CRC STANDARD
CURVES and SURFACES David von Seggern
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CRC Press Ann Arbor London
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CRC STANDARD
CURVES and SURFACES David von Seggern
Boca Raton
CRC Press Ann Arbor London
Tokyo
Library of Congress CataloginginPublication Data
von Seggern, David H. (David Henry) CRC standard curves and surfaces / David Henry von Seggern. p. cm. Updated ed. of: CRC handbook of mathematical curves and surfaces. c1990. Includes bibliographical references and index. ISBN 0849301963 1. Curves on surfacesHandbooks, manuals etc. 1. Von Seggern, David H. (David Henry). CRC handbook of mathematical curves and surfaces. II. Title. QA643.V67 1993 516.3' 52dc20
9233596 CIP
This book represents information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Every reasonable effort has been made to give reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Direct all inquiries to CRC Press, Inc., 2000 Corporate Blvd., N.W., Boca Raton, Florida, 33431. © 1993 by CRC Press, Inc.
International Standard Book Number 0849301963 Library of Congress Card Number 9233596 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acidfree paper
PREFACE Mathematical functions are a fundamental and prevalent ingredient in the lendeavors of scientists and engineers. The conclusions, predictions, and analyses of such professionals are most often concisely contained in these abstract relations, which are commonly referred to as "curves" when illustrated. Many special curves can be found in mathematical tables, such as the CRC Handbook of Mathematical Sciences, 1 and in mathematical dictionaries (for example, James and James 2 ). The National Bureau of Standard's handbook (Abramowitz and Stegun3 ) is the acknowledged Englishlanguage source for special functions in physics and engineering. The recent work entitled A Catalog of Special Plane Curves 4 is an excellent source for illustrations of interesting functions in two dimensions. Yet, i~ spite of the frequent and widespread use of mathematical functions, there has been, to date, no volume in which a diversity of curves appear in graphic form. Unexpectedly, there is no work which illustrates the spectrum of simple functions found in most integral tables. Thus, there is not a single reference work which draws together the entire gamut of forms which the modern scientist or engineer uses within a career. Such a reference volume is long overdue, especially in light of the fact that "curves" have become the ready tool of many other disciplines due to the computational and storage powers of modern electronic computers. Lastly, most of the curves appearing in older reference works show the imprecision of hand drafting methods, and the reappearance of familiar curves in precise, computerplotted form should serve a useful purpose in itself. Curves are abstractions of the form and motion of the physical world. Scientists have analyzed this world for millennia in order to render these abstract expressions in the most minute detail, from gross astronomical movements to infinitesimal atomic phenomena. It is now possible for a remarkably detailed synthesis of natural phenomena to be created by the proper use of these abstractions. Some such synthetic renditions have emerged from the field of computer graphics already (mountainous terrain, cloud formations, trees, to name a few) and ,are nearly indistinguishable from reality. Modern scientists' skillful mathematical description of the motion of nature, coupled with modern computing power, has also enabled them to make increasingly accurate predictions of natural events, such as weather, earthquakes, and oceanic currents. All such endeavors involve, as the quantitative basis, functions whose curves are the visual representation of the predicted motion. Scientists and engineers can use this reference work in two ways to aid their work. In the forward manner, they can look up the equation of interest and see the corresponding visual form of the curve. In the inverse manner, they can select a particular curve visually to serve in data fitting or in computer modeling exercises. This handbook, however, purports to serve a larger audience than those engaged in mathematics, science, and engineering. Architects, designers,
draftsmen, and artists should benefit from this reference book of curves. New expressions of form can be imagined through even a casual scanning of the contents of this volume. And if one has general notion of the desired appearance, the appropriate curve can be located in this volume and its mathematical expression noted. The mathematical expressions given here can be readily translated into highlevel programming languages (for example, FORTRAN) in order to generate a given curve in a particular environment of application. Recent graphics languages enable cells, segments, or symbols to be created once and stored for future use. These abstractions, which can be composed of one or more curve segments, may be placed in a computerbased design at any scale or rotation angle to achieve the desired effect. The computer revolution indeed makes curve generation easy and rapid and eliminates the former laborious hand calculations necessary to graph even the simplest curves. Achieving the most intricate and subtlest abstract forms, as well as the simple and plain, is possible for those who have only a rudimentary programming knowledge. Properly designed computer programs can open up this possibility even to those who have no grasp of the underlying equations. This work is intended to contain all curves in common use in applied mathematics. In order to be comprehensive, the notion of "curve" has been extended beyond its usual connection with algebraic or transcendental functions. Here "curve" means any line or surface in two or three dimensions which can be generated by a rule or set of rules expressible in mathematical terms. Such rules may be entirely smooth and deterministic, and the first part of this handbook is devoted to the curves represented in this way: algebraic forms, transcendental forms, and special integrals. Here mathematicians, scientists, and engineers will find those curves familiar to them. Selection of functions for curve fitting can be eased by use of this handbook, and questions concerning the form of a given function can be quickly settled. Designers can find curves appropriate to their design goals. The latter part of this handbook comprises curves and surfaces which are not smoothly generated by a single relation, such as piecewise continuous functions, polygons, and polyhedra. When the generating rules include random components, a new series of curves and surfaces emergesthe subject of the final chapter. The need for cataloging such curves is due to the work of Mandelbrot,S who has shown that the study and description of the random component of natural phenomena is as important, if not more important, than that of the deterministic component. A future volume will collect together many interesting and unusual curves which are not normally considered in pure mathematics. These curves will be most useful to artists and designers who are able to employ modern computerassisted art and drafting systems. This handbook begins with a chapter containing a qualitative summary of deterministic curve properties and a classification of such curves. An explanation of the means and conventions of presentation in later chapters is also
given here. This first chapter is meant to acquaint the reader with fundamental mathematical properties of curves in order that application of the material of the handbook can be more knowledgeable and meaningful. Those with a solid background in calculus will find little new information here. A section on matrix transformations has been included to indicate how a given curve can be made to appear in many different forms. The following chapters are organized so that similar curves are grouped together· for easy reference. Early chapters deal with curves in two dimensions, progressing from the simple to the complex. Later chapters extend the notion of curve to curves and surfaces in three dimensions. Final chapters deal with piecewise continuous functions in two and three dimensions.
