cllrve, We bricl-ly rev iew th e salient characteri st ics, in turn , of the 11"llllilil' curvc , natur,d and B- (for h,/s ic) splin es, and Bezier curves, I ;, 'lll'l :dl y spc
Coordinates and splines : line repre se ntation

Geometries

150

Renault car company for designing the outer panel of cars as a smooth surface, are generally similar in concept, but can also be effective at higher order polynomials , and are easy to split into several other Bezier eurve pieces (Figure 4.22f). In the few spatial information systems that provide smoothing tools, apparently for graphical purposes and not for reducing data storage, it is the Bezier one that is the most commonly encountered.

4.5.3 Some realities of line and polygon representation In practice, several spatial information systems oriented toward mapping features at a high precision offer tools for line simplification by weeding out points, often using the Douglas-Peucker algorithm. The chain codes are rarely used, and smoothing facilities are more common for computer drawing or mapping purposes than for spatial information systems, although as noted above, a few commercial products of the latter type do provide users with capabilities of line smoothing using Btzier curves. The different represe ntations resulting from different degrees of generalizing are well illustrated by the public domain data sets World Data Bank 1 and II , containing digital coordinate data for major features of the world, especially coastlines. The two versions, different in the number of points captured by a digitizing process, represent natural features curvatures quite differently. The more detailed of the two, World Data Bank II, also provides a good example of the impact of line generalizing to different degrees. Figure 4.23b shows the loss of detail at steps of different minimum point separation distances. Line simplification procedures aim to reduce the amount of data while retaining a high level of information conte nt. The data reduction may __:s ';, ''Ii'

).f~ ~ ;J /)~~~ ~1

I

(~

\

1

(~h ,~tv ~

(~\1 c;~

~ :\I

~'Y ~'c;§ ~,

~Uc ~,-

~

(a)

(b)

Figure 4.23 An example of line gc ncrali/.a tioll . (a) 1\ rcsolution or 0.01 inch; (b) A resolution or o.m inch: (c) 1\ rcsolllliOIl or 0 . 10 illch . (Extracted rrOIll Figurc 4 in I\ntlcrso ll cl
(C)

151

serve to reduce storage space needs, and reduce access or drawing time, but will be achieved at some cost of loss of precision and occasionally of form of entity . Overall, different forms of represe nting irreg ular linea r features or polygon boundaries are better or worse depending on the uses of the data and the computer resources available. Using coordinates is simple but requires data for many short lines if the features are smoothly curving. The success of generalizing by eliminating some details by point e limination or chain coding depends on the kind of curvature in t.he lines as well as how much reduction might be wanted in data volume. Natural entities like coastlines and rivers may well be represented by successions of spline curves , but they are unlikely to be as useful for boundaries of administrative units or railroads. 011 the other hand. complicated curves ean be represente d by only a few pieces of data, the parameters of the equations , rather than by a long string of coordinates.

4.5.4 Intensional and extensional representation of objects III general, recalling some comments of Chapter 3, the indiviuual entities Illay be represe nted in practical terms in two contrasting ways. rur "x:llnple, a straight line can be con structed from an ordered se ries or points or may be created via a mathematical formula. The arc or spline Iype of e ntity implies the latter case. So we recognize these two .SI t lid I ions:

I. The extensional character.

The intcnsional character.

hll tlte intensional state, we deal with a small amount of data explicitly, :111t! store the rules, procedure, or method for obtaining other information. l'II:lt is , t!l e n , ror the river the coefficients of a mathematical equation are ·.I(\lnl al'lng with the end points of the lin e feature. A more simple "~: lIl1pk is the represe ntation of a triangle by three vertices and rul es to I (llllJlIlk segmcnt length s and /or angles (figure 4.24). A s we demonstrate 1. 111'1 , wc :lIe not restricteu to geometric objects, but can also deal with 1 : llllllk~ or space in this way. hll the 1I1011lL'1I1. wc summari/.c Illal intensional data require:

Th l.· Stol:l gL' or sonIc privil egl' d I.' klllenl s 01' (Li(a for the object. Till' SIIlTili,':llioll' 01' :1 IlIk 1',)1 .1"'lll·I:ltinl: :111 po.s.sihle clements, the !'\.' II VI:IIIVl'rlik. I'll(' (\l'l illilillll ,,1';1 lId" 1(11 1(' ,' .1111 /1 II .111 ,' l"lll('llt, 1'01 l:X:\llIpk , " lin c ·,q;III'·llt . I ~ :I 1IIt'IIIh. ·! 01 th., "IIP 'I 'I, IH 111.1(1'.:1 JI"lv g(lIl .

Geometries

152

Fractals:

Y

a way to represent naturdl objecls

153

(a)

A

A, B, C coordinates are stored in the database.

Y A

Rules are used to create the side lengths and angles for the triangle, for example:

Y

c

c

a

2

=

(\'>t)

2

2 +(YC·YB )

o (b)

or, by the law of cosines: a

a

Y B

2

=

2 2 b +C

•

t· i)

0.2

0.00 0.02 0.000 0,002 0.020 0.022

2 bc cos a, and

2 = arccos « b +

0.0 0.200

0.22

0.202

0.220

0.222

etc.

I 2bc)

B

0.20

Figure 4.25 Ca ntor dust. (a) Genera tio n of th e dust. (b) Encoding method.

x

Xc

XA

XB

4.6.1 Creation of fractal objects

Figure 4.24 Intensional representa ti on-rules for a triangle . The third aspect is req uired for , among o th er things, asce rtainin g if a particular coord in ate position falls inside an obj ect (say a polygon) or ou tside, as well as on the boundary. Thus, as we discuss more fully later, there is a way to circumvent the practical difficulty in storing the infinite number of points that li e in side a polygon.

4.6 FRACTALS: A WAY TO REPRESENT NATURAL OBJECTS

Su far we hilve assumed that sharp boundaries or smoo th shapes exist for rca l ent iti es . This assumpt ion reflec ts a map model or geometric bias rat her than a necessa ril y approp ri ate model for nature. In spatial inform ation syste ms we have two kind s of entit y to model: 1. Natural earth feature s such as terrain and coast lin es. 2. Perso n-m ade objects like buildings and roads .

Fractal geometry has scope enough to represent more adequately than Euclidean geometry real world entities th at are not smoothly formed , as is the case with most natural objects. For examp le, a tree would be modelled in the latter domain by sets of co nn ected cy linders, cones ,Ind other regular geometric primitives; but fracl"al geometr y approachcs thc situ ation by a different route giving LIS more acceptable shapes, especi:dly for the leaves and bran ches.

Fractal geometry (Ma ndelbrot , 1982) is an atte mpt to synthesize v: lriou s mathematical work s at the turn of the twenti eth century. The word fractal implies properties as in fraction or fragmented; in esse nCe fract.I1 gco metry has ideas of fragmentation and se lf-s imilarity . Even though objects may be rough o r irregular, that is fragmented , Ihey may at the same tim e have so me similar se mblance of shape o r pattern whcn viewed from different distances. Self-similarity is symmetry across tlillcrent scales; th ere are patterns within patterns. Or, as Mand elbrot says, l!act,Iis are geo metric shapes that are eq uall y co mpl ex in th eir details as III their overall form. An appreciation for these concepts is best :Ipproached by seeing how particular types of obj ect can be crea ted, herore demonstrating th e use of fractal geo metry for natural landscapes. ('ollsider a line segment sp lit into three eq ual parts with the middl e "I'Clioll discarded (Figure 4.25a). An indefi nitely long co ntinued process pi splittin g th e lin e segmen ts will in the end produce a set of very small , ,dl )..'. lled segments ca ll ed Cantor dust. Since a point can be defined as a IlIll' wi lose len gt h is tending to ze ro. we ca n say that we end with a set of ;di!~l\cd (C
,') lIclI ;1 PIOCl'SS 01 paUel'1l gcmTatill1l II :IS Iwo componcnts : /\11 illilLII(l1 UO! (';\lltO! dll sl til" I ~ 11I l' lillt· Sq~I\Il'llt) . /\ n 'p('liIO! ( IWII 1)\ 111 ' 1' ,~ I'I', III1'II ") III 1':111 1' 111 i:I'I I('r: llol' ,

/\il (·ll l ·lIdilil'. 1I11'IIII,d wd l

~l' I V\ l III
Illl' dillnl' lIt ' pOilll s' ,

Ceometries

154

/\

,

!\

,

,-j

/\ )' ;\ ./\

)

(

1\ . ~

./

"-

C

?' '"( (

\,

i \ /

,\

.' ' ,

-, A

"

-J '\,,;.'

7

'-

\

\ '.

~ '"

(;.~ ) " L"

I"' ;' - ( '" )

: ":)

I'

(,..J

Fractals: a way to represent natural objects

,

v

~

lr.

/

{~,~

i.A ( '"I "{

<W"c,, /

Ide ntifying the initiator as 0, at the first subdivision we have pieces 0.0 and 0.2, but no 0.1 which is the gap in the line (Figure 4.2Sb). The digit 1 ne ver appears in the encoding, which proceeds to produce numbe rs as shown. At th e second step there are four segments, e ncoded as shown; at th e nex t step th e re are eight, and so on , always e ncoded by only digits 0 a nd 2. At step N , the set consi sts of 2N segments, and e ach segm e nt has a numerical code with N te rnary digits after the 'decimal point'. The segment length is 13 - N , whe re I is the unit length (perhaps in centime tres) of th e initi ator , here a le ngth of 1. Thus , for the third leve l of a pplication of the recursive process, if the original line was 4 units, then each piece will be ti , that is just under half a unit. For the re pe titor, its self similarity ratio r is J" id e ntifying the subdivi sion of the original line into three pieces. A somewhat more intere sting patte rn results from th e use of the von Koch curve which is similar in con struction to the Cantor dust exce pt that the middle piece of the split line is replaced by two join ed segm e nts with the S(lIllC length, making an equilate ra l triangle (Figure 4.26). After ,\ppl ying the repetitor for 'a while , the pattern takes on a curved appearance. It is e asy to imagine that , at the infinitum , there will be produced a continuous curve form e d by the succession of small angles (Figure 4.26) . Even if this curve is continuous , it has no tangent or derivatives because a tangent is defined by a line corresponding to th e limit of a chord betwee n two points taken in the curve. If we are at a certain leve l of resolution we can define a chord and take its limit ; but if we change the resoluti o n th e chord angle alters as we make ne w fracta l segm e nts along the von Koch curve. If the line segm e nt initi a tor of the von Koch curve is replaced by a n e quilateral triangle, a snowflake is produced hy tile rec ursive proce ss (Figure 4.27) . At step one, the initiator is repl,\ced by ;I si x-pointed s(;\r corresponding to ,\ pol ygoll with 12 sid es I\t Sll'P two Ihl'l'l' an: 41-: sJ
\

.

!

,J\_rClj 2')

~, ~)~ '"

(J'''-C",

S

~Jj

v-

Figure 4.27 The Von Koch snowflake.

t(,..;"r-A _

Figure 4.26 The Von Koch curve .

.A

155

at each step the number of sides is mUltiplied by four. So, for an initiator perimeter of le ngth 1, the pe rim e ter becomes 1(1)N which numbe r evidently tends to infinity , although th e area te nds to a finite limit. The se lf- similarity ratio is again i. An initiator and re pe titor as shown in Figure 428a will , by self similarity , pro duce ne w smaller islands (Figure 4.28d). A tree form , perhaps a lso a rive r system , is produ ced as shown in Figure 4.28e. Many other patt e rns can be produ ced via fractal geometry, but this group of dust , snowflakes, islands and trees apply to most spatial information syste ms. Polygon s created via fracta l geometry, called teragons (
[,

(a)

• • , 1 I. , _,~

' I

r

fl,

I) I /11

[I

"--'.Jt'[.'{~J t 1-'

;J

.t· .'-t .", "-~}.:j~~

I'" .

. .... " -:':I. ('--::'./

rr (b)

'r.'

(e)

(d)

'iJ

\~l (e)

•

I I

Fi~lIn' 4.2X 1\ few oliln 11';1('1;" oili""" , (;1) IlIili ;llm ;lIId rc pelilor for Ih(' isi;'IHI !',l' II(' r; llor. (Ill SI;"lilll~ Ilw "'/ lI lId W' II t' I ;,IOJ, (e) Second SI,·p ill ):(,(I<'IOIlilll', I s l;lJId~ , (d) Tilll.! ' ,\<, I' (, . ) 1' \ 1111'1>1(' (If" frilclal 11 (,('

Geometries

156

\~ -

E

- ~--,, ,i -

1

(JA \1 "-,c ... ]

-------------

j ,/

l

"

] I'

.__ , '

(

--

"

/

/

157

Space-filling curves and dimensionality

\

' C

/ /\-

\

/

F

0)

/1)

/ '/f / 1"

/ /'

(/'

(

/

(,/

(b)

Figure 4.29 A stochastic interpolation for a line. (a) Displacement of the middle point along the y axis. (b) Displacement of the middle point along the segment perpendicular bisector.

4.6.2 Stochastic fractals So far we have produced only regular objects. To create more 'natural looking' shapes, that is, involving variety in the sense that the leaves of a tree species may generally have the same form, but are individually different, we use randomization. The fractional Brownian model. a family of one-dimensional Gaussian stochastic processes of value in analyzing time series for natural phenomena, incorporates a curve which is selfsimilar. This kind of fractional movement is difficult to simulate. approximatc solution is given by the formula: = :i(YI

jjJ>f /1

[~

y

i /( // r / ( I It· / ) /t /

(b)

Figure 4.30 Stochastic interpolation for terrain. (a) Procedure. (b) Repeated application of the procedure (after Fournier, ct ai., 1982).

" (a)

YIICW

Ii /1 /1 /T /1 ·{,(/l/I ..

G (a)

j,~

,/

+

Y2)

+ uo()2

An

eh

where u stands for a random number, 0 is the parameter for a Gaussian (normal curve) distribution, I is the level of recursivity, and h is a fractal parameter specifying the roughness of an object. For example, for coastlines or tcrrain generation, this is often near 0.8. This formula means that instead of having the pure middle of a line segment one-half of the addition of the two Y values, we add a small error term which has a Gaussian distribution, and fading with the level of recursivity. In other words, the stochastic quantity is less important whcn wc incrcasc the level of recursive splitting. In practice, an acceptable approximation of the Brownian motion (curve) can be achieved by using a recursive suhdivision process
linear interpolation; afterwards we take the middle and we move it a little bit, so giving two segments. In sequence, we take the middles of these two segments and move them a little according to the distribution noted above. A pseudo-random number based on the Gaussian dist"ibution is generated and used as a distance in the y-axis to displace the middle point of the line segment. In this way, the equation provides a stochastic interpolation, as discussed in section 7.1.1. A slight variety of this process displaces the point along the segillent mediator (Figure 4.29b), the line perpendicular to the linc joining the Iwo points at the ends of the starting line segment. Some authors prefer to move the middle along the perpendicular bisector, and some prefer movement along a line parallel to an axis. For the mediator thc formula is more complex but truer. The process can also be used for two-dimensional contexts, to produce simulations of terrain (Figure 4.30a). Starting from points A, B, C and 0 we generate midpoints on each side of the original rectangle (or square), ;Ind then get I from the midpoints E, F, G and H. Gaussian displacement v;ilucs are again generated, producing an effect as shown (Figure 4.30b). ('ontinuing this process to small line segment lengths will produce a terrain simulation from the non-systematic irregularities in the height values (Figure 4.31). The surface modelling, varying according to the v;ilues of the h parameters, can be made to produce rough or smooth tnrains as illustratcd. As h increases towards 0.9, the surface roughness ivssens and the surface approachcs a plane surface.

4.7 SPACE-FILLING CURVES AND DIMENSIONALITY I );I\;I " : 111

pn)(' L' SSIII ~~ ;111(\ sl()I;I! ~ "

Ill.'

nS l'd I()

IIIlTt

till"

III;I Y

... ;IIIIl·

11<'

lilill"

,·, '(II\(IIlIlC;iI

P·'I IIII,·l ll l ·II(' .

if Ic,ss

l'IiIIS ;lre;1

IlirOrilliitioll

1IIIILs III;I,Y

Ill'

Geo metries

158

159

Space- filling curves and dim ensionality

H = 0.3 D = 2.7

H = 0 .5 D = 2.5

(b) Row- prime Order

(a) Row Order

H = 0.7 D = 2.3

~ Figure 4.31 T e rra in re prese nta tio n using the fra c tional Bro wni a n process. This example shows th e three surfaces that are outco mes of diffe re nt H pa ra me te rs. As H inc reases so the surface ro ughness lesse ns , a nd th e fr ac ta l dim e nsio nality tends towards that fo r a pl a na r surface. (E xtracte d fro m a figur e in Goodchild and Mark (1987), a nd re prin ted with th e pe rmi ss io n o f The Association of Am e ri ca n Ge og raph e rs, Was hin gto n , DC , USA.)

represented by a ce ntro id , a zero-dimensional object or by parametri c curves. A data reductio n ca n also occur if objects could be pos iti o ned in only one dimensi on rath er th an two or three. Indeed , the ques tio n also arises as to how m any dimensio ns a material object ha s.

4.7.1 Paths through space The matter of dimensionality i s encountered in spatial info rm ati on systems in different ways. In an earlier chapter we di scussed th e identification of m ateri al entiti es as h
(c) Cantor-diagonal Order

(d) Spiral Order

Figure 4.32 Th e co nce pt of pa th s through space. (a) Row o rd e r. (b) Row-prim e o rd e r. (c) Ca nto r-di ago na l orde r. (d) Spiral o rd e r.

k llld oy a on e-dim ensio nal sys tem . Fo r example ( Figure 4.32a), we ca n 1I11agine we can refe r to the sixtee n adjacent map sheet s in a sin gle ord er Ilv I r" versin g from o ne to others along rows from bottom to top. There arc differc nt o rderin gs, that is one-dimensional paths , thro ugh Ill,' two-dim ensinn al tiled space. If we think of a regular distributi on o f 11! 1l k' S ror th c til es, th en we have to solve a location probl em of th e I (Iii I ill )'. k i 11(\ , cxce pt wc arc no t constrai ned by a network. Path s co ul d 1 ,,'/ :'.1'., cOlild go ;tl o ll g a row in one direction and in a reverse directi o n in II,, · " "x l row, like ;1 hid irec lional C()lllplitcr printer (Figure 4 .32b), o r ' "\lld f'lli nw ;1 p;llh Ih ;11 reduce ., 111(' 1()lal d'SLIIICC oj travclthrough goin g III ,", II!;IIIV illllllvdi;lk Il c i)'.ld )Ollr., ; 1., I,",.·,ihlv, ;IIHI il;] vin)!,; 1 small /lumber "I IIIII)e,\' r (·'lIllltTlioll ' , or (·'ll".! Ii:lv, ' d"'1'
160

Ceom etries

Total path length

Spa ce-filling curves and dim ensiona lity

161

mixes up corne r and side joins , while the spiral ord er terminates in th e middle , making it impo ssible to connect to other blocks of space . An ot her path (see Figure 4.33a) , represe nts a Z or N, d e pe nding on the direction of view ing; this pa th mixes a few lo nger jumps with mostly steps from one til e to an imm e di a te ne ighbour. B eca use it joins , sometimes a t edges, and some times at co rne rs, it has three differe nt leng th s. A comparison of different paths (for a given reSO luti o n) ca n use several m eas ures (Figure 4. 33b):

1. Total le ngth of the p a th. 2. Vari a bility in unit le ng ths, wh e re unit length is the dis ta nce from o ne point on th e path to the next in sequence. Different unit length S Total length of path sum of - - segments unit lengths ~

?

(a)

- ••- - 6 I

(b)

Distance to neighbours

Figure 4.33 Measu res for curves in two-dim e nsional space. (a) The N curve. (b) Measures.

so m c convenicnces in single dimension address ing fo r two- or three dim e nsion a l sets of regul a rly shape d tiles. The path should pass only once to each til e in th e two- or N-dimensional s pace, and ne ighbours in space should be adjacent on the path. The path sho uld be useable even if there is a mixture of differe nt size d spatial units, a nd sho uld wo rk e qually well in two o r three dimensions, and for connecting to adjacent blocks of space. In reality there is no ideal path ; there are just ord e rings with some of these properties. The row o rder, the kind used fo r building Lip ,I te lc vi sion picture. has a longer to ta l path than the ro w prim e se que nce in which evcry other lin e is traversed in a reverse direct ion . and ha s seve ral pla ces in which ne ighbours on the pa th, for e xample 1 ,Inti 4. ,Irc nol adj,lcclll ill Sp'ICC. The diagonal a nd spiral orders arc lik c lllC row-priillC Sl'qll l'II Cl' ill il ;IVill).', the imm ediate ,ldjacl'llc y propert y. Ilo wl'v("\" . titl' di;l gO ILd SV <jIWIll"<.'

3. Th e ave rage distance o n the pa th from tiles to their four ne ighbours in space . For a n exa mple of a given bl ock of space divided into sixtee n tiles , the row order ha s a total le ngth o f approximately 22 (four rows of horizo nt a l lines o f length 3, plu s three diagonal lengths each of v'TIi. Th e row prime and spiral have shorter length s , each of IS. Th e C anto r order, madl' up of unit le ngths and the 1.414 for the hypote nuse of triangles with two l'qual unit le ngth sides, is approximately 18, and the N path has a le ngth of ahout 20. The third measure represe nts how easy it is to re ach neighbours in space vi a the different paths. Fo r example for the tile ide ntified as point () in the row ord e r map , it has neighbo urs with path order values of 5, 8, 11, and 10, giving four distances of 4, 1, 1,4, a total of 10 (average = 2.5) Other di sta nces sums for this one tile a re 10 fo r row-prime , 16 for .s pira ls, 14 for diagonal, and 12 for N order. Comparative ave rages for th e

central block o f four squares a re shown in the ta ble below along with

ot he r properties of a sixteen-tile mosaic.

Path type

Le ng th (approximate)

Vari a bility

Row Row-prime Dia gollal Spir:tI N

22 15

2 1 2

lR 15 20

Average distance

10 10

18 13

.1

12

!\ I\lOI",' c()l1lpicll' ; Ippr;\i .~;d cOllld ill<"iwk IIIl' dis tallces for corner and ('d gl' c('lis. ()lit('1 ('ollsid(,I;lII(IIIS IId g lll I>\' IIIIPOII;!111 ill addition to path kill'.t II S, slldl ;IS till' poilll (II' ('lill Y ,lIl d ,' \I I 1111111 III(' lI\(l s;lic .

162

Geometries

Space-filling curves and dimensionality

163

4.7.2 Space-filling curves More especially we talk about space-filling curves rather than paths through space. These curves are special fractal curves which have characteristics of completely covering an area or volume. While they have a topological dimension of two, their fractal dimension is two when filling an area, or three when completely occupying a volume space. If we define a point as a zero-dimensional object , theoretically speaking it is not possible in the spirit of Euclidean geometry to find a one-dimensional curve passing through the infinity of points. However, if we think of a point as a (small) two-dimensional square the side of which tends towards zero, then, in the spirit of fractal geometry, it is possible to find a curve filling a two-dimensional space. So now the curve is defined as a sort of ribbon , the width of which tends towards zero. Similarly, at three dimensions, a point is defined as a small cube for which the side length tends towards zero, and the curve as a three-dimensional curve, a sort of spaghetti noodle for which the width also tends to zero. Consequently, thinking of paths in space now as space-filling curves, lines that pass to all possible points in space, we see they should have the following properties:

~

(a) Initial steps of the Peano N curve

-rR

1. The curve must pass only once to every point in the multi-dimensional space. 2. Two points that are neighbours in space must be neighbours on the curve. 3. Two points that arc neighbours on the curve must be neighbours III space. 4. It should be easy to retrieve the neighbours of any point. 5. The curve corresponds to a bijective mapping from a multi- to a one dimensional space. 6. The curve should be able to be used for variable spatial resolution, that is, a mixture of different-sized ' points'. 7. The curve should be stable, even when the space becomes very large or infinite. In reality, we do not possess such ideal curves, but there are some with valuable properties for our purposes. The original space-filling curve was exhibited in IW)() by the Italian mathematician Giuseppe Peano (Peano, 18(0). A later variety, now known as the Peano or N ordering (Figure 4.14a) , facilitates retrieving neighbours, and although neighbouring points in "pace Me not always neighbours on the curve, they generally arc. II is .dso possihle 10 de. II with different resolutions 'IS shown, and the UIIVl' is st;lhk . Tile I-lilhcrt

(b) Initial steps of the Hilben curve

FiguI'c 4.34 [nitial steps of the Peano and Hilbert curves. (a) Peano N. (b) Hilberl.

sOllletimes known ,IS the Peano-Hilbert curve, passes through alJ points ill a set hy nlcans of single length steps only , having the form of a (,rq'" Ieller :n: (Figurc 4 ..1411). This curve meets IllOSt' of the conditions II(lt"d ;Iilovl'. Illit docs lIot providl' ;III (';ISV way 10 retrieve neighbours and I', 1101 ,,1;1)1,',
III pl':ll'Iin' , IIIe' l'IW(ldill.l', (If ,~p;I(', . filllil l ', ,'111\,',; II\l '" Olll' coordinate ,.c1kd :1 key Ilv 111(1\1 pi :ICllllCIII\'I" , I" '.1.111<1 1"1 IIV(I 'II Ilion' ('()oJ'(lill
Geometries

165

Space- filling c urves and dimensio nality

164

x=

~ 1_

I0

y=

\ 1.)

-

:

- :::. ...---...:-~... "' -""'-"

I1 I0 ~~ _r _'' " ,..". ... '"

:/..

/'

10 10 10 10 1 1 ~ 1 10 J

p=

(a) y 03

02

01

00

01 11 (7)

11 0 1

(5)

( 13)

1111 ( 15)

0 100

0 11 0

11 00

111 0

(4)

(6)

( 12)

( 14)

0101

100 1

lOll

(9)

(1 1)

000 1

001 1

( 1)

m

0000

0010

1000

(0)

(2)

(8 )

vc rsely, wh a teve r is th e eve n numb e r of binary decim a l digits of a number, p , it is always poss ible to c ha nge it to two o the r real numbe rs, onc by tak ing digits with e ve n rank to ge t x , and odd ra nk digits to ge t y . So, this me chani sm a llo ws us to cove r space with a ny reso lutio n. For nega ti ve coordinates the prefe rred solution is to tra nsform the va lues to pos itive coo rdin a tes by origin shifting, that is, geometric tra nsla tion . T o dem o nstra te the awkwa rdness in using Hilbert keys , we present o ne .i1gorithm (Ma rk a nd G oodchild, 1986). W e will use H (X, Y) to de note Ih e Hilbe rt key of x a nd y .

PR OCE DURE H to X , Y (H , X, Y) U nsigned H ;

Integer X , Y ;

Begin

base - 4 P[8], Q[8] 1* handl es grids up to 256 bin a ry X[8], Y[ 8];

integer N ;

integer i;

N = 8;

1010 ( 10)

X

25() */;

1* first co nstruct vecto r P from H a nd make anoth er co py () X

00

01

02

03

00

01

to

II

dec ima l bin ar y

( b1 Figure 4.35 O btaining the Peano key by bil inte rl eaving. (a) T he bit interl eaving fo r x and y. (b) Keys for a bl ock o f sixteen spaces.

For the Pea no N curve the ke ys are easil y o btained ; it is quite compl ex to gel the keys for th e Hilbe rt curve, as no te d below . Fo r the form e r, the bin a ry di gits fo r th e x a nd y va lu es a re interleaved , as shown in F igure 4 .35a . For exa mple, x = 3, a nd y = 2 in bin a ry are respecti ve ly 0011 and 0010 . The key, p , thus is 00001110 , correspo ndin g to the decim a l number 14. A set of keys for a block of sixtee n squ a res is sho wn in Figure 4 .35 b . (There is a special arithme tic called tesse ra l a rithmeti c th a t se rves to look after the peculiariti es of mec hanisms such as bit inte rl eav ing.) It is a little mo re awkward to dea l with real a nd ne gati ve coo rdin ates compa red with the case of intege r coo rdinates just descrihed , hut th e idea is th e same. With fo ur digits aft e r th e decim a l po int for x and y, we get e ight di gits aft e r th e decimal po int in thc Pea no key, fo r eX <1lllp":, for x = 11.0101 a nd y = 10.1011 , I) = 111(l.(III()OIII . It there ,1I C 1eJ1 digits ,ill e r the decimal point, th e Pl'
for i : = 1 to N do

begin

P[iJ := (H mod 2 Q[i] := P[i]; e nd

**

(2

*

(8 - i

+

1») /2

**

(2

1* nex t rea rra nge th e bit pa irs in Q *1

for i : = to N - 1 do

beg in

if P[i] = 0 the n

fo r j : = i + I La N do

begin

if (Q Ullllod 2) = 1 then QUI := 4 - Q[j] ;

end

el se if PI iI = } th e n

for i :'-' i + I to N do

heg in

"1 ., Oli I; if (Oli I 111<)(1 2 ) OtlwJlOlil CJld

n I
I I() N dp

*

~"/

(K - il l;

Geometries

166

begin sw itch (Q[i]) begin case 0 X[i] case 1 Xli] case 2 Xli] case 3 Xli] e nd 1* finally assemble X

167

particul ar circumstances; but, in ge nera l, it appea rs that th e Peano and Ililbe rt orderings a re more robu st, that is bette r performers over a range o f requirements.

.- 0; Y[i] - 1; Y[i] - 1; Y[i] 0; YP] -

0; 0; 1; 1;

break; break; break; break;

and Y from their binary vectors X[ ] a nd Y [ ] *1

X:= 0; y := 0; fo r i : = 1 to N do begin

Space-filling curves and dimensionality

X : = X + Xli] * 2 * * (2 Y : = Y + Y[i 1 * 2 * * (2

4.7.3 Dimensionality Rece nt adva nces in this area of fractal geometry ha ve allowed us to Illod c l nat ural objects , as shown ear li er, but also give us diffe rent ways to Ihink abo ut dimen siona lity. For exa mple, the le ngt h of a coastline can

4

* *

(8 (8

i)) ; i));

3

end e nd e nd Generally, th e ordered paths have similar shapes at different sca le leve ls (Figure 4.34). They are said to be self-similar, that is the entity is ca lled self-similar if any part of it , afte r e nlargeme nt , cannot be distinguished from the who le object. H owever , the particular place of a point or tile in th e sequ ence for a pa rti cul a r curve type may not be consistent across scales. Whil e the N curve does hav e suc h stability, as revea led by th e coded numbe rs (Figure 4.36b), this is not so for the Hilbe rt curve (Figure 4.36d). That is , for t he Peano case, if the space is ex te nded by doubling each side for the b lock of four quadrants , we see th at the order of squares is not perturbed. The square numbe red 1 keeps its sa me numbe r; it is stable. Space-filling curves have two princip a l, practical uses in the dom ain of spatial info rm a tio n systems . Firstly, they provide so me efficienci es in sca nning opera tio ns , e ither hardware devices or searches through datafi les. Secondly, they are used as spa ti a l indexes, simplifying two dimensional add ressing as single dimensi o n addressing. Th e United States Burea u of the Ce nsus uses Pea no keys for this purpose, and the Can adia n Land Informa ti o n System , ofte n regard ed as the first working geographic informatio n syste m , used the co ncept of o rderin gs und e r the labe l of M o rton sequences (Morton , 1966). Ore nste in (1l)~6) ca ll s th e two dimensional reverse Pean o o rd e r thc Z -or<.ier. (We avoid referen ce to th e Z-o rder, recogn izing thaI Z is ll.~lIally lI se d to indi cat e the third sp
o

I

o

3

>

(a) Peano N space-filling ordering

I XJ~' ~' ~

~l:

(3)

· ,0

{)(~)

(b) OrdO fln g s tot)ility will) IIl() N ard or "'i~~ III T

"1.'\(' (

(1I11111111'd \1 \ 1' 1 "",,I)

0

0 '

Geometries

168

r- '

1- . '

L1Ti0/-1):, rtf I I

I

' .

( "\ I \ 4)

1

\

1

'I' ( (2 )

I

,

I

o

,

I

"

I

(

I

1~1 ,

I

I

I ' / I

./

I I

I 1

l

d

i

( 14rt-,/ 15)'

'

I

I

2

>

I

3

4

(c) Hilbert space-filling ordering

( 5)

!(~ ~,)II

I

(

, j

( 6)

,

I

I

() I

I

( 7) 2) ---

I

(

)

l (J

1C)

() I

()

-

1

169

whi ch it is located, It is not dependent on th e m easurement ullits use d or alter atio n of the sp ace by stre tching or cond ensin g, D iffer ent objects m ay be id entified by a parti cular fractal dimen sio n . T he fractal dimension of m an y, but not all , enti ties ea n be obtain ed by th e f o rmula ( from M andelbrot, 1982):

I ( 13r--~-(1 2

I

\----1-11 \( 0 ;-: \ )\ ' .'

--<, 8

I

I (3 \ I o

I ,' \

Space-filling curves and dim ension ality

C)

(d) Ordering instability when space extension with Hilbert ordering

Figure 4.36 Orderin g stabili ly fo r the N and Hibert ord erin gs. (a) Pea no N space-filling ord ering. (b) O rd erin g stability with the N order. (c) Hilbert space-fillin g o rd erin g, (d) Ordering instability when space extension with Hilbert orderin g, vary dep endi ng o n scale, rangin g fro m an apparently infini te ly high length (assuming no limit to the precisio n o f measurem ent ) to a ve ry short dist ance if w e highly generali ze the shape. It is int erestin g th at fra ctal geometr y ca n gi ve us measures o f the dimensiollalit y 01 o hi ects that are different fro m th ose we ex peet fro m Et lclidean geolll el ry . The rr;lcLtI dim ensio n tell s us how dell se ly " ph l' IlOIlIl' 1I011 ol'nlpi l's I Il l' Sp;ll'l' ill

=

log n l (log (1lr »

[o r log n /log s]

w here n is the num ber of pi eces in the repetito r , r is the self- simil aril y r11 1.1,, 1/1,' 1 .. I'l n '\'-, 1t:I V( ' ' 11I :d"', "

170

Geometries

numbers, and dimensionality values of 1.5 or 2.5 are more appropriate for highly indented lines and rugged terrain, respectively. Such natural objects are not pure fractals, in the sense of having a constant dimensionality, but are approximately so, in the sense of being represented by a non-integer dimension. Overall, fractal geometry has promise for some of the requirements of spatial information systems. Two-dimensional stochastic interpolations (discussed more fully in Chapter 7) are useful for terrain modelling; one dimensional applications are for coastlines or boundaries of entire continents. As noted earlier in this chapter, fractal geometry gives us a way to indicate the dimensionality of specific objects and structures, and underlies the concept of space-filling curves. Fractals may also be used for cosmetic embellishments for cartographic and other graphic displays, for example, image generation.