REFERENCES 1. Beyer, W. B., Ed., CRC Standard Mathematical Tables, CRC Press, Boca Raton, 1978. 2. James, G., and R. C. James, Eds., Mathematics Dictionary, Van Nostrand, New York, 1949. 3. Abramowitz, M., and I. A. Stegun, Eds., Handbook of Mathematical Functions, National Bureau of Standards, Department of Commerce, Washington, D.C., 1964. 4. Lawrence, J. D., A Catalog of Special Plane Curves, Dover Publications, New York, 1972. 5. Mandelbrot, B. B., The Fractal Geometry of Nature, W. H. Freeman, San Francisco, 1983.
PREFACE TO THE SECOND EDITION Since the publication of the first edition of the CRe Handbook of Mathematical Curves and Surfaces, an extraordinary tool in mathematics has emergedMathematica, the work of Stephen Wolfram and Wolfram Research, Inc. Without question, I knew, immediately upon discovering it, that I would prepare a second edition using this program to plot the functions. The plotting capabilities of programs such as Mathematica remove much of the timeconsuming work of preparing a volume such as this. All plots of the second edition were done using Mathematica, except those few figures of Chapter 1. In this process, a few minor errors were found in the original version, and two serious errors. More importantly, Mathematica enables this hardbound book to be accompanied by an electronic notebook which may in fact be more useful than the hardbound version. The electronic version is a "dynamic" reference work, while this hardbound version is a "static" one. Only those who have used the Mathematica notebook format can appreciate this fully. The author hopes that those who are using this hardbound version will also have access to the accompanying electronic version. This allows them to redo the plots for the exact parameters of interest and in a style consistent with their needs and preferences. The general content and form of the first edition have been preserved while the number of plotted functions has increased by 30%. An entire new chapter has appeared to show functions of a complex variable. Another new chapter is devoted to curves with a random element, such as autoregressive processes. All the curves of one former chapter entitled "Miscellaneous Curves" have been moved to other chapters where they logically belonged. Several new functions have been added to the chapter on "Special Functions in Mathematical Physics." Several algebraic and transcendental surfaces have been added to the appropriate chapters. In all, nearly every chapter includes significant new contributions. The surfacerendering capabilities of Mathematica were a welcome tool for improving the presentation of the character of the functions over the line rendering used for 3D in the first edition. The only problem with surface representation is that I had to choose one particular view orientation which best illustrated the surface when, in fact, many different views are really needed. This is, in all cases, a subjective choice. Those with access to the Mathematica notebook version can easily rectify this inflexibility of a hardbound book. Many people have commented on the first edition or suggested new curves to include in a second edition. In this regard, I must especially mention Richard A. Skarda, who sent me a large number of interesting curves, and Oscar L. King, who allowed me to see a collection of curves in the trochoid family. The people at Wolfram Research, Inc. who have kindly helped me were Kevin McIsaacs and Steven Adams.
THE AUTHOR David H. von Seggern, Ph.D., is a geophysicist currently with Phillips Petroleum Company of Bartlesville, Oklahoma. He previously worked for Teledyne Geotech in Alexandria, Virginia, on numerous aspects of undergroundnucleartest detection. During that time, he authored or coauthored numerous professional papers and company reports on the subject. He completed his education at the Pennsylvania State University with a dissertation on earthquake prediction which included an early application of fractal theory in seismology. At Phillips Petroleum Company, Dr. von Seggern has specialized in applying computer graphics to the problems of processing and interpreting seismic data, has promoted seismic modeling as an aid in data interpretation, and has done research in seismic imaging methods using supercomputer technology.
TABLE OF CONTENTS Chapter 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1. Concept of a Curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1 1.2. Concept of a Surface .................................. 2 1.3. Coordinate Systems ................................... 2 1.3.1. Cartesian Coordinates ........................... 2 1.3.2. Polar Coordinates .............................. 3 1.3.3. Cylindrical Coordinates .......................... 4 1.3.4. Spherical Coordinates ........................... 4 1.4. Qualitative Properties of Curves and Surfaces ................. 5 1.4.1. Derivative .................................... 5 1.4.2. Symmetry .................................... 6 1.4.3. Extent ....................................... 7 1.4.4. Asymptotes ................................... 7 1.4.5. Periodicity .................................... 8 1.4.6. Continuity .................................... 9 1.4.7. Singular Points ................................ 9 1.4.8. Critical Points ................................ 10 1.4.9. Zeros.. .................................... 11 1.4.10. Integrability .................................. 11 1.4.11 Multiple Values ............................... 12 1.4.12 Curvature ................................... 13 1.5. Classification of Curves and Surfaces ...................... 13 1.5.1. Algebraic Curves .............................. 14 1.5.2. Transcendental Curves .......................... 15 1.5.3. Integral Curves ............................... 15 1.5.4. Piecewise Continuous Functions ................... 16 1.5.5. Classification of Surfaces ........................ 16 1.6. Basic Curve and Surface Operations ....................... 17 1.6.1 Translation .................................. 17 1.6.2 Rotation .................................... 17 1.6.3. Linear Scaling ................................ 17 1.6.4. Reflection ................................... 18 1.6.5. Rotational Scaling ............................. 18 1.6.6. Radial Translation ............................. 18 1.6.7. Weighting ................................... 19 1.6.8. Nonlinear Scaling .............................. 19 1.6.9. Shear ...................................... 19 1.6.10. Matrix Method for Transformation ................. 20 1.7. Method of Presentation ................................ 21 1.7.1. Equations ................................... 22 1.7.2. Plots ....................................... 22 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Chapter 2 Algebraic Functions ..................... . ................. 25
2.1. Functions with xn/m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2. Functions with xn and (a + bx)m . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3. Functions with a 2 + x 2 and xm . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.4. Functions with a 2  x 2 and xm . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.5. Functions with a 3 + x 3 and xm . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.6. Functions with a 3  x 3 and xm . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.7. Functions with a 4 + X4 and xm . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.8. Functions with a 4  X4 and xm . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.9. Functions with (a + bX)1/2 and xm . . . . . . . . . . . . . . . . . . . . . . . 58 2.10. Functions with (a 2  x 2)1/2 and xm . . . . . . . . . . . . . . . . . . . . . . . 66 2.11. Functions with (x 2  a 2)1/2 and xm . . . . . . . . . . . . . . . . . . . . . . . 70 2.12. Functions with (a 2 + X 2)1/2 and xm . . . . . . . . . . . . . . . . . . . . . . . 74 2.13. Miscellaneous Algebraic Functions ........................ 78 2.14. Algebraic Functions Expressible in Polar Coordinates .......... 88 2.15. Algebraic Functions Expressed Parametrically ................ 94 Chapter 3 Transcendental Functions . . . . . . . . . . . . . . . . . . ................. 97
3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. 3.9. 3.10. 3.11. 3.12. 3.13. 3.14. 3.15. 3.16. 3.17. 3.18.