4.8 SUMMARY Spatial data are quite special compared with data usually stored in databases. We have to deal with an infinite number of points in space, not a fixed number of entities like bank transactions or people in a survey. We have both intensional and extensional forms of representation; and there are varying degrees of precision of representation. We also have, and will continue, to use geometry deterministically. That is, we are interested in the representation of particular rivers, terrain or buildings, not in discovering laws about all rivers, all buildings, or all terrains, the realm of statistical geometry. Ignoring the trivial case of point entities, our discussion of geometries can be summarized primarily as a need to be sensitive to various ways of representing line entities. These one-dimensional objects in Euclidean geometry may be represented by:

1. 'Real' position, using Cartesian or polar coordinates. 2. Some generalization of the vertices for the line feature

using

procedures that depend on coordinates. 3. Topology, ignoring geometry. 4. Parametric representation. S. Fractal geometry. Some data are discarded if only gcncralized versions arc maintained in a database. But the data volume can be lowered hy a reduced nUlllher oi' coordinates, or, better still under sOllle conditlolls, hy thl' p;lr
Summary

171

i'unctions or equations, or the IllItlators and generators for fractal representations. Coordinate-based representations are associated with digitizing or line-following forms of data capture. Procedures of simplification and data reduction usually are applied to those data, unless legular geometric figures are the entities. Otherwise, detailed geometric representations are dealt with by stored (\;Ita for positional coordinates. Absent good reasons for representing phenomena intensionally, that is by rules, as for regular geometric ligures, then devices are needed for adequately and effectively recording IHlsltion extensionally, The many spatial databases existing today are (ILlracterized by Euclidean measures, and projective geometry, as well as Il'y extensional data. ('olltinuing a theme first developed in Chapter 3, spatial entities can be vII'wed from several perspectives regarding dimensionality, The spatial 1IIIih can be dealt with in terms of Euclidean dimensionality, or thought 111 vi;1 i'ractional geometry. For some needs in spatial information systems ,11'11II<:;lIioflS the reduction in dimensionality associated with spaec-filling .111 ve s is important. W nrkillg at a global scale necessitates certain sensitivities and practices III 111l' usc of map projections to avoid excessive and mi,skaliillg .II ', IUIIIOIlS, but also makes it challenging to create spatial rdcrencing ',I IWIIil'S ,suitable for a sphere, and w,lys to store very large volullles of' l i,II:\, l'll v lise oi' Cartesian coordinates in a digital environment necessitates, I." I :111 ;Ippreciation for errors that may arise for data for position, and , 11 ',1,1111'1', espeei,t1ly whcn approximations are used, no matter how " i'l li' : dllll~ Whcther a reflection of computer machine precision or a user's , II. Il l " ()f' mcasurement needs, the limitations in recording are best 11'1' 11),111 (>I vi,l interval geometry rather than as a continuously retlned 1" \ ;1'-1III'IIIl'1l1 cOllcept. Scalc of observation, cmbodied in the concepts of II I,JIIIII!l1 ,111(1 precision, means that not all the digits for <1 positional I Il ll l tllll
I'"

172

Geometries

4.9 BIBLIOGRAPHY For more ex tensive trea tments of Euclidean geometry, the reader is referred to Gasso n (1983) and Gat re ll (1983). A comprehe nsive survey of line ge nerali za ti o n is tllat by McMas te r (1987). Elaborations on spline curves are provided by Foley el al. , and by Barte ls el al. In add itio n to th e treatises by Mandelbro t, fractal geo metry concepts are cove red extensively by Barnsley. The book by C larke provides th e ca rtographi c pe rspective on spa tial obj ects, oriented es pe cially to vector geometry~ whil e eit he r Robinson e t al. or Muehrcke pro vioes a n ex tensiv e review of spa tial referenci ng systems a nd map proj ecti o ns. Abbott, Edwin A. 1952. Flalla nti: A f? omartce of Many Dimensions. 6th edn . Ne w York , USA: Dover publications. Anderson , D e lmar, Jam es L. Angel and A lex ander J. Go rn y. 1977 . WDBll: Conlenl, Slructure and Appliwlion. Pa per presente d at the First Inte rnational Adv anced Stud y Symposium o n T o po logical D a ta Structures for G eographic I n[orma tion Syste ms . Harvard Unive rsity, C amb rid ge, Massachu se tt s , USA. Ange l, Ed ward. 1990. Compuler G raphics. R eading, Massachussetts, USA: Add ison- W esley. Armstrong, Marc P. and Brian T. D a lz ie l. 1989. Digiti zing and di sta nce estimation e rrors in spalia l analysis: sinuosity and network co nfi gurat ion effects. Proceedings oj the International Geographic lnformalion Systems Syrnposium, Baltimore, Maryland , USA , pp. 75-84 . Barnsley , Michael. 1988. Fractals E verywhere . San Diego, Ca lifornia , USA : Acade mic Press. Barte ls, Rich ard H . et al. 1987. An Introduclion 10 Splines for Use in Compuler Graphics and Geo metric M odeling. Los A ltos, California, USA: Mo rgan Kaufmann. Hell, Smah B. M. 1986. T esse ral addressing and arithm e tic - practical co ncerns. I'roccedings of th e Wo rkshops on Spalial Data Processing us ing Tesseral Mcfho ds. Swind o n . UK: Natura l E nvironm ent R esearch Council , pp. 11-1 6. But L, A. R. 197 1. A lte rnativ e algo rithm for Hilbe rt 's space-fillin g curve . IEEE

Trllll.lactiol1s OIl Computers (April) : 424-426. Clarke, Keith C. 1990. Allalytical and Compuler Cartography. E nglewood C liffs, New Je rsey: Prentice H a ll. Co mmittee o n Map Proj ec lions. 1989. Geog rap he rs and ca rtograph e rs urge end to popula r u se of rec tangular m a ps . The Am erican Carlographer 16(3): 222-223. Davi s, J o hn C. 1973. Sta tistics and Dala Analysis in Geology. New York , USA: Wiley . Di az , Bernard M. 1986 . Tesse ral arithm etic - overvie w. Procecdings of the Worksh ops o n Spatial Dala Processing using Tesseral Methods. Swinoon , UK: Natural Environment R esea rc h Council, 1-10. Douglas , David H. and Thomas K. Pe ucker. 1973 . A lgorilhills lor Ih e reducti o n of th e number of points required 10 rep re,c nt a dit!,ili'l.ed lin e or ils C
Canadian Cartogrlll1hcr Itl( 4): I 10-- 122. Dutton . Gcollrey . II)SI) l'I;IIll'\; lr y IIlIHkllillg vi;I hi"I,m 'hi e d \L'sSl'll:!li')I\. l' ro(,(,l'Ilill,!;.I' "f' II,, ' .. \111" ('"rl /l I) ( '/1 11./"1',.11.1'1 ' , Hllllill/"r,· . I ;dl s ('h\lrch , Vill'.illi;!,

173

Bibliography

USA: A me rican Socie ty fo r Photogrammetry and Re mote Sen si ng! A m e rican Congress for Surveyin g a nd Mapping, pp. 462-471. Dutton , Geoffrey. 1990. Location a l properties of quate rnary tri a ngular m es hes . Proceedings of Ihe 41h International Symposium on Spalial Data Handling, Zurich, Switze rl a nd, pp. 901-9 10. I· pste in, Earl F. and Thomas D. Duchesneau. 1990. Use a nd value of a geode tic re ferenc e sys te m . Urban R egional Informa tion Syslem s Associatio n Jo urnal 2( 1): 11-25. I.'o lcy , J ames D. el al. 1990. Computer Graph ics, Prin ciples and Practice, 2nd edn. R eadi ng, Massachu sse tts, USA : Addison-Wesley. I.'ll lllnier, Alain , el al. 19S2. Computer re nd e ring of stoc has ti c models. Cnmm unications of the Associarion for Computing Machin ery (6): 37 1-384 (;:)sson , Pete r. 1983 . Geomelry of Spll/ial Forms. C hi ches ter, UK: E llis Ilorwood . (;;!Irell , A nth o ny. 1983. Dislllnce and Space. Oxford, UK: Cla rend on Pre ss. (;illies. C ha rl es F. 1989. T he motor fre ighl syste m a nd port of Baltimore, Mdryland [950-1980: thirty years o f spatial cha nge . Thesis, Unive rsil y of M;!ry land, College Park, Maryland , USA. ( ;kick , J;lllles. 1987. Chaos, Making a New Science . Ne w York: Penguin . I ;,)I,bchl age r , L. M . 1981. S ho rl algorithms fo r space-lillin g, cu rves. SO/III 'I//('. I'me/i('(' and Exp erience 11: 99- 100. (i')lll icilild , Mic hael F. and Da vid M . Ma rk. [987. T he rl,lc (; 1i 1l:l(UIC 01 )',(" )gr:lphic phenomena. Annals of the Association o f' Americun (ieil~l'IIl lll ('fs (') 265-278. " ""Ilc- hil d, Micha e l F. a nd Yang Shire n. 1989. A hierarchi cal spalial d ,ll:I ',1 I IIl' (lIrC lor global geog rap hi c informalion sys te ms. NC G IA Tec hni ca l Paper ~;'!', N
1'):-\2. The Fractal Geom etry of Nature. San Francisco,

\ .<111' "111;', li SA: W. 11. I<'!·ccillan .

l\ l lll< kll," d, H"I)oil B . ;II HI .I. W . V"n Ness. 196X. Fraclio nal brow nian moti o ns , 11 .1 ,1.<1 1I0!,",', "'I'HI "pplic'llioll s. Socil'lv fill' IlIdll.l'lrilll lind Applied Math emalics 1""' 1, '1\ ' ItI( ·I) : ·122 ,117 . ~ l. i ,1 1).I vid M ,)I It! Mi l' lI :ll' l I:. ( ;ood l' lIild . I\)Sh. ()I) IhL' ord c rin g of two '. IIIII ,' II ·"')II ;iI "1';" " " illlll"llIc(io)) :tlld 1I 'l illi lll ) 1<1 1,' '''; \.' 1": <1 plillciplcs. l>rocccding5 " I Ii'l ' \\ ' ,,/ /,1 /["1' (II I .\'t "Il;,,1 /),,111/ '111( '1'.111 11,': 111/1/, : I nl l,/,,,II\,/I·t!I/Jt!s. Swintioll . 111\ N !ll lIl: d 1'11 1'''<1 11111 ,' 111 1\' '',1' :", '11 ( \ '"11,01 1\ 11 1\ 1" I ); ,v," II) :~: I 1'1' ,1111' !-I'I'" I"IJ" I III ' ' 1 11 1,, 1 :,1 III, ' .\p; lli;
1 74

( ,( '( 1/11{ '// 11'<,

R efere ncin g Syste ms A ssocia ti o n Fo rulll . 1n- 14 OU o ile l I()X .l, Co ll ege I'a lk , Ma ryla nd , USA. Mo rto n , G. M . 1966. A Compute r-o rie nted Geode tec Dat a base alld a New T ec hniqu e in F il e Sequ e ncin g. IBM Co rpo ra tion , Ca nada - O nt,ni o Re po rt . Ontario , Ca nada : IBM . Muehrcke , Phillip C. 1986 . Map Use, Reading, Analysis, and Interpretation . 2nd edn. Madiso n, Wi sconsin , USA: .IP Publica tio ns. Orenstein, Jack A. 1986, Spa tial qucry process ing in a n o bj ect-oriented data base system. ProceedingS of th e Association fo r Computing Machinery, Sp ecial Interest G ro up, Management o f Data , Washington, DC. USA, pp. 326--336. Pea no, Giu se ppe. 1890 . Sur un e cO Ul'be qui re m pli t to ute un e aire pla ne. Math ematisehe Annalen 36(A): 157-1 60. Peano, Giuse ppe. 1957. edn . La curva di Pea no ne l ' for mula ri o ma thema tico' . In Giuse ppe Pea no, Opere Seelte di G iusep pe Pecmo. Ro me. Ital y: Edizio ni C remon esi , vo l. I , pp. 11 5- 11 6. Poe lstra, T. J . 1990. Fra nki e goes to town , o r how to cap ture realit y ill ur ba n areas. In Ni cos D. Po lydo rid es (ed.), Data Acquisitinll. fo r Spatial information Systems. Ath e ns: Urba n a nd Regional Spati al A na lys is Ne twork for Educa ti o n and Trainin g, Com pute rs in Planning Seri es, vol. 7 , pp . 56- 66. Quinqueton, Jea n an d M. Berthod. 1981. A loca ll y ada ptive Peano scannin g algorithm. IEEE Transactions on Pal/ern Analysis and M achine Intelligence (4) : 403--4 12 . Ro binso n, Arthu r , et 01. 1984. Elem ents of Cartography. New Yo rk , US A: Jo hll Wil ey. Ro binso n , Arthur H . 1990 . Rectan gul a r wo rl d mclps - No' Professional Geographer 42( 1): 101-1 04 . Sie rpin ski W . 19 12. Sur un e no uve lle courbe co ntin ue q ui re mplit to ute un e a ire pl a ne. Bulletin de l'Academie des Sciences de Cracovie, Se ri e A: 462-478. Snyder, John P . 1983 . Ma p pro lections - used by the US Geo logical Survey . US Geological Survey Bulletin, P-1532. Washin gto n, DC , USA: US Governm e nt Printing Office. Va n Est, Jan and J a n Sche urwa ter. 1983. Th e geograp hi c base register and a geographi c base fi le: th e use of post codes as segme nt id e ntification . Pa per presented a t th e SO RSA Forum, /0-1 4 Octobe r ; Co ll ege Pa rk, Ma ryla nd , USA. Va n Gelder , B. H . W . 1990. G lo ba l Positi o ning Sys te m : sta te o f th e sta te. III Nicos D. Po lydorides (ed .). Data Acquisition for Spa tial Info rmation Systems. A thens : Urba n a nd Regio na l Spati al An alysis Netwo rk fo r Educa tio n a nd Training, Co mpu te rs in Pla nnin g Series, vo l. 7, pp. 36--42. Veregin, Howa rd . 1989. A tax onomy of e rro r in spatia l da tabases. Technica l Paper 89-12, Nat io na l Ce nte r for Geogra phi c In fo rm a tion and Analysis, University of Ca li fo rn ia, Sa nta Barbara , Ca li forn ia, USA . Warntz, Willia m. 1964 . A new Ill ap of the surface of popu lat io n potentials fo r th e United Sta tes, 1960. Th e Geographical R eview 54(2): 170 ·1 84 . Wilso n , Ala n G . a nd Mic hae l J. Kirkb y. 1975. M athematics for Geographers and Planners. Oxfo rd, U K : Cla re nd o n Press.

5 Topology (~r;:lp hs,

areas, ordering

"'.wldi e Z ip delivered the mail to Ihe boxes o n his hig hway roule. Freda Johnson lII i." ed her ride home 10 school because s he did not rea lize there were two crossin gs of Ridgc Roa d a nd C rescenl Avenue , a nd wa iled a l the wrong place. Inge Svenssen lI('cded to ma p the vacant spaces in her forest zone. Gerry M a nder tried to ta me a lig<'r 10 help him persuade his polilica l clients thaI Ihey could do their OWl) legisla tive dislrict ma p drawing on their own persona l com puler. The French police could nol app rehend Ca rlos COl'a because he disappeared inlo a hou se in Spain even Ihou gh they w m p letely s u rrounded the house. F"an ~ oi s Reclus was nol quite sure 10 which alllhnr ily 10 pay laxes because his kitchen was in one cily and his bedroom in anolhe r.

I'k lati o nshi ps betwee n pl aces and spaces a re exa min ed in thi s chap te r lro m th e po int o f re la ti ve pos itio ns. Co ncepts fro m the ma the ma tica l ;!leas of to po logy a nd graph theory a re vCl lu ab le fo r revea lin g th c spa ti a l ~ 1 ructurc of c otiti es see n as po ints, lin es. areas a nd so lids , afte r the de ta il s ()j' geo me try are stripped awa y. Afte r rev iew in g so me properties of graphs 1m point a nd line e le me nts, thi s cha pte r th e n introduces co ncep ts fo r dealing with area o bj e cts by me a ns of topo logy. Spatial prope rti es of ;Idpccncy , con nec tivi ty and containm e nt a re exa m incd conceptu ally , a nd 1hen an introd ucti o n is given to d a ta mode ls fo r spatial relationships .

5.1 NETWORKS AND GRAPHS

Space ca n be viewed as a th eo ret ica ll y infinite se t of po ints with o bse rva tio ns possible fo r one or mo re a ttri butes or spati a l propert ies anywh e re . B ut so me phenomena a re not · de fin e d for all points in thi s continuo us space, o r cannot correctly be rega rded as sampl es fro m an infinite se t. Fo r insta nce , for studying ri vers a nd transportation , we d ea l only with a su bse t of po sitions , constrain ed by th e predetermined loca ti o n of the associated ze ro- dimension al objec ts. In o th e r word s, the positi on of the nodes is t he do minant el e me nt . Fo rm all y, we defin e an isotro pic co nd ition as the prope rty of

176

Topology

(Cl l~1 (a)

(b)

• Main cilies and lowns which are nol always intersections

(dl l

1J

"'111 .1~ r,II)I.,

I

id entical ness in all directions. Anisotropic refers to the ex iste nce of properties that differ with direction in space. Physical netw o rk s, m an made o r nat ural (Figure 5.1 and Figure 3.11), in two or three dime nsion s represent an anisotropic state: Virtual networks or channels ex ist for air masses , ocea n currents and air ro utes, am o ng ot her exa mples. Many sch olars and practiti o ners work with phenomena of thi s kind , fo r example , transportat io n planners , air traffi c co ntro ll ers, water and sewer se rvice m ainten ance crews, marine biologists , hydro logists and geo m orph o logi sts.

5.1.1 Graphs The co mbin ation of lin e seg ments, the network, may be viewed as a graph (Figure 5.1). Assuming we begin with a m ap, showi ng the exact locat io n of the transportation fea tures , as for the railwa ys of New Zealand, we will ge nerally throwaway the lin e shape info rm atio n , th e compass orientation, and the lin e length S," co ncent ratin g o n the structural components , the jun ctio ns and co nn ections. The principal elem ent is co nn ectivity , but orie ntation , the recogn ition of a begin and end direction to a lin e, is som et imes important. Several steps of genera lizatio n and ext ra ction m ay be in vo lved , as show n . The resulting gra ph has o nly a few basic elements (Figure 5.2):

1. Th e intersections or end points of lines , usuall y referred to as vertices

177

B;il tll"lI1< '

College Park

(a)

Figure 5.1 Graph for a rail transportation network . (a) Map based on geograp hi c coo rdin ates. (b) Staightening of links. (c) Removal from co ntext to e mph as ize spatia l struct ure. (d) Vertices a nd edges.

or nodes, depending on co nt ex t.

N"tW('II"

~G,,\,)Y"k

London

•

Washing ton, DC

Lyon

.L:

~. '

(ll)

lJurham

Separa te su bgraph

-

Unconnecte d (not part of any grClpll)

~.