Trigonometric Functions with sinn(ax) and cosm(bx) (n, m Integers) ...................................... 98 Trigonometric Functions with 1 ± sinn(ax) and 1 ± cosm(bx) . ... 106 Trigonometric Functions with a sinn(c.x) + b cosm(c.x) . . . . . . . . . 112 Trigonometric Functions of More Complicated Arguments ...... 114 Inverse Trigonometric Functions ........................ 118 Logarithmic Functions ................................ 120 Exponential Functions ................................ 124 Hyperbolic Functions ................................ 130 Inverse Hyperbolic Functions ........................... 136 Trigonometric and Exponential Functions Combined .......... 138 Trigonometric Functions Combined with Powers of x ......... 140 Logarithmic Functions Combined with Powers of x ........... 146 Exponential Functions Combined with Powers of x ........... 150 Hyperbolic Functions Combined with Powers of x . ........... 154 Combinations of Trigonometric Functions, Exponential Functions, and Powers of x . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Miscellaneous Transcendental Functions .................. 158 Transcendental Functions Expressible in Polar Coordinates ..... 164 Parametric Forms ................................... 172
Chapter 4 Polynomial Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
4.1. Orthogonal Polynomials ............................... 184 4.2. Nonorthogonal Polynomials ........................... 194 References ............................................. 198
Chapter 5 Special Functions in Mathematical Physics . . . . . . . . . . . . . . . . . . . . . . 199
5.1. Exponential and Related Integrals ....................... 200 5.2. Sine and Cosine Integrals ............................. 204 5.3. Gamma and Related Functions ......................... 206 5.4. Error Functions .................................... 208 5.5. Fresnel Integrals .................................... 210 5.6. Legendre Functions ................................. 212 5.7. Bessel Functions .................................... 216 5.8. Modified Bessel Functions ............................. 220 5.9. Kelvin Functions .................................... 222 5.10. Spherical Bessel Functions ............................. 226 5.11. Modified Spherical Bessel Functions ...................... 228 5.12. Airy Functions ..................................... 230 5.13. Riemann Functions .................................. 230 5.14. Parabolic Cylindrical Functions ......................... 230 5.15. Elliptic Integrals .................................... 232 5.16. Jacobi Elliptic Functions .............................. 234 References ............................................. 242 Chapter 6 Special Functions in Probability and Statistics ................... 243
6.1. 6.2. 6.3.
Discrete Probability Densities .......................... 243 Continuous Probability Densities ........................ 248 Sampling Distributions ............................... 258
Chapter 7 ThreeDimensional Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
7.1. Helical Curves ........................... . . . . . . . . . . 262 7.2. Sine Waves in Three Dimensions ........................ 266 7.3. Miscellaneous Spirals ................................ 270 References ............................................. 272 Chapter 8 Algebraic Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 8.1. Functions with ax + by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 8.2. Functions with x 2ja 2 ± y2 jb 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 276 8.3. Functions with (x 2ja 2 + y2 jb 2 ± C2)1/2 . . . . . . . . . . . . . . . . . . . 278 8.4. Functions with x 3ja 3 ± y3 jb 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 282 8.5. Functions with x 4ja 4 ± y4jb 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 284
8.6.
Miscellaneous Functions .............................. 286
Chapter 9 Transcendental Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
9.1. 9.2.
Trigonometric Functions .............................. 292 Logarithmic Functions ................................ 294
9.3. 9.4. 9.5. 9.6.
Exponential Functions ............ : ................... 296 Trigonometric and Exponential Functions Combined .......... 298 Surface Spherical Harmonics ........................... 300 Miscellaneous Transcendental Functions .................. 302
Chapter 10 ComplexVariable Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
10.1. Algebraic Functions ................................. 310 10.2. Transcendental Functions ............................. 316 Chapter 11 Nondifferentiable and Discontinuous Functions . . . . . . . . . . . . . . . . . . . 323
11.1. 11.2. 11.3.
Functions with a Finite Number of Discontinuities ............ 324 Functions with an Infinite Number of Discontinuities .......... 326 Functions with a Finite Number of Discontinuities in the First Derivative . . . . . . . . . . . . . . . . . ............... 330 11.4. Functions with an Infinite Number of Discontinuities in the First Derivative ................................ 332
Chapter 12 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
12.1. Regular Polygons ................................... 338 12.2. Star Polygons ...................................... 338 12.3. Irregular Triangles .................................. 338 12.4. Irregular Quadrilaterals ............................... 340 12.5. Polyiamonds ....................................... 342 12.6. Polyominoes..................... .................. 342 12.7. Polyhexes ......................................... 342 Chapter 13 Polyhedra and Other Closed Surfaces with Edges . . . . . . . . . . . . . . . . . 345
13.1. Regular Polyhedra .................................. 346 13.2. Stellated (Star) Polyhedra ............................. 348 13.3. Irregular Polyhedra .................................. 350 13.4. Miscellaneous Closed Surfaces with Edges ................. 356 References ............................................. 358 Chapter 14 Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
14.1. Elementary Random Processes .......................... 360 14.2. General Linear Processes ............................. 362 14.3. Integrated Processes ............................. .... 372 14.4. Fractal Processes ................................... 378 14.5. Poisson Processes ................................... 380 References ............................................. 382 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
CRC STANDARD
CURVES and SURFACES
1
Chapter 1
INTRODUCTION 1.1. CONCEPT OF A CURVE Let En be the Euclidean space of dimension n. (According to this definition, E1 is a line, E2 is a plane, and E3 is a volume.) A curve in nspace is defined as the set of points which result when a mapping from E1 to En is performed. In this reference work, only curves in E2 and E3 will be considered. Let t represent the independent variable in E1. An E2 curve is then given by x
=
f( t)
y=g(t)
and an E3 curve by
x=f(t),
y
=
g(t),
z
=
h( t)
where f, g, and h mean "function of." The domain of t is usually (0, 21T), (  00,00), or (0,00). These are the parametric representations of a curve. However, in 2space, curves are commonly expressed as y
=
f( x)
or as
f( x, y)
=
0
which are the explicit and implicit forms, respectively. The explicit form is reducible from the parametric form when x = f(t) = t in 2space and when x = f(t) = t and y = get) = t in 3space. The implicit form of a curve will often comprise more points than a corresponding explicit form. For example y2 _ X = 0 has two ranges in y, one positive and one negative, while the explicit form derived from solving the above equation gives y = X 1/ 2 , for which the range of y is positive only. Generally, the definition of a curve imposes a smoothness criterion, 1 meaning that the trace of the curve has no abrupt changes of direction (continuous first derivative). However, for purposes of this reference work, a broader definition of curve is proposed. Here, a curve may be composed of smooth branches, each satisfying the above definition, provided that the intervals over which the curve branches are distinctly defined are contiguous. This definition will encompass forms such as polygons or sawtooth functions.
2
CRC Standard Curves and Surfaces
1.2. CONCEPT OF A SURFACE This reference work will treat only surfaces in 3space (E3). Therefore a surface is defined as the mapping from E2 to E3 according to x=f(s,t),
Y =g(s,t),
z=h(s,t)
As for curves, the conversion from this parametric form to more common forms z=f(x,y)
or f(x,y,z)=O
may not be possible in some cases. Again, a smoothness criterion 1 is desirable, but the generalized definition of surface requires only that this smoothness criterion be satisfied piecewise for all distinct mappings of the (s, t) plane over which the surface is defined. These generalized surfaces are termed manifolds. Cubes are examples of surfaces which can be defined in this deterministic manner.
1.3. COORDINATE SYSTEMS The number of available coordinate systems for representing curves IS large and even larger for surfaces. However, to maintain uniformity of presentation throughout this volume, only the following will be used: 2D
3D
Cartesian, polar
Cartesian, cylindrical, spherical
The term "parametric" is often used as though it were the name of a coordinate system, but it really means a representation of coordinates in terms of an additional independent parameter which is not itself a coordinate of the space En in which the curve or surface exists. 1.3.1. Cartesian Coordinates The Cartesian system is illustrated in Figure 1 for two dimensions. This is the most natural, but not always the most convenient, system of coordinates for curves in two dimensions. Coordinates of a point p are measured linearly along two orthogonal axes which intersect at the origin (0, 0). The Cartesian
3
x
",,,,,,,,,,,,,,,,,,,,,,,,,,,,,.P
I
y
FIGURE 1.
FIGURE 2.
The Cartesian coordinate system for two dimensions.
The Cartesian coordinate system for three dimensions.
system is also called the "rectangular" system. For three dimensions, an additional axis, orthogonal to the other two, is placed as shown in Figure 2. 1.3.2. Polar Coordinates Polar coordinates (r, e) are defined for two dimensions and are a desirable alternative to Cartesian ones when the curve is point symmetric and exists only over a limited domain and range of the variables x and y. As illustrated in Figure 3, the coordinate r is the distance of the point p from the origin, and the coordinate e is the counterclockWise angle which the line from the origin to p makes with the horizontal line through the origin to the right. ClockWise rotations are measured in negative e relative to this line. Transfor
"7f.~\\ :
FIGURE 3.
~
The polar coordinate system for two dimensions.
4
CRC Standard Curves and Surfaces
FIGURE 4.
The cylindrical coordinate system for three dimensions.
mations from polar to Cartesian, and vice versa, are made according to x = r cos 8,
Y = r sin 8
8
=
arctan( Y Ix)
1.3.3. Cylindrical Coordinates Cylindrical coordinates are used in three dimensions. They combine the polar coordinates (r,8) of two dimensions with the third coordinate z measured perpendicularly from the (x, y) plane at Cr, 8) to the point p at Cr, 8, z) as in Figure 4. The normal convention is for z to be positive upward. Transformation from cylindrical to Cartesian coordinates involves only the polartoCartesian transformations given above, because the z coordinate is unchanged. Cylindrical coordinates are often appropriate when surfaces are axially symmetric about the z axis, for example, in representing the form r2 = z. 1.3.4. Spherical Coordinates As illustrated in Figure 5, let a point in £3 lie at a radial distance r along a vector from the origin. Project this vector to the (x, y) plane, and let the angle between the vector and its projection be cPo Now measure the angle 8 of the projected line in the (x, y) plane as for polar coordinates. Then (r, 8, cP) are the spherical coordinates of p. The transformations from spherical to Cartesian coordinates, and vice versa, are given by: x
=
r cos 8 sin cP,
y = r sin 8 sin
8 = arctan(Ylx),
cP,
z
=
r cos cP
cP = arctan [ (x 2 + y2) 1/2 Iz ]
5
FIGURE 5. The spherical coordinate system for three dimensions.
Spherical coordinates are often appropriate for surfaces having point symmetry about the origin. The usual coordinates of geography, which refer to points on the earth by latitude and longitude, are a spherical system.
1.4. QUALITATIVE PROPERTIES OF CURVES AND SURFACES Curves and surfaces exhibit a wide variety of forms. Particular attributes of form are derivable from the equations themselves, and many texts treat these in rigorous detail. The purpose here is not to duplicate such explicit and analytical treatment but rather to present the properties of curves and surfaces in a qualitative manner to which their visible forms are naturally and easily related. Understanding of these properties enables one to choose the appropriate curve for a given purpose (for example, data fitting) or to modify, when necessary, an equation given in this volume into one more suitable for a given purpose. 1.4.1. Derivative A fundamental quantity associated with a curve, or function, is the derivative, which exists at all continuous points of the curve (except singular points as described in Section 1.4.7). Although the definition of derivative can be made with analytical rigor,! in graphical terms the derivative at any point is the slope of the tangent line at that point and is written as dy / dx for twodimensional curves. For threedimensional curves, the tangent line is along the trajectory of the curve, and three such derivatives are possible using the three pairs of the coordinates x, y, and z. Closely associated with the derivative is a curve's normal, which is the line perpendicular to the tangent. In two dimensions the normal is a single line, but in three dimensions the normal sweeps out a plane perpendicular to the tangent to the curve.