•

•

• Node link <1 Space (region or fClCP)

Fi!-(ure 5.2 Compo ne nts of a g rap h . (a) EX(tmplc . (b) T Ci'lllill(llol'Y

, Tlt c lin es, ge nera lly ca ll ed edges, o r links , but SOlllc f illl\' s ; 1I l'~ 1>1 dlains. \, SL'paratc single link s o r disconnected sets of lin es, ca ll ed stlilg r:'1 l lt s I, V;lcant sp(l ces (also referred to as faces o r regions) betwccn or ()tli.,;id(' L·dg('s. (iC(lIllctrically spe(lking, it vertex is a unique point positioll w lti('11 1t :1.' III ifller size nor dimension, and an edge, a non -i ntersec tin g curved 01 ', II ;Ii!',ltt line , w ith end s that terminate in tw o, not necessarily dillL'I '\'III, ~I ' lti n's If Ihere
Topology L

K

'Figure 5.3 Two sets of isomorphic graphs. harder to visualize. Ground transportation systems a re e xamples of non planar graphs if they have over- or underpasses, bridges or tunnels. In nature, subterranean river syste ms can be viewed as three-dimension a l graphs, while aircraft flight paths a re virtual three-dimensional graphs in the sense that there should not be any physical inte rsections (plane collisions). For a solid complete three-dim e nsional object, sometimes termed a cell, the graph consists of a series of e dges bounding faces, plane , concave or convex, and conve rging, at least three in number, at the vertices. A solid object al so has an in side , and an outsid e, but no connections to other objects. In a case of jOined solid objects any edge may have several facets converging at it. In that graphs are dev ices for dealing with certain kinds of spatial relationships, it is pos sibl e to use them to reveal structural similarities, such as comparing linkages by diffe rent modes of transportation , or changing patte rns of connections over tim e . Two graphs a re said to be isomorphic if there is a one-to-one correspondence be twee n their vertices and their e dges. That is, in other words, the conditions of connectedness and a djacency correspond, even though shapes may be quite different (Figure 5.3). Graphs may have loops or circuits or cycles (Figure 5.4a). If they have none, they are called tree graphs, or just tre es, for example , a star-shaped patte rn of conn ections in a local a rea electronic communications network. Structures that have no circuits but which have directed e dges , or directed acyclic graphs, are very common for sewerage and some wain supply sy stems. Graphs with circuits , Ihat is, cyc li(,' g r;lphs , ILlve al le;l sl Olll'

Networks and graphs

179

ve rtex connected to itself without the need for trave rsing one e dge in both directions . Highway and air transportation systems have many circuits; river systems genera lly a re tree graphs, except in the case or braided stre ams. Both acyclic and cyclic g raphs may be directed or oriented, as shown by arrows for dire ctions of movement . The network configurations re present different condition s; th esc range from simple patte rns such as all vertices singly connected to a ccnlral node, or singly connected to a pe ripheral node , to complete linkagcs , tlial is , having all poss ibl e connection s. Som e special pattern s of conneeti()1l are th e spanning tre e, in which each ve rtex must be connecte d to at least one other vertex , the travelling sa lesman configuration, for whieh thl'le must be a way to re turn to a sta rting point without re tracing an e dge, or a radial pa ttern (Figure 5.4b). Conditions of accessibility for the set of edges, (lnL! for particular node s may vary substantially for diffelTllt co nfigur a tions. Some represe nt e conomi es of building; others, like 111(' minim a l spanning tree represent economies of effort in use with cosls ()f lise associated wi th the edges. We can analyse the graph to reved I lie s( '1 of link s that has a minimum cost of connecting all place s.

~ tj

Unoriented (e.g, railways)

tt Ltt

Oriented (e,g, rivers)

Unoriented with loops (e.g . highways)

(d) 2

'I ~

, ~~ ( I, )

. ,'[glln·

Number 01 co nneclions al a node

~

2

2~

) SrC1nn ing truLl

=

~

Travellino salesrnnrl

,~

~\~

2 N() (;,rC(J,1

S.4 S" 'Il(' tYI'l''; "I' '_;"11,(' 1,.,11"11" , 1';1111'111' 1

2

I~

2

"""III" 'd 1',I;ll'iI,;, (1)) 11(,\1, ·,.,

Oriented with loops (e .g, highways)

"I "JlIII "

Spanning troe radial

(.,)

!"',,',

()' ;( ' llk(/

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' ,;1111('

'.1

I "I'

180

Topology

Networks and graphs

5.1.2 Properties of graphs Various measu re s ha ve been devised for revealing properties of particular graphs. In addition to counts of the number of vertices, edges and subgraphs, th ere are measures for properti es of connectivity th at help in an swe ring questions like:

1. What is th e relative level of accessibility of the different cities se rved by air routes ? 2. How will travel be affec ted if a new bridge is built over a hitherto un crossed estuary ? 3. How m any alternative w ays might I travel through a city street syste m from my home to workplace to avoid traffic jam s or road works existing on my usual route ') The measures relat e to particular elements , th e vertices or edges, or th e entire graph. Besides comparing graph s by their counts of edges, vertices and circuits; it is possible to dev ise deriv ed and nOflned meas ures as useful descriptive tools. The particular vertices may be co mpared by co unts o f the number of edges (total, or in or out if the se ar e different), often being referred to as the degree of a vertex. In the exa mple, Fi gure 5.4, for a skeletal highway system, it is clear there are major differences in accessibility to nodes. The conditions of conn ectivity for the whole network m ay be summarized by a count or average nodal deg ree, or may use measures based on edges, like the minimum or maximum number of edges fo r a given number of ve rti ces ; th e number o f edges relative to the maximum possible number ; the number of circuits relative to the maximum possible; the ratio of edges to vertices. Not only are there such descriptive measures for networks , but, as made apparent by the Swiss mathematician Euler a few centuries ago , graphs have some special properties, representable by a si ngle eq uation. For th e case of two-dimensional graphs, following the style o f Wil so n (1985) , we have:

V+F=E+S wh ere V and E indicate vertices and edges, and F denotes the spaces between a cycle of edges and 5 is the number , called th e Euler number o r ge nus (often denoted G), that provides numerical balan ce to the equation. This number also va ries depending on whether or not Ihe outside r egion around a graph or set of polygon s is counled ilS a face. As demonstrated in Figure 5.5<1, if it is, Ihcn S is 2: olilClwise il IS I. Th ese Euler (someli mes ca IIcd Fulcr-I'oi 1lc;lrc) equalities speci ry ;1 COIlS\;11l1

181

~7

V~4

F~2

E~4

5~2

7

I

V:6 E=5 P=2

F=2 S=1

/V+F~E+S/ (4+2~6 4+2 = 6

S~2

S= 1

where F is defined as inside and oUlside polygons

IV+F : E+S+

if F is defined as only the inside region (lace)

(b)

(a)

fS':/ /V+F=E+S/ if F is delined as only

(e l

V~4

E~5

F~2

5 : 1

4+2=6 5+1~6

the inside polygons

pi

6+2:8 5+2+ 1 =8

where P is defined as a SUbgraph and

5 is defined as 'he interior space

~

Iv

V=9

E=l1 5=1

F=3

+

F

~ F is defined as only

=E

+

9+3= 12 = 12

sl ( 11 + 1

the inside polygons

(d)

Figure 5.5 De monstration of th e Euler equality for the planar two dim e nsi o nal case . (a) Simple case of four connected vertices. (b) Example of s ubgrdphs. (c) The addition of one edge [or exist ing vertices. (d) An exa mple of a more complex graph .

rclation ship for a set of links on a plane surface. 'That is to say, as dcmonstrated in Figure 5 .5 , the number of vertices and fa ces IS dependen t o n the Ilumber of edges. Adding edges for non-existing ve rtice s add equal numbers of edges and vc rti ce s, thus balancing th e eq uations (Figure 5.5d). An addition al IlollCollnected edge (Figure 5.5b) will add two vertices and o ne edge and ()Ill' subgraph , again balancing the two sides of th e equality. Additional nlgcs for existing verti ces do not change th e count of vertices, but do add IIlOlC loops , as show n (Figure 5.5c); that is, on e extra circuit (face) for (' :I Lh ex ira edge, agaill providing balance to the eq uation. III dca ling with graph representations of natural or person-made plt \.' llolllena, we havc both elemental level entities, the vertex, and edge, 1)111 ;lis() several cO fllpound entities. The combination of edges and vnlicl' s givcs LI S Ihe graph, or network; the selection of certain edges 1lllHlllcl's :1 path . which lTlay Ih en be se lected as a rout e. A change o f , c li" 1l\;IY Ill'Cl' SS ;LIIC lI SC or 11 ,",\ r(l1llc Irnlll nile 111:)('1' 1(1 all0111<:J . 01 ;1, 11;tlllrally rc prese nted by a 11\"1 dl:IIII:I,I'" '. \,Sll ·lll . 1\11 IlI'il'lIl1'd path j., :III ,1r
Topology

182

specified origin to a specified destination vertex. A path may be non oriented, reflecting an ordered set of connections, without specifying origin or destination ends. If the graph is more than a tree, then different paths may exist for any pair of vertices depending on thc number of circuits (loops) , which reflects how generally connected the graph is. An oriented cycle, a path from a vertex to itself, is made up by a set of edges arranged in sequence. Consequently, in working with paths, the topological property of sequence (of edges) is necessary, along with the property of connection. Moreover, there are many diffcrent contexts for laying out paths between nodes. As demonstrated in Chapter 17, we may place paths over terrain to avoid steep gradients, to circumvent hostile land types like swamps or lakes, or to take advantage of relative costs of different forms of transportation. In a graph sense, the nodes representing end points may be fixed, but a practical solution may requirc that a particular e dge be subdivided to recognize points standing for the facets of land of particular terrain slope, land cover or freight rate, respectively. Note, too, that the space within the set of edges making a circuit is not regarded as an object with particular attributes of interest. Identified for purposes of mathematical completeness , as in the case of the Euler equation, the area element can serve a useful role; but for the application of graphs to real world phenomena, the vertices and edges arc the entities of concern. For example , with reference to Figure 5.6, we attach no significance to what lies within the triangles between the air rOLltes, in contrast to the case of discretized space for land use. For graphs, DISCRETIZED SPACE

GRAPH 1 __

,,

• GR

(b)

(a)

•

•

3

GR

•

•

Land use

Air routes } Number of flights

FOR

WH

3

•

•

WH

,,

WH

FOR GR

} Type of land use

Number of destinations

Figure 5.6 Graphs, areas, and attrihlltes . (a) Attributes lor Ilodes and links of a graph. (h) Attrihlllcs for polygoll 'paces

Graphs and areas

183

attributes can be associated with edges, to produce weighted graphs, for example one showing the number of planes travelling on particular routes, and the time taken to travel each route. Vertices may have measures like the total number of air passengers, and both edges and vertices may have spatial properties derived from the graph components as illustrated earlier, and again in section 5.3.2. In the history of spatial information systems the two-dimensional units were given more prominence at first in comparison to one-dimensional entities, reflecting applications oriented to natural resources and thematic mapping for population data. More recently, spatial analysis involving networks and the interest by public utility companies in computer based inventories has led to a greater attention to line phenomena. Graphs not only are valuable tools for representing structure in physical entities, but are also used for dealing with connections among other phenomena or abstract entities. Patterns of interpersonal communication among people can be modelled as networks. The intellectual building blocks for conceptual models are often shown by graphs (see Cha pter 9); connections among apparently disparate pieces of information
5.2 GRAPHS AND AREAS I{ecal\ that contiguity of polygons is another topological property, and rcc(J1\ that one type of discretization of space produces a complete set of polygonal entities without holes or overlaps. Here we turn our attention I() concepts and tools that treat the spaces within networks as well as the verlices and edges . A set of polygons representing, say, the three lOllntries of Great Britain, comprising vertices, edges and spaces, can be Ilc,\tecl as a planar topological map, provided that there are no (lvcrlapping regiolls , and that there are no line crossings that do not pi odllce vertices.

:' .1.1 Digital line graphs

hi il.~ !lI'CP:II:llioll 01 digilal enco(\ings of topographic maps the United "'; 1:lll'S (il'ol(»).,ic,d Survey dcvised ill Ihl' l:tlc 1l)70s thc concept of the dil:ilallim' graph (1)/ ,(;). TilL' hydl'ol())',ic:d, 1I:IIlsportali()ll, land use and 111111'1 ph"n
184

Topology

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Graphs and areas

and later to support cross-refe re nces to demographic statistics for city blocks a nd other units, this topological e ncod ing , using a n unde rlying graph theory concept, focused on the city stree t segme nt as the principa l graph e dge. Streets, tra nsportation features and other ma p e le me nts were broken into straight segments for which address ra nges, a nd identifiers for areas on left a nd right were encoded, as we ll as the nod es at beginning and end of the line segme nts (Figure 5.8 b). Both of these pract ical applications used co ncepts from the mathe matical fie ld of topology. Geometrically speaking, this pe rspect ive treats th e polygons as connected cells, like in a bee's hon eyco mb. Most primitively, the set of cells is a tessellation of (irregular) triangles , where each cell has only three edges. In the natural world, where we represent phenomena as discrete units such as soil types, we usually have area e ntiti es with more than three edges, and generally we do not encounter vertices with more than three edges converging. For exa mpl e, for the conterminous states of the USA there is only one vertex where there are more than three edges - the Four Corners point where Arizon a, New Mexico, Utah and Colorado touch (Figure 5.9), and where man y people

F'igure 5.7 The concept of th e Digital line grap h. (a) Different lypes of area l spat ial unit. (b) One in stance of a line graph. (Based on material in US Geological Survey, 1986 .)

Node- Extra node in curved street

area objects, as Figure 5. 7a shows. Several kinds of area phenomenon were recognized :

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1. Polygo nal units of data , such as ad mini strative areas. 2. The area forming a background to po int or lin ea r objects. 3. The background a rea for polygon spatial units. The pol ygo ns were see n as a set of boundary edges; and lines were viewed as having beginning a nd ending ve rtices, called nodes in the co ntext of topology (Figure 5. 7b). Polygons have single specia l refere nce points associated with them, not necessarily contain e d within the boundary of the polygon , acting as a sun-agate, or polygon seed, to which attribute data for the areal units may be associated (Figure 5.7b). Individual point features were seen as degenerate nod es , that is, lines of ze ro length. Somewhat earlier, the Census Bureau of the USA had used concepts from the field of topology to organize the data for the st ree t segments of urban areas, via the line segment or D I !VI E st ructure as used for th e creation of digital files, officially callcd thc GI3F/DIME (Gcographic Base File/ Du al Independent Map Encoding) SySklll (l'igllrc )X). Dcsigllcd 10 help produce mars for licld CllllIlH.: rators ;tlld Il'l ;t ll' ;tddrc sscs to sl ree ls ,

185

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Graphs and areas

Top ology

187

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Alaska and Hawaii

(001 drawn 10 scale)

Figure 5.9 The example of th e states of the USA.

U nited States. We also cite the differe nt but often used te rminology ;id op te d by th at organization for a topo logical represe ntation of a pl a na r se t of polygo ns . That is, for a comple te tesse ll a tion , we have a group of live conditions for the vertices, edges, and areal cells (Figure 5.lOa):

seem to wish to ha ve photographs taken of th e mselves in a n unflatte ring spread-eagled pose. Incid e ntall y, the map of the fifty states of the USA ca n be viewed as a graph oT conn ecti o ns of bo undaries , but has the special cases of several pieces separat e d by water (Alaska, H awa ii) and a ho le for the District of Co lumbia (Figure 5.9) . A las ka and H awaii may be conceived of as be ing co nnected via telecommuni cations, but are o th erwise not ph ysically part of the graph ; whereas the feder a l government territory , the District of Columbia. is a po lygo n which does not have three corners - o nl y two states , Virginia an d Maryland, touch the District of Columbia . While we can observe adj ace nt po lygo ns in thi s case, the situation does not completely meet the form a l requireme nts for a tesse llation o r to pological map , and is m o re correctly a connected graph.

I. Every ONE-CELL has two ZERO-CELLS.

Every ONE-CELL has two TWO- CELLS.

\. I ':ve ry TWO- CE LL is s urrollnded by a cycle of ONE-CELLS and ZE RO-CE LLS. I . I ':very ZE RO-CELL is surround e d by a cycle of ONE- CE LLS a nd TWO- CE LLS.

". There ,lre no int e rsec tions th a t a rc not Z E RO-CELLS .

Th e le lO-, on c or two-cell s correspo nd to the geometric e ntities of 1'(lllli S, lines and ;lreas, but ha ve definitions particular to th e co ntext of \' \ISll' llCC ill ;l tessdl;ltion . Thu s. iso lated points will still be points, wl\('r,'; ls poillt s tll;l( ;II C t()pological jllnction ~ are zero-ce lls (nod es) . I ) 1.'(,O IlIl~TI,' d ()J llilr,'l;lt ed lill c; lr i'catllrcs , ;tl so re presc nt a

5.2.2 Topological consistency Fo r planar graphs, recallin g that ea ch c dgc bas two vertices and separalcs two voids. wc now prese nt the con ce pts uscd by the Census Bureau or th e f

188

Topology

Graphs and areas 189

special condition of the duality of zero- and two-cells for the edges. The third and fourth conditions refer to the completeness of the mixture of entities: a cycle around every vertex should e ncounter a success ion of edges and spaces; a cycle around a polygon should consist of a series of alternating edges and nodes (Figure S.lOb). The fifth condition means th a t the geometry of the set of polygons does not contradict the topology: if two lin es cross, then there should be a one-cell defined. Otherwise, a non-planar condition exists, there is an erro r in position of one or more lines , or there is incorrect topological data e ncoding. This subject of error detection is developed fully shortly. Not all features on cartographic ma ps are topological. The curvature of lines, the shape of polygons, and the labe ls for places are not. More over, several different maps may have the same topological structure. A s Figure 5.3 demonstrates, sha pe can vary, and the absolutc, but not relative, placements of points can differ. The properti es of rclati ve position are said to be invariant under change (deformation) in the base. For exa mple, the writing on an inflated balloon alters shape as the balloon is stre tched and twi sted , but still retains its pa ttern. Lengths and angles can change, but four e lements must not be altered:

1. 2. 3. 4.

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These properties arc referred to as the map properties that are invariant und er continuous deformation of a surface , the map basc. Notice that tearing and puncturing can alter some of these in . particular cases , but stretching will not do so. This state of affairs proves quite valuable for map transformations of certain kinds and for substantiating checks on the integrity of data. The topology may not necessari ly be logically inte rnally consistent. For instance, an edge may be missing for two ve rtices, or there may be no occupied area within three edges , as in, by analogy , a badly treated umbrella that has a panel of cloth missing. We defi ne a topologically consistent map (or database) as having all spatial entities projected upon a plane surface, with no freestanding fe a tures, and having complete topology. That is, the five rules identified above are obeyed. Complete ness of inclusion means that there are no iso lated non-connected points and that all lines are parts of boundaries of polygons. The completelless of incidence means that lines interse ct only al points ,Issoci,ilcd wilh the ends of lines. These cOllsistency condilions m;IY he see ll ;IS cOllSlr
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Figure S.lJ Topological inconsistency. (a) Data for three one-cells. (b) Graphs for three one-cells. (c) Only two-cell data a re used . (d) Ambiguous situa tion. (e) Unambiguous data.

A few e xampl es demonstrate the Occurrence of inconsistency or ilillbiguity in a topological structure (Figure 5.] 1). Consider the case of Iwo polygons, I and II, apparently touching at one common edge, and /t;, ving two other topological one-cells, as recorded by the data table (l:igure 5. [la). Each of the lines has two nodes and two polygons , Illcluding the exterior region , O. But if we had only zero-cell inform a tion I hc gnlph would consist of three arcs as shown, or all coincident (Figure '). llb); if we had only two-cell information , then th e next diagra m results. Working with only polygon ooundary lin es again produces ambiguity, as "("curs even iI we li se both node and polygon information (Figure S.lId). We have lorgollcn thai a Iwo-cell has ;1 millimulll of three sides and three vnlin's. h)/' Ill,: l'x;\11lpk, Illl' cOlllpkk ,,' 101 d;lI
•

190

Topology

Error identification

191

certain kinds of spatial relationship implied by the m a p. Of what zero-, one- or two-dimen sional elements do es the map consist? Which areas d o particular Jines se p arate? Which points terminate which lin es? For particular vertices, which lin es are incident, and for particular areas, which lines bound ? Is an area completely bounde d by a set of lin es, and is the set of lines and areas for a particular point complete? What is the set of areas adjoining a given a rea? Creating a topo logica l s tructure from geometrical inform a tion is not that simple in practice. Imagine tracing map fe a tures as part of a digitizing process in which the way individual elements are connected is not recognized . Today most spatial entity oriented software takes thi s 'spaghe tti ' of lin es and builds a topolog ica l structure from it. Conse quently, the software must know where lines cross in order to establish vertices, must create all the separate edges, mu st th e n asse mble ordered sets of edges to make regions, a nd must assoc iate edges with vertices. The topological information is deduced from the met ri c data in this process, or it can otherwise be provided by the crcator. In either case, e rrors can anse.

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I

I

I

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+ +

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~~t (unintentional line crossing)

Badly placed reference pOint

/ OverShoot

Duplicated arc or missing node? Undershoot. no matching node

Missing line

Figure 5.12 Some conditions for digitized maps.

Some geometric conditions are: 1. A nod e is missing or misplaced. 2. An edge is miss ing or misplace d. 5.3 ERROR IDENTIFICATION

J. An edge has a bad shape or too many (or too few) points On it; or coordinates are missing or incorrect. 4. A node has more than one position.

Various tools have been devise d for checking digital c ncod ings repre senting pl a nar enforced surfaces. The US Census Burea u has a series of edits to assist it in identifying errors in its 1990 TIGER file s (and for the 1980 DiME files). Some map conditions are best seen and corrected manually, but some can be more e asily dealt with by computer routin es.

Some topological co nditions are: I . Unconnected e dges exist.

2. A polygon has a gap between two edges, that is it is not close d.

.I . Duplicate arcs are prese nt.

4. A polygo n has more th a n one or no refe rence point associated with it. S. A node has only one or two e dges, rath e r than at least three.

I). A polygon may be missing.

5.3.1 Possible conditions in digital maps A summary of possible conditions that may result from preparing digitized versions of maps is provided by Figure 5.12. Typically , hand digitizing produces some over- and undershoots of lin es beyond other lines or a map edge; some edges or nodes may be completely omitted or duplicated. and line shapes may be bad. Whe ther or not so me conditiolls ar e errors depends on external requirem e nts or validity. For exa mple, an overshoot co uld rep rese nt a cul-de-sac in a street sys tem, or an unconnected edge might be a railway line that d()e s n()t join all Y other line .

Such topological conditions may arise from imperfect geometry, as when there is a line undershoot, with or without a node identified on a Ilcilrby line; or when a nod e is misplaced. At times, unless ex te rnal data :Irt.: available for checking, it may not be easy to validate whether a ('()IlLiilion is inhercntly gcollletrical or IOf1ological , or inde ed, whether or 11l)1 it rcprcs(' llls ,Ill t.:rror (Figllre S 12) . Topol()gic;J/ crrors will have to be IIIlUlVl'rt.:d hy ,1I1lollLllcd s p;lti;iI (i:Jr;1 ("(Iililll~ pr()l"Cdlll"e s or visual ma p III~Pl'cli()II.';: ()I Ilh'Y 111:IY 1>1' (k;dt willi \'i :1 oilln l\'("iJlliqlll'S associated wilh d:lt:1il;I',( , 111:11):1.1'.,'111('111 SY.\ IVIlIS, :1 1"1'1< ' Illl ('11:11)(("1" 1.'1 .

•

192

Topology

Error identification

193

For a planar , complete surface, a topologica l map will have at least three edges intersecting at topological junctions . Nodes with only one or two edges (sometimes referred to as dangling and pseudo-nodes respectively) represent singly connected edges or artificial points neces sary for allowing duplicate arcs between the same pair of points (as neede d, for example, for the boundaries of the District of Columbia), or for dividing one line based on different attributes. But they may represent a failure to close a polygon by reason of an undershoot in digitizing. Gen e rally speaking, it is relatively easy to detect errors for areal spati a l units; but sometimes it is not straightforward. For instance, a missing polygon in a cadastre may not be cl ea rly visible from a plotted map th a t shows boundaries for all neighbouring parcel s of land .. Line or point entity phenomena are not so easily dealt with when detecting errors other than by graphical means. For example, powe r transmission lines may cross each other without any connection , or they may meet at substations. A planar graph will need to recogni ze intersection s in both cases, and differentiate betwee n the two - but this is not possible directly using only spatial data. An indirect method is to providc additional information that says even though a topological junction exists, it is not possible to travel from all edges to all others at the intersection. Individual point entiti es, not connected tLl other ph e nomena as part of graphs or topological maps, may be positioned on thc wrong side of line entities or polygon s, possibly because data come from differe nt sources. For example, an obstacle to air navigation , like a radio tower, on a map may be placed on the wrong side of a major visible ground feature like a highway, cau sing problems for Aight via visual rules. Topology and geometry can be in conAict not only at line intersections, but for other spatial relationships and at thc boundaries of adjacent tiles or map sheets.

5.3.2 Some procedures for checking for errors The Euler equation pro vides an initial simple tool for checking, re lying as it does on only counts of the basic e le ments in a graph or map (Figure 5.14). If the Euler number test fails , then a graph is not topologically internally consistent. On the other hand, therc is no guarantec that the reverse is not the caSe because there can be compensating mi scounts. For example, on e fewe r node and one extra edge will bal,lIlCe in the equation (Figure 5.13f). In the case of Figure :'i. IJil compared with S.lla , " verlex is not recorded in a data tahle ; or,
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Figure 5.13 Use of the Euler equality for detecling topOlogical inconsistencies. (a) Original graph and data. (b) One vertex nol encoded. (c) Double encoding for one vertex. (d) Onc edge is nol connected. (e ) One edge is missing. (f) One edge and o ne vertex an: missing.

dangling node. In Figure 5.13e the missing line reduces the edge count by one, although the number of vertices and regions is correctly recorded. Various possible situations are shown systematically as Figure 5.14 for

hoth polygons and graphs. Notice th e variation in the S number for the

differe nt cases; of course, if S is incorrectly specified, then the equality is

in;lerently incorrect, even before other quantities are examined.

Topological editing is based on the five rules presented on page 187. A

surface can be represented as smooth and continuous if there are positive

,Inswers to a series of questions. Is everyone-cell incident with two zero

cells ? Is everyone cell incident with two two-cells? Is every two-cell

hounc\ed, and is the neighbourhood of a zero-cell equivalent to a disc?

l',lIticular routines for topological editing make use of Some of these

properties identir-icd earlier. For example, there should be a sequence of

vdges ,1I1d nodes ,Iround a polygon, and any labelled node should Occur 'lilly Iwin: . Th e assessmenl, excluding the line intersecting which is done ('OIl1)lll(;llioll;dly, rl.'qllires a mixture of encoded data and checking IOlllilll'S. 1'01' L'X;lIllpk , usillg lille segilleni ellcoding (as with the DIME .s lnll"lllll·) 11ll" firsl Iwo ljlll'sli()IIS :1It.' ;IIISW,·!"l·d hy rccords which contain d;II;, f()1 lit,' f'''"1 :llId 1<1 lIod,'s ;lIld "'fl "lid ri}'.ltilliliHI sidc polygolls,

Topology

194 (b) Exam ples for graphs

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all ow ing the dual incidence to be determined, whil e the third and fo urth questions are addressed by algorithms, as now described, A polygon chain r outine can detect certain kinds of errors, and a nod e chain cycle routine can detect others (Thompson el at" 1982), Thus an error in o ri entat ion of lin es ca n be detected by a node chaining edit which recognizes the to- and from-designations (Figure s,lsa) - the cycle of one- and two-cells does not produce a chain of repea led polygon labe ls. The third record should be CB, not BC, ref-lecling th e inad vertent switchin g of the direction of the lin e, A node ed it usin g both o ri entation and adjace nt polygon data operates as show n in Figure s,15b, The edges at a specifi c node are co ll ected , and then gathered so that o ri ent ation is co rrect; this is done by writing out duplicates for the o ri ginal three lin es, with the from- and to-node designat io ns re versed, If there is a chain through the pol ygon labe ls. th en no error is detected, but if one polygon is badly labe lled , or if another , non-adjace nt polygon has the same label , then the chain will not work co rr ectl y, Thus, additional chec ks are needed beyond both the co untin g to see if each edge has two nodes and the ad j ace nt polygons because it is possible to make labe llin g errors, A polygon chaining procedure works in a sim il ar way to the node chaining edit , undert ak in g a cycl e ar o und particular polygons lookin g at the successio n of one- and two-cells, This procedure looks for mi ss ing or redundant nodes; the nod e chainin g proced ure look s for mi ss ing or redundant lines, The United States Census Bureau uses a ser ies o f such automated checks on its digital boundary data files (Marx and S
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,Irea unit has a co mplete boundary, An intersect ion detection routine is used to look for intersections that are co rrectly coded as topological junction s. Using an exa mpl e from th e US Census Burea u (Figure 5,16), we emphasize that there can be different conditons and, th erefo re, different solutions , represe nted by o ne instan ce, The geometric and topological resoluti ons produce different maps, Of course, the structural chec ks app ly to o nly internal consiste ncy, While eve ryt hing appears to be fine with th e logical co nditions of the 1',I
Topology

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tolerance ite m fo r the acceptable distance apart of the different represe nta ti o ns of the same node. Line overshoots ca n be a djusted or not , accordin g to a d a ngle tolerance. And a t est applied to all features, a fuzzy tol e ra nce, works globally, so metimes with und esira ble o r unforesee n results, on the distance betwee n points, whether they are nodes o r lin e vertices. As such , thi s tolerance , by the value assigned to the smallest distance acceptable, establishes the resolution for entity o ri e nted spatial geo metric d a ta . Tolerances ca n be esta bli she d for o ther spa ti a l con texts ; we have presented th e m here in some d e tail only for irreg ul a r tessellations.

H r:: ' 5 td C

,\

GI-;OMETRI C/ To rOLUClCfl l .

Polygons and areas

Reca lling our earlier discu ssion (Chapter 3) , we know that po lygo n e ntities a re e nco unte red in dealing with m a ny natura l a nd man-made ph e no m ena. Polygo n d a ta m ay rep resent naturally occu rrin g e ntiti es like Likes or forest s ta nds or person-made e ntiti es like administrative unit s. or they Illay be discre ti za tions of continuously varying phe nom e na . A rea lInits are most often asse mble d from line data , a nd , less commonly , made lroill point data. In th ese cases th e y ge ne ra lly ta k e th e form of irregular tesse lla ti o ns with no hierarc hica l ordering o f polygons a t different levels. ;\ s such, they d o not dea l ex pli c itl y with the property of containment. Fo r

C G

C

cJ

(C)

Figure 5.16 Re solution of a topol ogica l and geomctrical inco nsis tency. (a) Original conditions. (b) Geometric rcso lution of the pro bl em. (c) Topological resolutio n of th e problem. Nodes are here called points; seg refe rs to lines. (From Marx a nd Saalfeld, re printed with th e pe rmission of the US Bureau of the Census, Washing ton. DC, USA.)

The design and ran ge o f functions ava ilable in software for ca pturin g spati a l data for entities seen as points , lin es a nd areas are impo rt a nt for aiding th e efficient production o f e rror-free databases, a problem which will be re-exa min e d thro ugh the conce pts o f integrity constraints in Chapter 15. Systems with a utomated procedures based on to po log ica l concepts a re like ly to be mo re effective th a n those that rely solely on co mparisons of original a nd digitized maps. Furthermore, because o f hum a n imperfe ctions in the ha nd-digitizing process, it is helpful to have to ols that a re ad a ptable to the different levels of e rror that c a n be tol e ra ted. Fo r exa mple, undershoots m easured in hundre dths of milli metres , but noneth e less ca usin g a common node to be re prese nte d by two diffe re nt coordinate values, may be dealt with in the di g iti z ing process by a utomatic snapping o f the nodes toge th e r. The varying tolerances th a t may be invo ke d as editing co ntrols in buildin g spa ti a l databases address co nditi o ns for the different spatial units (Figure 5.17). Line genera li z ing ma y acce pt diffe renccs be tween ve rtices grea ter than a ce rtain distance, or angle , depending Oil the weeding procedure used . Nodes Gill he sll;lppcd togethcr, :Iccordillg 10 ; lIlOlhcr

·I~

L illO weeding

(.Ippllos 10 points 11 1( Ikim,IIp linos)

·I~

Node matching (applies 10 nod es only)

·I~

Dangle truncation (applies to linwi only)

1,'i/:III'(' 5.17 '<';" 111" "1' .111 :11 101\. ·1: 111 ' ·' ... .

•

Fuzzy tOlerance (applies to any coordinates]

Topology

199

Polygons and areas

198

pedagogical purposes , we treat two-dimensional units in two sections, based on whether or not containment is a factor.

5.4.1 Types of areal spatial unit We separate area phenomena that have no overlap, no holes and no isolated pieces from the additional fe atures of enclaves and exclaves. Referring to Figure 5.18, we define a simple polygon as a single unit of space bound ed by three or more lines, generally having an irregular shape, and containing no holes. More generally, we can use th e con ce pt of a cell to refer to any unit of space, regular or irregular , bounded by three of more lines and not containing holes. A polygon is complex, sometimes referred to as an area, if it contains one or more holes. A region is made up of two or more separate polygo ns (or cells) , not necessarily connected although , in the field of geography, the term region has other defmitions. Any point position is in only one space (polygon), unless it lies on a boundary or in an overlap of two or more polygon s or areas. It is not just a matter of recognizing that encl aves and exclaves can occur, it is also a matter of co mpleteness geographically, Many countries or parts of countries have land separated partially or comple tely by water. How does one establish contiguity across interve ning water? Is Alas ka adiacent to the state of Washington? As Figure 5,19 suggests , conceptually there are diffe rent boundaries for administrative and physical representa tions, In the case of political redi stricting in Maryla nd in 1981, for in stance (Thompson and Slocum, J982), it was necessary to assign a s pecial code to contiguity ' relationships across the Chesapeake Bay, Topologica l and geographic views can be in conflict at times, Much of the difficulty associated with polygons is practical, not conceptual. That is to say , it may be necessary to intersect many polygons

D~ Simple polygon (no holes or exclaves)

Complex polygon (has enclaves) [ = area I

Non'co nnected polygon

(has exclaves)

[= region J

Figure 5,18 Definiti()n s luI' arl'al SI1;lli;!I ullits,

Boundaries for natural features

Physical and person made boundaries

Official districts ignoring water separations

Figure 5,19 Different versions for e ncoding areal phenomena,

to create homoge neou s units; it may be desirable to combine several similar units to make a more-or-Iess homogeneous la rger unit, or they may have to be created from point or lin e data, For example, land parce ls ;Ile both a basic geometric two-dimensional unit and a useful statistical 1I1liL yet homogeneou s terrain units are made up from se parate polygonal IInits for a mixture of soil conditions, slope, land use and other variables, Wh ile the basic areal spatial units for which attribute data e xist may s;ltisfy many user requireme nts , they may not be th e most basic spalial rl'cord ing unit. So we define additional objects: a basic statistical unit is l ile smallest unit for which attribute data of a given kind are collected. T h is may be the housing unit , as in the case of d e mogra phic data, or field (1Pint samples aggregated to represent areas of space, As is the case with '"'IlSUS data. re porting units may not be the same as data collection units II I llI'der to protect individual privacy, A n() ther fundamental m a tter is the extent to which there is a one-to tllll rc\1' (I V\, " :qllll/I)', ;111 '; 1:-- (l:igIIlL' 'i.2(k),

200

Topology

Polygons and areas

201

OJJ[QJQ]

A, 8, C = 1, 2, 3 =

polygons \ } primitive spatial units ~

Three separate regions

Disjoint region

Overlapping regions (A and B) (A)

A: 1

A: 1 A:2

8:2

8:2

8:2

C :3

A :3

8:3

A: 1

County A State X

( a) One-to·one relationship (b) One-to-many relationship

(C)

\

State

%

-+-+

Many-Io·one relationship

(a)

Figure 5.20 Primitive spatial units and polygons. (a) Three areal units. (b) Disconnected region. (c) Two overlapping regions. (I3ased on material in Kirby , and used with the permission of the Cambridge University Press.)

River County boundary Railroad

--+--t ( b)

Y

County 8 State X

Election precinct Election district Legislative district (and election district) (c)

Figure 5.21 Examples of inclusion relations. (a) Line coincidences for rive r/boundary and railroad/boundary. (b) Common boundaries for e lection precinct and district, and for election precinct, election district, and legislative district. (c) City split by a county boundary.

5-4.2 Containment and coincidence

republics of the USSR, and then into progressively smaller administrative territories. Generally, but not always, there is a one-to-many rela(ions hip of the larger unit to the smaller constituent pieces. Sometimes excepl iOlls occur, as in the case of the city of Takoma Park near Washington DC _ Ihis city lies in two counties of the State of Maryland (Figurc 5.21c). On the other hand, the Vatican City in Rome is not in any hierarchy at al/. While a dministrative hierarchies with only a few exceptions have a propcrty of nesting, that is boundaries are coincident (Figure 5.2] b), Ihere are perso n-made spatial units that are dominated by overlapping, t hat is have only partial ordering. The market areas of consumer products, for example, regional newspapers, can have major areas of overlap, meaning that the one small piece of territory falls into more than ()lle larger unit.