6
CRC Standard Curves and Surfaces
As for curves, the derivative of a surface is a fundamental quantity. The derivative at any continuous point of a surface relates to the tangent plane of the surface at that point. For this plane, three "partial" derivatives exist, written as 3y /oz, oz/ox, and ox/oy (or their inverses), which are the slopes of the lines formed at the intersections of the tangent plane with the (y, z), (z, x), and (x, y) planes, respectively. The normal to the surface at a point is the vector orthogonal to the surface there. It is defined at all points for which the surface is smooth by the partial derivatives
oy Tt oz Tt
OZ
os
ox 8S
oz Tt ox Tt
ox 8S
oy 8S
ox ot oy Tt
j p
using the parametric representation equations. If the surface can be expressed in the implicit form f(x, y, Z) = 0, then simply
The above definitions give the (x, y, Z) components of the normal vector; it is customary to normalize them to (x', y', z') by dividing them with (x 2 + y2 + Z2)1/2 so that X,2 + y,2 + Z,2 = 1. 1.4.2. Symmetry For curves in two dimensions, if y =f(x) =f(x) holds, then the curve is symmetric about the y axis. The curve is antisymmetric about the y axis if y
=
f(x)
=
fe x)
A simple example is powers of x: y = xn. If n is even, the curve is symmetric; if n is odd, it is antisymmetric. Antisymmetry is also referred to as "symmetry with respect to the origin" or "point symmetry" about (x, y) = (0, 0). For surfaces, three kinds of symmetry exist: point, axial, and plane. A surface has point symmetry when Z
= f(x, y) = f( x, y)
Simple examples of point symmetry qre spheres or ellipsoids. A surface has
7
axial symmetry when
z = f(x, y) = f( x, y)
An example of axial symmetry is a paraboloid. Finally, a surface has plane symmetry about the (y, z) plane when z=f(x,y) =f(x,y)
Similarly, symmetry about the (x, z) plane implies z = f(x, y) = f(x, y) Examples of plane symmetry include z = xy2 and z = eX cos(y). 1.4.3. Extent The extent of a curve is defined by the range (y variation) and domain (x variation) of the curve. The extent is unbounded if both x and y values can extend to infinity (for example, y = x 2 ). The extent is semibounded if either y or x has a bound less than infinity. The transcendental equation y = sin(x) is such a curve, because the range is limited between negative and positive unity. A curve is fully bounded if both x and y bounds are less than infinity. A circle is a simple example of this type of extent. For surfaces the concept of extent can be applied in three dimensions, where "domain" applies to x and y while "range" applies to z. Surfaces formed by revolution of a curve in the (y, z) or (x, z) plane about the z axis will possess the same extent property that the twodimensional curve had. For example, an ellipse in the (x, z) plane gives an ellipsoid as the surface of revolutionboth have the fully bounded property. Similarly, any surface formed by continuous translation of a twodimensional curve (for example, a parabolic sheet) will have the same extent property as the original curve. 1.4.4. Asymptotes The y asymptotes of a curve are defined by Ya
= lim f(x) x~
±oo
Although this definition includes asymptotes at infinity, only those with
IYa I < 00 are of interest. Asymptotic values are often crucial in choosing and applying functions. Physically, an equation mayor may not properly describe real phenomena, depending on its asymptotic behavior. Note that even though a curve is semibounded, its asymptote may not be determinable. An example of a semibounded function with a y asymptote is y = eX, while one without an asymptote is y = sin(x).
8
CRC Standard Curves and Surfaces
The x asymptotes of a curve may be defined in a similar manner: xa
=
. lim y) ±oo
f( y)
when the function is inverted to give x = f(y). An example of a curve with a finite x asymptote is y = (c 2  X 2 )1/2 , whose asymptotes lie at x = +c and x = c. In addition, curves may have asymptotes that are any arbitrary lines in the lane, not simply horizontal or vertical lines; and the limit requirements are similar to the forms given above for horizontal or vertical asymptotes. For instance, the equation y = x + Ijx has y = x as its asymptote. 1.4.5. Periodicity A curve is defined as periodic in x with period X if y = f(x
+ nX)
is constant for all integers n. The transcendental function y = sin(ax) is an example of a periodic curve. A polar coordinate curve can also be defined as periodic with period a in terms of angle 0 if
r
=
f(O
+ na)
is constant for all integers n. An example of such a periodic curve is r = cos(40), which exhibits eight "petals" evenly spaced around the origin.
Surfaces are periodic in x and y with periods X and Y, respectively, if z =
f( x + nX, y + mY)
is constant for all integers nand m. A surface also may be periodic in only x or only y. A cylindricalcoordinate surface may be periodic with period a in terms of the angle 0 if
z
=
f( r, 0 + na)
is constant for all integers n. Another type of periodicity expressible cylindrical coordinates is in the radial direction with period R, when
z =f(r
III
+ nR,O)
is constant for all integers n. An example of such periodicity is given by z = cos(2'lTr )cos(O), which has a period of unity in r.
9
1.4.6. Continuity A curve is continuous at a point x o, provided it is defined at x o, when
and y= lim f(x) X~Xo
are finite and equal. Here "+" and "" refer to approaching Xo from the right and left, respectively. Discontinuities may be finite or infinite: the former implies y+* y even though they are both finite, while the latter implies one or both limits are infinite. For surfaces, the paths to a point Po = (x o, Yo) are infinite in number; and continuity exists only if the surface is defined at Po, and z= lim f(p) P>Po
is constant for all possible paths. When the curve or surface is undefined at Xo or Po and the above relations hold, it is said to be discontinuous, but with a removable discontinuity. Also, if the curve or surface is defined at Xo or Po
and the limit exists there but is not equal to the defined value, it has a removable discontinuity there. For any points at which the above relations do not hold, the curve or surface is discontinuous, with an essential discontinuity at such points. The curve y = sin(x)/x has a removable discontinuity at x = 0 and is therefore continuous in appearance, while y = l/x has an essential discontinuity at x = 0 and is therefore discontinuous in appearance. Curves and surfaces are differentiable (meaning the derivative exists) at removable discontinuities. 1.4.7. Singular Points Curves and surfaces of degree 3 or greater may contain singular points. Writing the function for a twodimensional curve as
f(x,y)=O
the derivative dy / dx can be written as dy dx
g(x, y) hex, y)
where g and h are functions of x and y. If for a given point p(x, y) the functions g and h both vanish, the derivative becomes the indeterminate form 0/0 and p(x, y) is then a singular point of the curve. Singular points imply that two or more branches of the curve meet or cross. If two branches
10
eRC Standard Curves and Surfaces
are involved, it is a double point; if three are involved, it is a triple point; etc. Singularities at triple or higher points are not as commonly encountered as those at double points. Doublepoint singularities for twodimensional curves are classified as follows: 1. Isolated (or conjugate) points are single points disjoint from the remainder
of the curve. In this case, the derivative does not exist. 2. Node points are where the two derivatives exist and are unequal, so that the curve crosses itself. 3. Cusp points are where the derivatives of two arcs become equal and the curve ends. A cusp of the first kind involves second derivatives of opposite signs, and a cusp of the second kind involves second derivatives of the same sign. 4. Double cusp (or osculation) points are where the derivatives of two arcs become equal while the two arcs of the curve are continuous along both directions. Double cusps may also be of the first or second kind, like single cusps. Curves having one or more nodes will exhibit loops which enclose areas. Curves having osculations may also exhibit loops, on one or both sides of the osculation point. The concept of singular points is extendable to surfaces. Many surfaces are the result of the revolution of a twodimensional curve about some line; such surfaces retain the singular points of the curve, except that each such point on the curve, unless on the axis of revolution, becomes a circular ring of singular points centered on the axis of revolution. Singular points appear on more complicated surfaces also, but an analysis of these possibilities is beyond the scope of this volume. 1.4.8. Critical Points Points of a curve y = f(x) at which the derivative dy I dx critical points. There are three types:
=
0 are termed
1. Maximum points are where the curve is concave downward and thus the second derivative d 2 y Idx 2 > O.
2. Minimum points are where the curve is concave upward and thus the second derivative d 2 y Idx 2 < O. 3. Inflection points are where d 2 y Idx 2 = 0 and the curve changes its direction of concavity.
For surfaces z = f(x, y), the critical points lie where ozlox = ozloy = O. Maximum and minimum points of surfaces are defined similarly to those of curves, except both second derivatives must together be greater than zero or less than zero. In the case that they are of opposite sign, the critical point is termed a saddle. Such critical points are nondegenerate 2 and are isolated
11
from other critical points. More complicated types of critical points occur for surfaces and are classified as degenerate or nondegenerate, depending on whether the determinant of
02 Z ox 2 02 Z oX oy
(52 z ox oy 02 Z oy2
vanishes or not. The surface z = x 2 + y2 has a single non degenerate critical point, while z = x 2 y2 has two continuous lines of degenerate critical points, intersecting at (0,0). 1.4.9. Zeros The zeros of a twodimensional function [(x) occur where y = [(x) = 0 and are isolated points on the x axis. (For polynomial functions, the zeros are often referred to as the roots.) Similarly, the zeros of a threedimensional function [(x, y) occur where z = [(x, y) = 0, but the locus of these points is one or more distinct, continuous curves in the (x, y) plane. The zeros of certain functions are important in characterizing their oscillatory behavior (for example, the function sin x), while the zeros of other functions may be unique points of interest in physical applications. Not all functions, as defined, have zeros; for example, the function [(x) = 2  cos x has unity as its lower bound. However, such a function can be translated in one or the other y directions to produce a function having zeros in addition to all the qualitative properties of the original function. The calculation of the exact zeros of a function is often difficult and often must be accomplished by numerical methods on a computer. Zeros of many functions are tabulated in standard references such as Abramowitz and Stegun. 3 1.4.10. Integrability The function y = [(x) defined over the interval [a, b] has the integral
The integral exists if I converges to a single, bounded value for a given interval, and the function is then said to be integrable. Note that the integral I may exist under two abnormal circumstances: 1. Either a or b, or both, extend to infinity. 2. The function y has an infinite discontinuity at one or both endpoints or at one or more points interior to [a, b].
12
CRC Standard Curves and Surfaces
Under either of these circumstances, the integral is an improper integral. Proving the existence of the integral of a given function is not always straightforward, and a discussion is beyond the scope of this volume. Transient functions always have an integral on the interval [0,00] and are often given as solutions to physical problems in which the response of a medium to a given input or disturbance is sought. Such responses must possess an integral if the input was finite and measurable. Examples of such functions are y = e ax sin(bx) and y = 1/(1 + x 2 ). Surfaces given by z = f(x, y) are integrable when
exists. Improper integrals of surfaces are defined in the same manner as those of twodimensional curves. Transient responses exist for three dimensions and are integrable also. A curve property which has an important consequence for integration is that of even and odd functions. Even functions have f(x) = f( x), and for such curves I=2[f(x)dx
o
if I exists over [a, a]. For odd functions f(x) = f( x), and 1= 0 over any interval [  a, a]. This concept can be easily extended to surfaces. 1.4.11. Multiple Values A curve is multivalued if, for a given (x, y), it has two or more distinct values. A simple example is y2 = x. Multivalued functions are not integrable in the normal sense, although one or more particular branches of the curve may have welldefined integrals. While a curve may be multivalued in its Cartesianform equation, the polar form of the equation may be singlevalued, in the sense that only one value of r exists for each value of angle 8. Compare, for example,
which is the equation of a quadrifolium, with its polar equation r
=
cos(28)
Integrability is affected by the choice of coordinate system; this example shows that, when an integral is not defined due to a function being multivalued, it may be well defined when the transformation to polar coordinates is made and the integral evaluated along the polar angle 8.
13
Similarly, surfaces may be single valued or multivalued depending upon whether z takes on one or more values for a given (x, y) point. 1.4.12. Curvature Given that a differential of length along the curve path is ds and that the tangent line changes its direction over ds by an angle de, where e is the angle of the tangent with the x axis, then the radius of curvature is given by
p
=1 ~~ 1
This radius can be expressed in terms of the derivatives of the curve also. If the curve is expressed implicitly as f(x, y) = 0, and if fx and fy are the first partial derivatives and fxx, f yy , and fxy are the second partial derivatives, then
When the curve is expressed in polar coordinates and the derivatives dr I de and d 2r Ide 2 are given by r' and r" respectively, then the radius of curvature is 2
p
=
2 3/2
(r + r' ) r2 + 2r ,2  rr"
~~~
The radius of curvature at lobes of polar curves is of interest in order to define the "tightness" of the lobes. At the peak of the lobe, r' = 0 and p = r 2/(r  r"), This reduces to p = r in the case of a circle, for which r" = O. Using the same formula as for curves above, the curvature of surfaces can be measured along any arbitrary linear arc of the surface made by an intersecting plane, where e is the angle of the tangent line relative to the horizontal in the intersecting plane. Thus the curvature of a surface is relative to the perspective it is viewed from.