We have just described a scenario about the property of containment, that is the inclusion of one spatial entity within another. A point may be in different areas; lines may be in different areas or parts of boundaries of different polygons; and areas produced by overlap lie within different polygons. Such situations arise not only for administrative or othcr forms of territoriality, but also for natural phenomena and for linear spatial units (Figure 5.21). A boundary of a county may also be a river or a boundary for electoral constituencies. An edge comprising part of a land parcel boundary may be part of several real features, possibly a river or road, or the parcel itself. Different spatial situations that arise for sets of spatial units that are completely or partially ordered by inclusion are:

These examples demonstrate that some types of spatial organization :lle not easily handled in a straightforward way by only the concept of a pl:lne surface. A single map showing non-nested many-to-one relationships l()()ks rather messy (newspaper circulation); several maps are needed (one I'()r each newspaper), or non-map representations can be used. Hier ;IIchieal tree graphs are frequently used for such situations (Figure 5.22). They can show the nverlilp (lr nesting quite we ll , although, in practice , (':! S(' S with 11l;IIlY IInih might require very large pieces of paper to show all r(·I:lliollsitips. lIne, ror Silllplicily, the p(llyg(lils are huilt up from regular Ill()('k:--, 11111 irn'!'-ul
1. There is a perfect hierarchy of small units combining at several levels to make larger units, with no overlaps and no holes, as in the case of election precincts combining to create legislative districts. 2. There is incomplete nesting in which, for at least one level of spatial subdivisions, some units are split. 3. Areal or linear subdivisions may be put into different larger units for different purposes. Man-made areal spatial units frequently exist in a vertically structured organization. Political units , for ex ample, on a worldwide scIie, IH:gin with nations, which are subdivided into ,I lew I;lIge territmies like tite

!\.~

:1

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•

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Topol ogy

202 1 2 E 7 11 12 1f. ,., 21 22 2E 2' 3132 3E 37 Ca)

3 4 8 9 13 14 1819 23 24 28 29 33 34 38 39

5 10 15 20 25 30 35 40

Ce)

c .. re ions

A

B

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__ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17.

•

A A ,I ii l

d'>too"

1 2 3 , 5 6 7 8 9 10 11 12 13 "

15 16 17

Pol ygo ns and areas

203

le ve ls. I t is poss ible to mode l the co ntainm e nt re la tio nships, oth e rwise re ferr ed to as own e rs hip orderin g , so th a t th e lo west leve l has a ll individual pi ece s, the small es t spa tial uni t, o r two -cell (in the te rmino logy of the US Cen su s Burea u). For exampl e , as Fig ure 5.22 sh o ws, two -cell s a re co mbin e d to crea te zo nes, regions a nd dis tri c ts at a highe r lev e l (Fig ure 5 .22 b). The re is no orde ring fo r di stricts , reg io ns and zones, so th e seque nce a t th e hig he r le ve l does no t m a ttte r. In this case, th e basic units a re establi s hed prior to combination s bein g m a de . If a subdivision process is ta ken, t he n the units a t the lo west le ve l will be produced by id e ntifying the units prod uced by ove rl ay ing th e district , reg io n and zone maps, as s hown in Fi gure 5 .22c. Th ese , th e minimum number o f units necessary a nd sufficient fo r pro ducin g the seve ra l co mbin ation s fo r di s tricts , re gion s , and zo nes, a rc ente re d in the tree diagra m as a n e w layer as s hown in Fi g ure 5.22d. So (afte r Visv a lingham el af., 1(86). we dis tingui s h am o ng (Fi g ure

). 23): , I. Largest common spatial units, resultin g from ove rlaps . 2. Th e sm a ll est units (cell s ) which are co mbin ed to produce th e COlll11Hlll spati a l units . .) . Th e spati a l primitive s (po ints a nd line s) whi c h mak e up th e SllliJik' sl are al units . -i. The spa tial units fo r whi c h d a ta are record ed (th e hasic spatial rccordin g unit).

zon es

II II

1 2 3 4 5 6 789 10 11 12 13 14 15 16 17 ..

CD) 2

Cd)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

18

Figure 5.22 Ownership ordering for area l spatial unit ,. (a ) Areal cells. (b) Ownership trees fo r three se ts or te rritori es. (c) COi11lllon geographic units. (d ) Own e rship order illg. (Base d Oil Me ixlcr ;1I1t! Saalfe ld , 19R5.)

I n th eory, th e re can be a s m a ny ce ll s as the re a re larges t common , pillial units . In laye re d data bases the se will be th e sam e if th e lowes t le vel co mm o n spa ti a l units a re pro duced by polygo n overl ay intersection s . II I theory , too, th e cell s es tabli sh the resoluti o n, the minimum sca le, a t whieh th e phe no men a a re to be o bse rve d. In prac tice th e re m a y he ("u lnoll1ies if th e d a ta an a lys is wo rk s at th e level o f comm o n spatia l units I:lIhcr than cell s. Fo r ex a mple , fo r its 1990 ce ns us , th e US C e nsus Burea u 11 : 1.~ abo ut ten million two cell s; it is h o ped th a t th e re will be man y fewer l:irges t co mmon spati a l units. I\ t the same time , a building bl ock struc ture may fa cilita te d a ta .1) ~J!, leg ; lti()n. Population record ed for indi vidu a l ho u se holds, as illustra ted IlI l' Ilollse boat s in Fi gure ).23 , is o rdin a ril y groupe d to data re portin g \lll ilS 10 prc'serve rrivacy of info rmation. Othe r them a tic att r ibutes, o r l ' V " llI~ , stich ilS Irafric accidcnt s, thund c rs torm s, or ho micid es associa ted Wllh ililsic s pati;d Ic(o nlill g IIllits , ca ll he aggre gatcd to specia l purpose ' '' II l"~ lik ,' slrvc' ls III It'icpllllll c s<'lvin ' :1I ,' :IS. 1'11(" 11 <' :1111)('111 o f lill L'i '" I"" illlm ·:, 1':II;Jilt- h Ihill of ;lle al units. The idl l', ( '~ 1 ("llllllllllll ~I'illl ; d IIllil ,<,; illI· Ih l':"· lViwl ' · Ih c·le is cO lll c idcllce, as for

205

Polygo ns and area s

Topology

204

'"----------

/.1':r---;7 ~7 ..... L_...

1

Rivers

Countries

~[J

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6 5

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o

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I: I 1_- "7-_":_--) I! :

I

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f 0r ho les fl Dundar y ror ellC lo:.ur es

(a)

1: ' ----J:

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1--_ =-=-/ '-<

I

'-

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,...

:1 ___ __ -':u _____ ______ _ _ __ _____

(b)

Spatial primitive units for geometry

Tree

level in tree

Type of boundary

The plane 'A' on which

boundaries exist

(c)

I

_

A

B

c

for river

:

-

Smallest areal spatial units = cells

u

i~, ~I::I -;;]1

I ~" :'11 I _ -'''\

,i

,

1_ _ _

H~"OO~ prioviding data

r

,---------(

./'

Thematic spatial data 0 recording unit

I

~

6

/\

Figure 5.23 Types of spatial unit acco rding to o wnership. (a) Two e xamples of sepa rate th e matic units . (b) Combination of polygo ns for two themes. (c) Geometric a nd th e ma tic primitives.

/

Enclosure Hole

)

1/\ J

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0

2

Enclosure

3

Hole

4

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/\ 10

(c )

th e pa rt of the northern boundary of Durham County , England, wh ich is also a rive r, and the limit of a n electoral district. There m a y no t be any one-dimensional cells, but the lines will ce rtainly be made up of other entities, either primitive segments, o r represented by mathematical e quations. Coincidence encoding for line objects can be a va luable check on data integrity, avoiding inconsi ste ncies that te nd to a rise with separa te digitizings , although there are limitations . For e xampl e, a boundary line b e tween nation s may re main stahle even as the rive r on which it is base d

Figure 5.24 Containme nt e ncodin g usin g boundaries. (a) A set of primiti ve a reas . (b) The boundaries of th e primitive a reas . (c) The tree represe nting th e co ntainm e nt hie rarchy o f th e boundaries . (Reprin ted from Kirby , 1989 with th e permissio n o f the Cambridge University Press.) II <.' c diagra m. This approach re quires a data table for holding th e 1)()lIndary data as well as tables, discussed earlier, for th e lin es , nodes, :1I1l1 intermedi a te points. Thi s orga nizati o n contrasts with others (see a lso ,«('lion ').')) in which there is no e xpli cit coding of th e boundaries, but in w hi( 'iI other de vices, slich as sp ecial tabulated numerical codes for lines , ," :Irlilieial eO llneclors call e d stalk arcs joining separated polygons are 1I',('d, M() r~: ()v<.'r, Ilwr" 1\I or I\'IIIP"1 ,lilli" :II\(I pr('(' ipil:lli o n condition."

changes over time . Coincidences of two-dimensional spatial units may be approached from the perspective of boundaries as well as th e interi ors of the areal units, e specially to be able to mo re effecti vely deal with hol e s in side spatia l units (Kirby et al., 1989) . That is , the geomctric and phe nomcnal parts are separated , a nd only the bound a ry line information is used. With refere nce to Fi gure 5.24, th e enclaves and island s require boundarie s for both insides and outsides , as shown by the do uhle lines around the e ntities . The containment hier ,lreh y or I)(HIIHLirie s e, 111 he see ll rnlill the

"I

•

206

Topology

boundaries are a rbitrar y. In oth e r cases boundaries ma y re prese nt the equi-probab ility o f membership of more than o ne area unit , so tha t th e concept of a m e mbe rship functi o n is mo re a ppropri a te th a n discre ti zation. If d a ta o rga niza tio n is co nceived as separa te maps or data laye rs of polygons for diffe re nt th e mes , the n it is d esira ble to be a ware of co ndition s for both boundari es a nd inte riors. M a ny combinations ex ist for th ese e le ments, rega rding the topo logica l pro pe rties of touching or ove rl ap. For boundaries two area units may be compl e te ly separa te , touch at one point, parti a lly overlap , or be iden ti ca l (perfect ove rlap). For inte riors , one unit may be ins id e th e other, o r the reverse, o r th e re m ay be overlap regard ing th e one or m o re ho le s . It is importa nt to no te th a t exclusion is only a se ma nti c property: unlike inclu sion it ca nn o t be computed. Diffe re nt lju estions as to containment ma y need to be answered. W e m ay wis h to kn o w th e po lygo ns includ ed in a gi ve n pi ece of arbitrarily d e limited territ o ry ; or which polygo ns overl ap with a ny piece of th a t territory. For exampl e , respectivcly, which reg io ns of E uro pe fall e ntire ly in th e circulation a rea of th e Manchesler Guardian newspaper , or whi ch regions include a portion of th a t circulation a rea ? Other queries may involve s pecificat ion of size of a reas, o r coin cid e nce ot bo und ar ies. Diffe re nt d a ta a rrange me nts a nd content may be necessa ry to adequately d ea l with all possibilities.

5.5 DATA FOR SPATIAL RELATIONSHIPS In working with digita l databases it is impo rta nt to recog ni ze tha t so me geometric or topological prope rties may not be directly e ncod ed , having to be derived from o ther data that a re. It is thus important to identify what to po logy is explicit, th at is to say, in a practica l se nse of loo king a t tab les o f numbe rs, wh a t is directl y recorded. Som e forms of s pati a l data organization do not e ncod e a ny topo logy prope rti es. The polygon model (appropri ate fo r lines a nd point fea tures too) h as tabul at io ns for the coordinates of each a rea unit taken sepa ra te ly (Figure 5.25a). M o re ge nera lly , this ap proac h has a o nc-lo-on e corres pondence between a map feature, po int , lin e or area. a nd a record in a t able. Entities are geom e trica ll y d efi ned . but no spatial relation s hips arc recorded. So to uchin g po lygon s o r co nn ec te d lines (Ire rcve al e d only in visual form wh e n th e e ntiti es arc dr
Data fo r sp atial relation ships

207

,

y •

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10 A

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y

15 17 7

14 6

•

17 14 10

POlygon boundary segments

9

-

POints comprising polygon (diagra m c) (one exa mple Only)

(Nole : Th e polygons in Figure bare separaled lor purposes of illuslrat ing the concepl of discrete units.)

(b )

Coordinates y X

Poinl S

A Po lygon 10

B

2

~<~.@~7\

,, 5, ' ,,

~

(b )

I

A

14 14 6

POlygon liSI 01poinls A B

1 2 3 8 9 I 9

(c )

Figure 5.25 Spatial data mode ls without ex plici t t:odi ng of topology prope rti es. (a) D ata for three po lygons. (b) Polygo n Ill odd concept. (c) Po ints list mode l.

lecord adjacency or co nnectivity directly , but it avo id s dupli cation o f feature coo rdin ates (Figure 5.25b) . Th e polygon or lin e units are made up hy indica ting which po ints a re ne cessa ry ; th e table for points has only u)o rdin a te d ata. Such non-topological data models a re th ose in which only position a l 1I l/'o rmati o n is reco rded . The recog niti o n o f individual spa tial units as ~e p"rate unconn ected e le m e nts, characteri ze d th e ea rl y data organization 1m digital cartographic data. This co ncept was impl e me nted in th e po lygo n mod e l o f th e SYMAP softwa re , deve lo pe d prim a rily at Harva rd l illi v<.: rsity in th e la te l %Os, the first co mpute r them a tic m a ppin g pack age II) gain in( c rnation al fame. I I is tori Gtily . the firs t topo log ica lly explicit data mod e l was the line structure (Figure 5.26a), devised a nd used by th e US Census I1l1l c; \lI lor th e 1()7 () ce ns us . Thi s DIME format had a basic reco rd for line ll W l'.I ClltS 0 1 lro Il I- nOlI<: • to-llodc . left rolygon, a nd right polygon , a nd ,I,sociakd illlollll ;tlioll lik <.: (il c coo rdin
Topology

208

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1 10

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7 Lis t of links

(a)

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3

4

1 0 1 - 1 0 1 -

0 0

2

1

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Connectivity matrix

2 1, 2,

3, 4,

5 5

I

List of nodes

CONTIGUITY Chain

III

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b

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A

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5

Node or X,Y Poinl

o

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dl A

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Right area C A B

List of links

Area

A B

A

B C

C

Area

Touches

1 0

A B C

B, C, 0 A A,D

0

Contiguity matrix

List of areas

(b)

Polygon lell righll

Figure 5.27 Recording dat a for connectivity and contiguil y, (a) Connectivity, (b) Co nti guity, Chain

a b

Node from to III III

IV I

Polygon left right

B

I

C B

Note: The 2 or 3 points for a parlicular node

(at leasl theoreticall y) have Ihe sa me

coordinates, They are shown differently lor

purposes of demonslraling Ihe data elem enls ,

(a)

b

e

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Chain

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a

f

2C

b

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From To Link node node

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II II II II

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A

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7

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b

5~

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5

~.~

1f____ 8~

/

IV

209

CONNECTIVITY

{- ~

h

Data for spatial relationships

I (b )

Figure 5.26 Explicit coding of topology , (a) Line segment encoding concepl. (b) Chain form a t concept.

recognize d as nodes as well as the street intersections, Moreove r, only line segment features were directly a ddressed . Other entities , particularly polygons, had to he assembled (often quite laboriously) from the data encoded in th e line segment records. Encompassing additional properties or s patial units directly genera lly involves a structure of seve ral tables of data. A chain model (Figure 5.26b), so me times known as the POLYVRT structure, used by the ODYSSEY system of Harvard Unive rsity from about tcn years ago , first explicitly recognizes intermediate points alon g lin es, as well as thc line end points. Data are stored in different tables for the chains, as for the line segments in the DIME model, with froill - node, to-node , lelt polygon, right polygon ; but unlik e the DIMI': ;lrr;ln gclllcllt, ;dso for lIode,s, fur

coordinates; for points, for coordin a tes; for polygons , and for Ilw CIl;lills making up the bound a ries . The US Geological Survey's original digilal line graph concept was encoded in the form of a set of tables , for nodes, with (x, y) coordinates; for areas , with coo rdinates for centroids; and lines, with sta rting and ending nodes, and coordinates for them, and le ft ,1I1d right-hand side areas. Connectivity (Figure 5 .27) is readily handled by a link-node list, showing from- and to-node labels, and a separate table for the coordin a tes of the lin e s. But it can a lso be dea lt with by a connectivity Illat ri x and separate node coordin a te list. Contiguity is directly addressed hy a table of lin es with data items for left and right polygons, and a se parate table for the coordinates for the lines (Figure 5.27); or by a IOlltiguity matrix or list, with separate data for coordinates for the polygons , either as polygons, or sets of lines . Lists may be be tte r than 111;ltrices for some purposes, such as for connections in skel e tal graphs :111\\ contiguity relation s for contour lines. In both cases, the matrices W()ldd he very sparse , containing many null entries. However, some ;II illlillclric oper;ltiolls are performed more eas ily on matrices, better than 1111 li ,s\.s, Are;1 k ;lllIn' s Cill he rL' prc;cI11I 'd Ily polygon- lill e topology , using I.illit- :, 101 II\(' lil\( '.s 111;11 IlLlkc IIpllle Jl<.lh',I'.'lII.S, ;111(1 Ihl' coordin;lles for the 1111<" " SpCli;d ('ode s III;l V Ill' Ilsl,'d ,,, lIi " llIily 1111' OriCIl\atioll of the

Top ology

210

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Figure 5.28 R epresenting polygo ns. (a) Islands encoded via a list of links . (b) Hierarchy show n by a label. (c) Polygo ns and islands represe nted by boundaries. (d) Encodin g for common spa ti al units.

topology edges, a nd th e existe nce of inside a nd o utsid e bound a ri es if enclaves exist (Figure 5.28a). Node fea tures ca n be tab ulated as se parate records with d ata ite ms for line s that co nve rge, as we ll as coordin a tes . A s discussed ea rlier , co nta inment has to be dealt with som ewh at differently if it is to be dealt w ith directl y, ra th er th a n indirect ly , via spec ial codes for bounda ries. A boundary o ri e nte d st ru ct ure. shown as Figure .'l .2Kc, adds records fo r bo unda ries to those for e d ges. nod es
Figure 5.29 Topologica l e ncod ing for lin ear entities. (a) Link li s" o ri ented grap hs. (b) Link and node lists for u norie nte d grap hs.

ror

that can be built up in a nested fashion from common units is unde rtake n h y labe lling of diffe re nt leve l units, po rtrayed in simplified fo rm as F igure .'l.2Kb. And the hi e ra rc hica l approach used by the Census Burea u , as no{c d ea rli er via Figure 5.22, by co ntrast is h a ndle d as in Figure 5.28d. Severa l poss ibiliti es ex ist , too , for graph e ntiti es. Each link in a ri ve r Ill' sewer o r highway ne twork co uld have a data ite m of the link before it ;llld th e link
212

Topology

Other considerations

5.6 SOME OTHER CONSIDERATIONS AND SUMMARY

213

only one, or not all, properties explicitly; and not all situations can be easily handled topologically. For example, discontinuous surfaces or non planar graphs require special treatments. It may arise that geometry is required at one level, and topology in another, for a layered concept for a database. For example, when several polygon type layers are overlaid, . only the combined topology is retained, not the new geometry. And , from a database point of view, topological data are redundant because they can be derived from the geometric data.

The structural properties of graphs and planar surfaces exist, minus any information for the attributes of the point, line, area or volume objects, Or for their positions. Depending on purpose, the geometric data may be necessary, the topological properties may be required, or, for many reasons, both are essential. The metric information is used for:

1. Drawing maps of edges, nodes, and surfaces. 2. Differentiating between duplicate edges. 3. Locating intermediate points on edges. 4. Measuring distances , perimeters, shapes, or areas and volumes. 5. Determining positions of entities.

The topological data model may have disadvantages for cartographic production. Having to deal with nodes and links can disrupt smoothness in digitizing activities, and the short lines associated with a graph structure (compared to unbroken features in a map model) can increase map drawing time because of many starts and stops. Unfortunately, a misleading distinction is sometimes made between cartographic databases and topological spatial databases implying that topology properties are not necessary for the former. On the contrary, thematic mapping, especially polygon shading, can be more effective if the boundary lines between adjacent polygons with similar attribute values arc known. And some topological properties are valuable for editing cartographic data used only for base mapping, even for line features. In any event, it is better to distinguish between geometric ;lIld topological spatial data rather than cartographic and topological databases. In a geometric (pseudo-cartographic) model, entities can be manipulated completely singly, not being assembled from separate disconnected pieces. For example, a long highway , such as Route 1, can be identified ;lIld stored as a single linear feature rather than requiring retrieval of a long set of separate line pieces created by the topological junctions at illtersections with other highways. The topological model does not allow (iircct reference to logical complex objects - these must be identified at

I he stage of data input.

The requirements for certain categories of retrieval of data for spatial entities are met only by positional information, although this does not have to be based on Euclidean geometry. There are several ways of spatial addressing as covered in both this and subsequent chapters. Topological information is valuable for accomplishing tasks involving relative positions of entities or for dealing with spatial relationships, although it is not necessary because visual representations , maps, contain data to establish those properties. Topology must be encoded explicitly, however, to permit numerical spatial data problem solving and reasoning, or, at times , to facilitate some data editing operations. The topological data are used for: I. Undertaking analyses that need information for connectivity of line elements. 2. Undertaking procedures that utilize data for sequences of line clements. 3. Determining the character of adjacent area units. 4. Automating some error detection procedures. 5. Facilitating searches in neighbourhoods. 6. Facilitating data retrieval for associated elements. 7. Making data updates more feasible by separating the metric information from the structural. 8. Reducing limitations inherent in the use of only metrical information. 9. Facilitating aggregation of primitive spatial units into larger units. 10. Providing a basis for automation in map matching and transformations. 11. Facilitating spatial reasoning.

Topologically structured data can be manipulated algebraically and ;Irithmetical'y to produce valuable derived statistics for revealing graph or 'illrLlCe propcrtics, or facilitating error detection. Many people favour the topol()~y model because it directly addresses entities in a phenomeno IDl!,ic;t/ sense, tltat is, as to what exists. At the same time, it does this illiperfcc1ly because the visiblc objccts may be polygons and networks, 1;lllter Ih;lIl the encoded primitives of points, lines and areas. However, III;IIIY sp;tli;t/ rel;ltiolisilips C;lnllOt hc (easily) computed and must be "Illicilly rlT()l(kd . S()llle rcl;lIiolis ('<111 he cOlllputed from just metric lIiI(IIIII;llioll, lik,' tlisLlllcl'
The two kinds of spatial data may not he completely rccordcd in spatial information system databases. Not all lopologic;t1 properties 11I
•

214

Bibliography

TOpology

215

international Geographical Union Co mmi ss ion on G eog raphical Data Sensing and Process ing , pp. 411--436. Frank , Andre w U. and Bruce Palmer. 1986. Formal methods for th e accurate d e finition of some fundamental term s in physical geography. Proceedings of th e Second International Symposium on Spatial Dala Handling, S eallie. Williams ville , New York. USA: Interna tional Geographical Union Commission on Geographical Data Sensing and Process ing, pp. 583-599. Haggett , Peter and Richard 1. Chorley. 1972 . Network Analysis in Geography. London, UK: Edward Arnold. Harary, Fra nk. 1969. Graph Theory. Reading, Massa chu sse tts , USA: Addison Wesley . Ka inz , Wolfga ng. 1989. Order, topology and me tric in G IS. Technical papers of

In summary, returning to the general subject of spatial relationships , geographers have known and worked with many concepts like contain ment , neighbours, and distances for a very long time . Today, various scholars in several fields are developing a more formal basis for understanding spatial relationships using the three categories of:

1. Metric relations (distance , direction). 2. Topological relation s (con nection, orientation, contiguity). 3. Order relations (inclusion). Metric relationships are based on (usually Euclidean) geometric measure ments for entities located in a two- or three-dimensional void. Topological spatial relationships are based on the notion of a continuous surface of space and conce pts of boundari es, and interiors of cells. While inclusion properti es can be dealt with topologically , they are directly addressed by th e order relations via concepts like partially ordered sets. Moreover, a strong foundation can be provided by use of the principles developed in various branch es of mathematics , geometry, graph theory and topology. While we have taken an intuitive a pproach to these matte rs , nonetheless th e practical implementations for topologically structure d spa tial data draw upon the formalisms of those fields to great advantage. Formal descriptions of phenomena based on mathematics a re very valuable for clearly, concisely and consistently de fining spatial

th e Annual Convention of the American Society for Photogrammetry and R emote Sensing/American Congress on Surveying and Mapping Annual Convention, Baltimore, Maryland, USA. vol. 4, 154- I(iO. Kain z. Wolfgang. 1990. Spatial relationships - topo logy vers us order. Proceedings of the Fo urth International Symposium on Spatial Data Hllndling. Z urich, Switzerland, pp. 8 14-819. Kirby, G. H ., M. Visvalingham and P. Wade. 1989. R ecog nition <\lId rcpll:scnl:l tion of a hi e rarchy of p o lygon s with holes . The COII/IJlIiI'r .!oUI'I/II/'I2( 1): 554-562. Mar x. Robert W. a nd Alan 1. Saalfeld. 1988. Pro g ram ~ for
relation ships.

5.7 BIBLIOGRAPHY For more extensive treatments of graph th e ory , the reader is referred to Harary, among olhers; while th e book by Haggett and C horley (1972) is a comprehe nsi ve revi e w, albeit dated , of network a nalyses for both social a nd physical phenomena.

Geographers, Baltimore, Maryland, USA. I\; ueker . Thomas K. a nd Nichol as R. Chrisman. 1985. Cartographic d a ta structures . Th e American Cartographer 2(1): 55-89. \aalfc\d . Alan. 19k7 . Stability of map topology and robu stness of m a p geometry .

Broome, Fre deri c k R. 1986. Mapping from a topologi ca lly encoded data base: the US Bureau of the Census example. Proceedings of the Auto Carlo Conference,

London. UK, vol. 1, pp. 402--411. Corbett, James P. 1979 . Topological Principles in Cartography. Technical Pape r No. 48 , US Bureau of th e Census. Washington, DC. USA : US Governllleni

I'roceedin gs of the Auto Ca rta 8 Conference, Baltimore, Maryland, USA, pp . 7i\-8(1. .-';:lllVY, .I c an ;IIHI Silllonne Sallvy. 1974. The Child's Discovery of Space. I·larlllond.o.;worlil, UK: Pcnguin. Sloclinl. Tnry 1\ .. I{ohnl .I . Il:lIli .o.;c h ;111(1 Derek Thompso n. 1984. The de sign and ;Ipplicalions or a c:lllo!',raphic d :lla \1;1.,,' r
Printing Office. Environmental Systems Research Institute. 1987 . A RCIIN 1'0 USI'I.\ (;l/ir/c. R edlands, California, USA: Environmental Syslelll s l{ csc;lrch Inslillilc . Fra nk , Andrew U. a nd Wern e r Kuhn. 1()X(1. C c il !',l
.'.: I .~ I 17(1. '1'11,>1111"",>11. Ikl ('"

•

:11\(1

T"IIY 1\. SI,)t'IIIII . I'il-:

I

.. \

!"'ll!'r:'1,hi,' illfoi'lilalioll .o.;yslclll

:nb

/CI/"J/CJ);Y

for political redi strictillg ill M:Il yl; lIld . !'ro("(,(,(!lIIgs. ( \!,!,lin ! ('-('(I,~IIiI )IIY Conferences (College Park, M:lrylil lld , lJSA) 'i: TJ-- ~n Tho mpson, Derek et al. 1YH3. Thc Ma ryland Reapportionl1lent Inio rillation System. Depa rtment o f Geography, University o f Maryland, College Park , Ma ryla nd , USA. US Burea u of th e Ce nsus. 1970. The DTME Geocoding System. Ce nsus Use Study Repo rt Number 4. Was hington, DC, USA: US Government Printing Office. US Geologica l Survey. 1986. Digital Line Graphs from 1 : 24,000-Scale Maps. National M ap ping Program T ec hnica l Instruc tio ns Data Users Guide 1. Reston, Virginia, USA : US Geological Survey. Visvalingham, M a hes, P. Wade and G. H. Kirby. 1986. Extraction of area topo logy from line geome try. Proceedings of the A uto Carta London Conference, pp. 156-165. Wern er, C hristian. 1985. Spatial Tra nsportation Modeling. Scientific Geography Series, vo l. 5. Beve rly Hills, California, USA: Sage Publicatio ns. White , Marvin S. 1984. Technical requ ire ments and stil ndards for il multi-purpose geographic data syste m . The American Cartographer 2( 1) : 15-26. Wil so n , Pete r R. 19H5. Euler formulas and geometric modelling. IEE E Transactions on CompUier Graphics an d Applications 5(8): 24-36.

6 Tessellations Regular and irregu lar cells, hierarchies Wong Lee prepared som ~ map overlay models to predict the elJecl of acid rain on natural vegetalion using JOO hectare square parcels of land . Carmen Adobe proposed 10 replace th e grid pattern of streets by a new fo rm with less dangerous inte rsections. Kiki VroomanoO' interpreted some LANDSAT images as parI of her sludy of v~gelalion change in tropical regions. Wendel C hrisloff saw se tll~ments in human space as sets of poinls within hexagons of differenl siz..es and sha pes. Eva Sanchez could not understand why or how th e marketing depa rtment of her company created postal zone boundaries artificially. Gco(rge) Control used a network of geoddic control points to eslablish surface elevations. And Olivous hko and Karinoushka. who plalonically lov ~ a dodecah~dron (or mayb~ a n octtree), go in search of the pyramid ' II address 003201320110230.

Th e th e me of pol ygo ns as spatial units is ex te nded in this c haptcr to the ge ne ral topic of tessc ll a tions, sets o f regul ar o r irregular pieces of space accoullting for all of that space. After presenting properties of geometrical figures, th e discussion turns to th e role of regul a r four-sid ed figures, in planar and hierarchical varieties. Irreg ular tesse ll a tions a re the n presented, espec ially the triangulated irregul ar netwo rks which ha ve several interestin g properties.

6.1 MOSAICS, TESSELLATIONS AND LATTICES Th e discretization of space, a to pic introduced in Chapter 2, takes many form s. Discretization is th e process of partitioning co ntinuous space, either lin es or areas, into pieces. This process is referred to as tiling in the case of two -dimensiona l spatial units, or segm e nting in the case of lin es. Mosaics or til es of zon es o f irregul ar and varied shape and size may be created via var io us processes. Origin all y conceived as polygons with four equal co rners, as implied by the Greek word tetara o r Latin word tessella , mosaics also can be made up of other regular geo me tric figures or a mixture of diffe re nt types. Mosa ics of quadrilaterals can be see n in

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combining photograpils ()f p;lpl'l IlLlp sheels to Illak c Ltrgcr , seamless, documents, or arithmetic graph paper with a square grid call be used to record the character of the earth repre se nted o n maps in large or small , consistently shaped and sized pi eces.

6.1.1 Tessellations

Land parcels

Thiessen polygons

IRREGULAR

Tessellations (sometimes in the computer graphics field ca lled meshes) , are se ts of connected discrete two-dim e nsional units (Figure 6.1). Tesse ll a tions may be regular o r irregular in geometry. A regular tessell a tion (for exa mple, grid squares) is a n (infinitely) re pea table pattern o f a regular po lygon (two-dim e nsional fi gure) or polyh edron (three-dimensional figure) . This cellular decompositi o n implies eve ry point in space is assigned to only one cell. An irregul a r tesse llation is an (infinitely) extending configura ti o n of polygon s or polyh e dra of varied shape and size. Irreg ular plan a r tessellation s may also be rega rded and represe nt ed as topo logical two-ce ll s. A tessellati on provides a way to deal with th e occupance of space , in contrast to dea ling with id e ntifiable entities lik e lakes or town s, However, some pe rson-made ph e nomena occur as tesse llations, for exa mple land ow ne rship parce ls. In practice, irregular tesse ll a tions are used in arcas such as:

r R4 R5 R6

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REGULAR

Figure 6.1 Reg ul a r a nd irregul ar tesseliatioJls.

For the context for this chapte r's discussion , we must remember thaI the und e rlying semanti c questions rega rding what is in (this) space or d for desc riptions of th e contents of space.

Zones for social, eco nomic, a nd demographic data Administrative or political units Surface modellin g using triangles Partitio ns of a large geographic da tabase Man-m ade phenomena lik e land parce ls Irregular sa mpling of a continuous spati a l distribution Wireframe mode lling of buildings Regular tesse llati o ns are encountered in: Image data from re mo te sensing Data compilations from maps by grid squares Organization o f map libra ri es Data generate d by photogra mmetric systems as lattices of points Uniform sa mpling of a continuous spatial di stribution

6.1.2 Lattices If we use point entities a rrangcd over space in a regular, perhaps sq uare or hexagon a l, pattern , we talk of lattices rather than til es (Figures 6.2a and b) . The la ttices may be see n as the inte rsec tions of th e g rid lines or as the centres of a set of squares (Figure 6.2c), Or, they may consist of a quinconcial patte rn made up of the points and middle for a se t of squares . Origin ally the word lattice refe rred to thin strips of wood o r metal, crossing diagonally (Figure 6.2d), a nd capable of forming different patterns or networks.

The former have th e advantage (discussed earli e r) of be ing ab le to be processed generally as topologically structured vector data. Howeve r , their crea tion may be base d on doubtful ass umptions for the discretiza tion process, or involve lengthy and cumbe rso me computations if diffe rent tesse llation s are overlaid , as di sc ussed in Cha pter 8.

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If th e qu esti o n , Wh a t is he re?, is a nswe red fro m th e fr a mewo rk of a tesse ll atio n (say a se t of squ ares) or a latti ce of points , the n d iffe re nt co nditio ns a rise. For th e regul ar units , th e attributes for a grid ce ll a re an average (o r othe r summ ary statisti c) of co nd iti o ns o ve r th a t space. If a pc.int fram ewo rk is used , the n the co nce pt of average may be in a pp ro pria te . T he zero-dim e nsio na l e ntity is best seen as a uniq ue positi o n a t whi ch some e ntity ca n be directl y o bserved . Ye t , the in sta nces in a se t of po ints may at tim es ca rry data fo r a s umm ati o n ove r a co ntinuous surface, as in the co nce pt of aggrega te access ibility space po te nti a ls. In thi s case, tho ugh, the re is no implica tion of a discre te a rea l unit be ing represe nted by a point ; th e data represent influe nces of a n aggrega ti o n of o th e r po ints in th e e ntire se t. Mo reove r , th e regul a r tesse ll atio ns may be see n as diffe re nt patte rns o f points. Th e patte rn s of the sets of po ints may cha nge via applicatio n of a linea r transfo rm ation , for exa mpl e , tiltin g th e y-ax is , just as a latti ce of wo od pieces ancho red by pins at th e crossings ca n be changed into diffe rent fo rm s by pushing o r pulling in one o r the other coordin ate ax is . Thus, sets of recta ngles o r p a ralle logra ms ca n be made fro m squ ares (Figure 6.2e) by re mapping th e centro id s of th e squares. A nd a hexagonal arra ngement of poi nts can become a squa re grid of po ints o r vice ve rsa. The re is not a o ne -to-one re la ti o nshi p between gri d squ a res a nd a square latti ce, fo r a squ a re mosaic ca n be changed into a set of equidistant po ints

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Figure 6.3 V ar ia ble reso luti o n a nd sca le . (a) Di ffe re nt sca les . (b) Diffe re nt sca les fo r hexago ns. (c) Varia ble reso lu tio n he xago ns. (d) V aryin g reso lut io ns at o ne scale fo r point da ta.

a~ra nged in a hex ago nal patte rn (or o th e r fo rm s), a lthough th ere is a dua lity of ze ro (l atti ce po ints) a nd two ce ll (tesse lla ti o n po lygo ns) units. 6.1 .3 Scale and resolution Th e basic spatia l un its of a tesse ll ation ca n va ry in size, shape, o ri e ntatio n a nd spacing . Also, fo r a give n theme, several tesse ll ati o ns ca n ex ist , re presentin g diffe re nt scales of obse rvati o n , o r agg regatio ns o r sub d ivisio ns fro m o ne initi al sca le o f o bservatio n (Figure 6,3) . Fo r regul a rl y sha ped two- dimensio nal units th e stru cturin g is a kin to a pyramid ; fo r irregul ar shapes , th e new la rger units produced by combining smalle r units , as in crea ting school distri cts from census tracts, will not have a shape consiste nt across th e diffe rent scales. Va rying scale is see n fo r la ttices in d iffe re nt po int de nsiti es pe r unit area. Th e basic unit may va ry in size for a give n th e me for a given scale. In

222

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this way the reeonlill)!, 1111;1 III;I V Ill' ltn Ill ;ilch tli~ ' s;/~' pi()pcni es or entities or the diversity or tlil' pliCIHllllell;! ill spacL'. Th l'se concepts ;Ire met in theoretica l formulatioll s like th e differe nt sizes or hexago ns for central place theory (Figures 6.3b and c) , a set of propositions whi ch, among other things, addresses the question of the size and shapc of market areas for services provided from towns. It also arises for practical app lications like surface modelling triangul a tions, as discus sed in section 6.6. Irregul a r tesse llations generally have implicitly variable spatial resolution (Figure 6.3c) , but regular tesse llation s with variable size units for one theme have to be produced from the smallest scale by subdivision, as is seen in the process of creating quadtrees by hierarchica l decomposition (section 6.3). Tesse ll a tions, mosaics a nd lattices are not only to be seen as spatial units for reco rding data, but also as devices for facilitating access to databases for co ntinuous space . We have already seen thi s use as a basis for space-filling curves. While any piece of the earth can conceivably be used for addressing and retr ieval , regular tilings have some advantages for this purpose as is demonstrated late r in this book.