1.5. CLASSIFICATION OF CURVES AND SURFACES The family of twodimensional and threedimensional curves can be displayed as in Figure 6. This schematic reflects the organization of this reference work, and every curve which can be traced by a given mathematical equation or given set of mathematical rules can be placed in one of the categories shown. There is a toplevel dichotomy between determinate and random curves; but, except for Chapter 14, no further reference will be made
14
CRC Standard Curves and Surfaces
ALGEBRAIC
mulNAL) NONGAUSSIAN
RATIONAL
POLYNL
LOGARTIHMIC
~OLYNOMIAL
TIUGONOMETIU~~GO~ REGULAR IRREGULAR FRAcrAL
FIGURE 6. A classification of curves and surfaces for this handbook.
to random curves in this volume. A determinate curve is one for which the functional relationship between x and y is known everywhere from the equation or set of rules in the abstract. No realization is required to produce the curve, for it is contained wholly within its defining equations or rules. On the other hand, a random curve will have a random factor or term in its mathematical definition such that an actual realization is required to produce the curve, which will differ from any other realization. For example, y = sin(x) + w(x), where w(x) is a random variable on x, defines a random curve. At the second level in Figure 6, the distinction is made between algebraic, transcendental, integral, and piecewise continuous curves as described below. 1.5.1. Algebraic Curves A polynomial is defined as a summation of terms composed of integral powers of x and y. An algebraic curve is one whose implicit function
f(x,y)=O is a polynomial in x and y (after rationalization, if necessary). Because a curve is often defined in the explicit form y
=
f(x)
there is a need to distinguish rational and irrational functions of x. A rational function of x is a quotient of two polynomials in x, both having only integer powers. An irrational function of x is a quotient of two polynomials, one or both of which has a term (or terms) with power p/q, where p and q
are integers. Irrational functions can be rationalized, but the curves will not be identical before and after rationalization. In general, the rationalized form has more branches; for example, consider y = x 1/2, which is rationalized to y 2 = x. The former curve has only one branch (for positive y) if a strict
15
definition of the radical is used, whereas the latter has two branches, for y < 0 and y > O. In this book, the rationalized curve will be presented
graphically in all cases, even though the equation is printed in its irrational form for simplicity. Besides simple polynomials, rational functions are often grouped into sets convenient for certain mathematical applications. Examples of such polynomial sets are Chebyshev polynomials, Laguerre polynomials, and Bernoulli polynomials. Many polynomial sets have the property of orthogonality, meaning that for any two functions II and 12 of the set,
over the defined domain of x for that set, where w(x) is a weighting function. This property ensures that the different curves within the set make distinct contributions to the set. 1.5.2. Transcendental Curves The transcendental curves cannot be expressed as polynomials in x and y. These are curves containing one or more of the following forms: exponential (eX), logarithmic (log x), or trigonometric (sin x, cos x). (The hyperbolic functions are often mentioned as part of this group, but they are not really distinct because they are forms composed of exponential functions.) Any curve expressed as a mixture of transcendentals and polynomials is considered to be transcendental. All of the primary transcendental functions can, in fact, be expressed as infinite polynomial series: n
00
eX =
L n~O
x
(oo<x
n!
(oo<x
SIll X =
00
n~o 00
log x = 2
1) n X 2n + 1 (2n + I)!
(_
L
1
(
XI)
2n  1 x
+1
( 00 < x < 00) 2n + 1
(x> 0)
n~l
1.5.3. Integral Curves Certain continuous curves are not expressible in algebraic or transcendental forms but are familiar mathematical tools. These curves are equal to the
16
CRC Standard Curves and Surfaces
integrals of algebraic or transcendental curves by definition; examples include Bessel functions, Airy integrals, Fresnel integrals, and the error function. The integral curve is given by
y(b)
=
tf(x) dx a
where the lower limit of integration, a, is usually a fixed point such as  00 or O. Like transcendental curves, these integral curves also have expansions in terms of power series or polynomial series, often making evaluation rather straightforward on computers. 1.5.4. Piecewise Continuous Functions Members of the previous classes of curves (algebraic, transcendental, and integral) all have the property that (except at a few points, called singular points) the curve is smooth and differentiable. In the spirit of a broad definition of curve, a class of nondifferentiable curves appears in Figure 6. These curves have discontinuity of the first derivative as a basic attribute and are quite often composed of straightline segments. Such curves include the simple polygonal forms as well as the intricate "regular fractal" curves of Mandelbrot 4 and the "turtle" tracks described in Hayes. 5 1.5.5. Classification of Surfaces In general, surfaces may follow the same classification scheme as curves (Figure 6). Many commonly used surfaces are rotations of twodimensional curves about an axis, thus giving axial, or possibly point, symmetry. In this case the independent variable x of the twodimensional curve's equation can be replaced with the radial variable r = (x 2 + y2)1/2 to form the equation of the surface. Other commonly used surfaces are merely a continuous translation of a given twodimensional curve along a straight line. Such surfaces will actually have only one independent variable if a coordinate system having one axis coincident with the straight line is chosen. If the two independent variables of the explicit equation of the surface, z = f(x, y), are separable in the sense that z
=
f(x)f(y)
then the surface is orthogonal. In such a case, the x dependence may fall in one of the classes of Figure 6 while the y dependence falls in another. Orthogonal surfaces require fewer operations to evaluate over a grid of the domain of x and y, because the defining equation only needs to be evaluated once along the x direction and once along the y direction, with all other points evaluated by simple multiplication of the x and y factors appropriate to each point on the (x, y) plane.