6.2 lHE GEOMETRY OF REGULAR TESSELLATIONS Of th e many regul a r geometric figures, from 3 to N sides , the sq uare , triangle and hexagon a re those currently most likely to be generally e ncountered in spatial data contexts. Some , like the pentagon , appear in special circumstances. The squme has so me practical ad vantages historic ally, being the unit used in graph paper a nd mechanical printing devices . The hexago n has utility in theory and practice wh e re adj ace ncy is a n impo rtant property or where packing of geometric figures into space is important. The triangle has a property of being the most primitive po lygon, that is, it cannot be furthe r subdivided into a different figure, and it can rea dily ha ve varying shape based on angles or length of sides conditions. The regular polygons all are characterized by identical lengths of sides, a nd equ a l angles subtend ed by lines from vertices to the centre of the figure ; and only the triangle , sq uare and hexagon regul a r figure s, unlik e circles, ca n fill a plane co mplete ly. Comparisons of the value of diffe rent tiling units a re best made on the basis of certain standard properties. In examining the geometry o f regul a r tilings we will co nsid er: shape adjacency

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Isoceles 2 triangle (4.8]

Nurroer of

at a comer

Squares [2

Equilateral 3 triangles [6 I comers

i\

Hexagons [1

6

1

Equilateral (2~ triangles

1

Number of instances

Figure 6.4 Shape, adjacency, and connectivity for some regular tessella tions .

connectivity

orientation

self-similarity

decomposa bility

packing properties

Shape is the geometry of the figure's different number of sides, albeit that particular instances of a particular type of figure may have different shapes, where there is no tight restriction on th e length o f sides o r thc angles, as for tria ngles. Shape for regular figures is identified by the number of e dges at a vertex and the number of vertices (Figure 6.4). For example, the hexagon has six in stances of three edges meet ing at corners , and the isosceles triangle has one ve rtex with four edges, and two vertices with e ight e dges converging. Adjacency refers to the conditions of to uching for repeated instances of the basic figure. The contiguity may be at the sid es (edges) , or corners

I,,,

224

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(vertices), or both (I,'iglll\' (, ,I) Tltl." ,,,ilu;III()11 Vdli\"., 1'11)111 Iii,' ,',Ilics oilly for hexagons, to both v<.:rllL'CS ;lml cdges 1'01' tri;lnglcs dil(l squ;lre,,, or rectangles. Metric measurements <.:an be used to distinguish difl'crenl figures - the adjacency distancc is the distance between the centroid of one tile and a neighbour, and the adjacency number is the number of different adjacency distances. The hexagon has one such distance, six in number (one for each equidistant neighbour). The square has two, and the equilateral triangle has three different distances. Related to adjacency, but considering only the conditions at the intersections, for some applications, like transportation, the hexagon has the connectivity advantages of having junctions at only three edges, unlike the four for a regular grid, as in an American-style city block street system (Figure 6.4). For most purposes, though, the polygon space adjacency is of more interest that the intersection connectivity patterns. The tile orientation, defined as the base relative to coordinate axes, is uniform for the hexagon and square, but not so for the triangle as not all instclllces of triangles in a tessellation can have the base parallel to either the x or y coordinate axes (Figure 6.5). Tiles with the same orientation can be transformed into each other by translations, that is, by shifting along the x- and y-axes, without necessitating rotation or reflection. If the basic figure shape is similar to that produced by aggregations to a higher level, then the property of self-similarity exists (Figure 6.5). This is

E

Opus quadratum

Self·similarity

Set of circles

IT I I I T- I I [II Opus lateritum (a)

Hexagon rosette (b)

Interlocking hexagons (c)

Figure 6.5 Self-similarity and packing. (a) Rectangles. (b) Triangles and hexagons. (c) Packing of hexagons.

(". '111111'111" 11/11'1:11/,1/ 11""oI,lI, iflllll',

Ll:'

lIlIlilllill'd 1'01' regui;lr liglll l'S wlti(·1t ILIV\' <.:;I<.:h side Oil ;111 IlIlillll<.:ly slraight lin<.: <':olllpos<.:d <.:ntlrely ()r ed ge ,.,. ThiS condition is seen clearly in the case of the tiles t()uching al cortlcrs (opus quadratum) as opposed to the bricklayer's scheme (o/JUS /aterilum) with only horizontal unbroken lines (Figure 6.5a). However, only two tessellation types, the square and equilatral triangle, with identical side lengths, and two tessellation figures with unequal side lengths, the isosceles and the 30-60 degrees right angle triangles, meet this condition. It is also of interest to examine changing conditions if the figures are decomposed into smaller units or combined to larger units. The square and equilateral triangle can be decomposed into units of the same shape, but the hexagon cannot; it can be split into six equilateral triangles. Indeed, as Figure 6.5b demonstrates, the aggregation of seven hexagons produces a rosette, while the aggregation of equilateral triangles produces polygons if pairs are taken, and higher-level triangles if four basic triangles are combined. While the hexagon is not decomposable with self-similarity, it can be used at different scales, rather as a lattice pattern, without nesting (Figure 6.3b). Only the square <.:an be de composed into smaller or assembled into larger tiles of the same shape and orientation, and be compatible with the orthogonal axes or C;lrle~i;\Il coordinate systems. The overall pattern of how the repeated regular figures pack space is an important condition in engineering for properties like stress, and is also important for living organisms in their effectiveness in the use of space. A structurally stable situation (Figure 6.5c) is demonstrated by the relationships of a set of circles at right angles, at oblique angles representing rotation of one of the two coordinate axes, and then displacement to oblique axes where the vertical spacing is half the horizontal. Essentially, the figures are locked together at this position. If squares or hexagons are substituted for circles, the same principle is at work, but it can be shown that the hexagon is more economical in the occupance of space. For example, there are gaps left if circles are used, but not if hexagons are used. This packing condition appears in the fields of geography and regional economics in central place theory, a set of propositions governing patterns of settlements and trade areas; in the insect world it is represented by the honeycombs of bees. It is not surprising, though, on the basis of this review of properties, that the sq uare figure dominates the world of regular tessellations for spatial information systems for two- or three-dimensional entities (except if a single sphere is to be partitioned). The square has simple and valuable conditions of equality of sides, decomposability, and stability for

226 I

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o rientation and aggreg; llioll , So I il l' sq llare appca rs ill lil l' rOrill 01 ;Irra ys of picture ele me nts, grid ce ll <1;1\;1 recording, 1,luiccs lor c iL' vd tioll data , and electronic d a ta capturc a nd di splays. The hcxago n is morc co mplex although it has nice properti es of edge to uching and eco nomy of packing space. In theory it h as had ma ny uses , but not in practice. H owever , rece ntly it has made a n appearance as th e data unit fo r one particul a r spatial database. Th e tri angle has adv a ntages of being th e most primitive po lygon, and does appear in several practical contexts, although in the form of irregul a r rather than regular tri a ngles , as discussed in section 6.6.

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6.3 FIXED SPATIAL RESOLUTIONS: REGULAR CELL GRIDS

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The commonly e ncounte red organization of data into an array of regular cell s, almost always, but not necessa rily , square. is rega rded as practically straightforward , but conceptually does no t ex plicity deal with quest ions of the type , Where do 1 find thi s particul a r e ntity? Th e bas ic unit o f the square ha ndles both loca tion, geographic or a rbitrary , and the attributes of the locatio n, Often known as raste r encoding o f earth features, this ce ll a rray approach contrasts with th e vector object-oriented e ncodin gs, having some practical advantages, as well as some limitations.

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6.3.1 Data encoding

Some fund ame ntal considerations to be examin ed rega rding grid-cell data organization a re: l. 2. 3. 4. 5. 6. 7. 8.

The ex istence o f an a priori fixed reso lution. Th e method of determining what attribute is in a ce ll. Th e le ve l of measurement used for th e a ttributes . Limitatio ns in recording precise ly point o r line featur es. The abse nce of reco rding coordinates for feat ures. The lack of ex plicit topo logy, The flexibility in und e rt aking man y operation s easily, A m atch with a field vi ew o f the world .

Clearly , the focu s on space rather than on features can produce problems in some cases (Figure 6,6). If a high degree o f precision is required for capturing data for point or lin e features, a very fine resolution may be required, at the possible expe nse o f havin g a very la rge amount of data. If it is not required, th e n the e ntire grid cell is used to

(d) tvb ryland

S tat~

Coo rdinate Gnd

l lll t-'S

Figure 6.6 Grid cell data e ncod ing. (a) The MAGI system grid for encoding. (b) Example of pol ygo nal da ta. (c) Grid overlay . (d) Resolutio n. The encoding unit is th e 91.8 acres block made from the 2,000 feet grid lin es on state base maps. (Ta ken from M aryland Departme nt of State Plan ning, 1981 , Figures I and 3, Reprinted with the permi ss io n of the Ma ryland Office of Planning, Baltimo re, Maryland, USA.)

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record that a pi ece (If Ili g hw:l y 0 1 1)()lIlld:lry ex ists ill p:11I (II tile ce ll , possibly ca using problelm UpOIl recollstru ction of <111 elltire line k~
Linear features

~ (a l

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je'l(

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19

FEATURES

A AI B E B AM Ale B A Cit C C/I

Areal features

III II III 1111 11/11 III III

PRESENCE

A

B

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A

. .

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.

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.

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c

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··· 0

DOMINANCE

A

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. .

cI

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.

. . .

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CELL CENTRE

Figure 6.7 Rules for e ncoding. (a) D ata for area l and linear fea tures. (b ) D omina nce-e ncodin g o f o nly th e principal a ttribute value . (c) Prese nce or im po rtanc e~ft e n recorded as multi va lued attri bute s. (d) Based on middle of cell (s hown onl y fo r aeri al fea tures) .

"ll')

I iX('(1 ' 1'.I li,I /II ·\II /lIl illll '

(a)

Vector

(e)

200 metre grid cells

(b )

(d)

100 metre grid cells

400 metre grid cells

Figure 6.8 Some de fi cie nces of grid cell data representation. Co n ti guity a nd co nnectivity rela ti o nships may be a ltered , sometimes substanti all y. (a) Vector re presentati o n. (b) E ncoding as o ne hundred me tre squ are ce ll s. (e) Two hundred metre squ are cells. (d ) Four hundred me tre square cell s.

Avoidin g the issu e o f domin a nce or impo rta nce is done by recording d a ta ~ a lu es fo r only grid cell ce ntres , but much info rma tio n ma y be omitted in this way (Figure 6.7d). If thi s procedure is used , then th e da ta are in e ffect a la tti ce , and no judge me nts can be co rrectly made about the occupa nce of space . Lattice re presenta tions are be tter for continuously variable quantiti es lik e temperature or precipita ti o n measure me nts, o r e levations a bove or below se a level. Generally , grid-ce ll re presenta ti o n is used for categorical d ata, either bin ary or multino mial, although it is possibl e to record scalar quantities (int e rva l and ra tio scales) in cells. Indeed , the re have been e xamples, as in the UK and Swede n , o f recording d e mographic data by grid cells. The

230

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practica lity of doing this rests lIIH)11 tile ilv;lil,lbility or disagglegated daLI, that is for individual houses 01 ' lddresses, which can he aec ulllu(;ltcd ror the regular tessellation units. At the same time, it is interesting to note that lattice representations are valuable for data from remote sensing for some purposes, rather th a n th e ra sterized version based on pixel cl assifica tion techniques. Each coordinate position on the earth can have scalar values for the multi-dim e nsional spectral signature, usin g several spectral bands. The e ncoding of point and line feature data has limitations , s ubj ect to th e grid-cell size used, in th e ability to show th e lo cations and shapes precisely. B o undaries may appear rather blocky , a po int may occur near the edge of a cell , some data may disappear, and lin e features that are close togeth e r m a y be shown as touching (Figure 6.8). Thus, some as pe ct of spatial relationships may be violated. Of co urse. th e precisi o n of repres e ntatio n is a function of the size of the unit in the regul ar tesse llation . A fine spatial resolution, defined as th e minimum di sta nce that can be recorded , and me as ured fr o m cell ce ntre to centre, produces m a ny more grid cells with which to work . Whether th e trade-off be tween . prec ision and amount of data is worth it depe nd s on various factors, not 'the least import an t of which is purpose.

6.3.2 Spatial properties Spatial relation ships are implicit in the data, but with only a few exceptions do the software systems for grid ce ll data allow direct handling of rel a tion ships bet wee n entities. Metrica l distance relations along orthogonal axes are readily derived, a ltho ugh not without limitations, fro m the differences betwee n x a nd y coordinates (row and column values), and other distances are co mputed as the hypotenuse of right angled triangles. While axial and diagonal lines ca n be measured by cell increments without loss of precision, this is not so for oth e r hypoten uses. Thus, perimeter me as ures for areal objects or le ngth s for lin ear features m ay be seve rely erroneous. Shapes can be approximated by observing orientations of lines of cells re lative to those of ne ighbours, assuming th a t the reso lution is good enough to produce su ffi cie nt unambiguous data . Metrical spatial properties can be readily dealt with by square cell tessellations, bearing in mind th e reso luti o n implied by th e cell size. Distances are harde r to obtain for hex agon s, because th e centres of the cells are arranged symmetrically alo ng three coordinate axes, but properties based on neig hbours are much easier because th e re a re no corner adjacencies. Distances and adj acencies are m ore aw kward for sets

/11, ·,1,./),1/1.11 " '" .,,111/11111"

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.d· ('(Illil,tlu;li Iri ;lllgiL" s, IJ,"(';III~l' till'y ILlve six IriallgiL"s I\JIIl'iling at \'(J rlll"l"S ;11)( 1 I InCl' di/Tell'lIt dI S!; llll'l' v; liu es for eac h triangle ccntre. (;I il[lil pmperties or connectivity h,Jv e to be ex pli citly record-::d for n ' II s l"Ontaining nodes, and details of direct io n of edges lea ving nodes (ncl'css ilry in order to create a precise graph) must also be e ncod e d . ( 'ollsequelltly, thc desirable situation is a combination of cells and vector represe ntation of point and lin e e leme nts within cells. Details for points i1lld lines within a grid ceU could be retained by particular coding Icc hniqu es, or by linking the grid cell to other dat a by refe ren ce to the gr id cell identifier. Some techniques have been d evised to record illformation ex plicitly in a grid framework for line and polygon e ntities. Thu s, the directional coding prese nte d in section 4.5 is some tim es used i"or boundary line or linear feature re prese ntation. T o polog ical properties are only indirectly dealt with by th e regular lessellation forms, because they do not tre at phenomena as entities. Of course, th e re are properties of adjacency and connectivity a mong cells, hut qualitative spatial re lati o nships for entities must be derived from Ihe data that are recorded . Th e topological properties of nOll-COil Ilecl ivi! y may be vio lated, such as the apparent joining of highways or touching or illner and outer bound aries o f po lygons. The graph propcrty or eX lsll'IlL"l' of hol es ca n be determin e d by calculating the distance to Ihe Ilearesl edge of a polygon (a set of ce lls) .

Ce ll encoding forces patte rn to be infer red from th e data values for a dja cent cells, but it does facilitate the process as it directly establishes neighbours and ne ighbo urh oods. If cells are square, th e n eight touching pairs ex ist in the immediate ring aro und a cell; other situations may ta ke in other cells, such as the neighbours of a block of four cells. Yet, unlike for to'p ological encoding, the id en tifi ca tion of neighbouring regular ce lls provides littl e information unl ess the attributes of th e cells are considered, such as in crea ting zon es by finding cells of a par ticular attribute value. H o wev e r, many operations, log ica l or arithmetic, are quite straight forward when working with grid cell data. Area meas ures simply involve counting sq uare s , not co mputational geometry; distances involve s ub tracting, a nd can be readily ascertaine d even 011 di agonalS. Aggregations can be made within prox imity zones, and map overlay modelling using attribute information essentially does nothing more than compare att ribute values for identical grid cells, as demonstrated in more detail in section 8.3 . At the sa me tim e, vector geom e try may have an advantage in that mueh of the space is void: for regular tessellations , the space is ' full'.

232

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Layer 10

Zone

At1ribule Colis value row col.

105 110 100 85 65 5S "0 JO

85 95 95 90 75 55 40 30

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16 5 19 8

2

A

65 70 75 80 75 65 45 35

(a)

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55 60 70 65 50 45

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45 50 50 55 55 55

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115--1

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I

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(a)

(bJ

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16

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Layer Row Column Attribute

10 10 10 value

5

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B New value is 97

105

1 2

1

5

2

8

fIJ II

Property

Smoothed value is

I 00

(105 + 110 + 110 + 100 + 95) / 5

=

104

(dJ

(b)

Geodetic control

Figure 6.9 Surface representations. (a) Values for grid square centres or lattice points. (b) Mesh-based interpolation. (c) Contour lines. Cd) Smoothing lattice or cell data.

6.3.3 Surface modelling from lattices The elevation dimension is readily m
(d) Figure 6.10 Structuring grid-cell data. (a) Grid-cell encoding. (b) Layer or matrix organization. (c) Clustering encoding. Cd) A layer encoded version of Figure 1.3.

6.3.4 Structures for grid cell data Just as there are numerous varieties of organIZIng geometric and topological properties in data tables, so there are different ways of structuring regular tessellation data (Figure 6.10). The common form for a basic tabular record of data for grid cells consists of a row coordinate value, a column coordinate value, possibly a cell identifier, in addition to the row-column combination, a series of values for attributes found in a cell, possibly a code value pointing to another table of data, a reference to the identifier for the nearest cell on the sides or corners. If only one attribute is of interest, then the basic table consists of a row identifier, column identifier (jointly making a space locator) and one data item: an organization reflecting each grid cell as an individual entity. The records, that is one per grid cell, can be listed sequentially in row order or column order, or, taking advantage of computer storage formats

234

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known as arrays, tltl' l'lllil,' 11I .. (rix (lln'lls CIIl Iw \ t(ll'l'll (l,'igIII'C (l.IOh). Th e matrix format may also he IIsed ir there ,m.; sevel';11 d

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