17
1.6. BASIC CURVE AND SURFACE OPERATIONS There are many simple operations which can be applied to curves and surfaces in order to change them. Knowledge of these operations enables one to adapt a given curve or surface to a particular need and thus extend the curves and surfaces given in this reference to a larger set of mathematical forms. Only a few of the most common operations are presented here. Of these, two (translation and rotation) are homomorphic, which means that the form of the curve is preserved and merely its position or orientation in space is changed. 1.6.1. Translation If the coordinates (x, y, z) of a point are changed to x' = x y'
+a
=y +b
z' = z
+c
the curve or surface undergoes a translation of amount (a, b, c) along the (x, y, z) axes. 1.6.2. Rotation In polar coordinates, if the angle () is changed by a positive a so that
()' = () + a the curve undergoes a clockwise rotation by a. This is convenient for polar coordinates, but the rotation can also be expressed in Cartesian coordinates as x' = x cos a y'
+ y sin a
= x sin a + y cos a
In three dimensions, a surface can be rotated about any of the three axes by using these equations on the coordinate pairs (x, y), (y, z), or (x, z), depending on whether the rotation is about the z, x or y axis. 1.6.3. Linear Scaling The relations for threedimensional linear scaling are x' = ax,
y' = by,
z' = cz
These stretch the curve or surface by the factors a, b, and c along the respective axes. When using polar, cylindrical, or spherical coordinates, a
18
CRC Standard Curves and Surfaces
similar relation r' = dr
stretches or compresses the curve or surface along the radial coordinate by the factor d. 1.6.4. Reflection A twodimensional curve is reflected about the x axis by letting y' = y
or about the y axis by letting
x'= x or through the origin by applying both these equations. In three dimensions, a curve or surface is reflected across the (y, z), (x, z) or (x, y) plane when
x' = x y' = y
z' = z respectively. It can be reflected through the origin when one sets
r' = r in spherical coordinates, and mirrored through the z axis when the same operation is performed on r in cylindrical coordinates. The application to twodimensional polar coordinates follows from the cylindrical case. 1.6.5. Rotational Scaling For two dimensions, let 8' = c8
for the polar angle; the polar curve is then stretched or compressed along the angular direction by a factor c. The same operation can be applied to 8 for cylindrical coordinates in three dimensions, or to both 8 and 4> for spherical coordinates in three dimensions. 1.6.6. Radial Translation In two dimensions, if the radial coordinate is translated according to
r' = r + a
19
then the entire curve moves outward by the amount a from the origin. Note that this operation is not homomorphic like Cartesian translation, because the curve is stretched in the angular direction while undergoing the radial translation. This operation can be performed on the radial coordinate of either cylindrical or spherical coordinate systems in three dimensions. 1.6.7. Weighting In a twodimensional Cartesian system, let
This operation weights the curve by the factor Ixl a , a symmetric operator. If a > 0, the curve is stretched in the y direction by a factor which increases with x; but if a < 0, the curve is compressed by a factor which decreases with x. Similar treatments can be performed on surfaces in three dimensions. 1.6.8. Nonlinear Scaling If in two dimensions the scaling
is performed, the curve is progressively scaled upward or downward in absolute value, according to whether a > 1 or a < 1. Note that if y < and a = 2,4, 6, ... , then the scaled curve will flip to the opposite side of the x axis. Similar scalings can be made in three dimensions using any of the appropriate coordinate systems.
°
1.6.9. Shear A curve undergoes simple shear when either all its x coordinates or all its y coordinates remain constant while the other set is increased in proportion to x or y, respectively. The general transformations for simple shearing of a twodimensional curve are x' = x
+ ay
y' = bx
+y
The transformations for simple x shear are x' = x
y' =y
+ ay
20
CRC Standard Curves and Surfaces
and for simple y shear are
x' =x y' ~ y
+ bx
Surfaces may be simply sheared along one or two axes with similar transformations. Another special case of shear is termed pure shear; the transformations for a twodimensional curve are given by x'
=
Joe
For surfaces, pure shear will only apply to two of the three coordinate directions, with the remaining one having no change. Pure shear is a special case of linear scaling under this circumstance. 1.6.10. Matrix Method for Transformation The foregoing transformations can all be expressed in matrix form, which is often convenient for computer algorithms. This is especially true when several transformations are concatenated together, for the matrices can then be simply multiplied together to obtain a single transformation matrix. Given a pair of coordinates (x, y), a matrix transformation to obtain the new coordinates (x', y') is written as ( x'
y')
= (
Y) (~
x
~)
or explicitly x' = ax
+ cy
y' = bx
+ dy
According to this definition, Table 1 lists several of the xy transformations discussed previously with their corresponding matrix. Translations cannot be treated with the above matrix definition. An extension is required to produce what is commonly referred to as the homogeneous coordinate representation in computer graphics programming. In its simplest form, an additional coordinate of unity is appended to the pair (x, y) to give (x, y, 1). A translation by u and v in the x and y directions is then written using a 3by3 matrix
(x'
o y'
l)=(x
y
1 v
~)
21
where, explicitly,
=x
+u
y' = y
+v
X'
l' = 1
With this representation, a radial translation by s units of a curve given in Cr, (J) coordinates is effected by
( r' so that r' = r
1) = (r
(J'
+ sand
(J
o (J
1
o
is unchanged. Table 1
Operation
Matrix
0
Rotation
(
Linear scaling
(~ ~)
Reflection
(~1 :1)
Weighting
(~
Nonlinear scaling Simple shear Rotational scaling
cos sinO
sin cos
Notes
0) 0
o is counterclockwise angle from positive x axis.
Use + or  according to whether reflection is about x axis, y axis, or origin.
xOa )
(~ yOa)
Un (~
n
Use with (r, 0) coordinates.
1.7. METHOD OF PRESENTATION This reference work is basically intended to be illustrative; therefore all functions, whether curves or surfaces, will have an accompanying plot showing the form of the function. The plot will in all cases be on the righthand page, while the equation will be on the facing lefthand page. Curves and surfaces and their plots are numbered for easy reference and grouped according to type. Wherever popular names exist for certain curves or surfaces, they are placed with the equations themselves.
22
CRC Standard Curves and Surfaces
1.7.1. Equations The equation of each algebraic or transcendental curve will be given in the explicit form y = f(x) or r = fee) wherever possible; similarly, surfaces will be given as z = f(x, y) or r = fee, z) or r = fee,
J
J
1.7.2. Plots Plots of twodimensional curves will be done in the (x, y) plane, with the x and y axes being horizontal and vertical, respectively. The domain of x and the range of y, unless otherwise stated, will be 1 to + 1, and the variable form of the curve will be adequately illustrated by a suitable choice of x and y scaling factors and of the constants in the equation. For example, the curve y = sin x can be illustrated for a domain larger than ± 1 by actually plotting y = sin ax, with a > 1, while still letting x vary between 1 and + 1. Similarly, the range of y can be limited to ± 1 by plotting y = c . f(x), where the constant c is suitably chosen. Threedimensional curves and surfaces will have the additional z axis, also from 1 to + 1, and will be plotted in a projection which satisfactorily illustrates the form of each function. Simple equations will be illustrated by a single plotted curve or surface, while more complicated equations may have two or more such plots with different constants in order to indicate the variation possible in a family of curves or surfaces. In the case of curves which are unbounded in y (for example, y = 1/x), the evaluation algorithm computes and plots the curve at exactly y = + 1 or y = 1. Curves expressed in polar coordinates (r, e) are similarly truncated at r = 1 in the case that r is unbounded. The implicit form of a curve will often comprise more points than a corresponding explicit form. For example, y 2  X = 0 has two ranges in y, one positive and one negative, while the
23
explicit form derived from solving the above equation gives y = xl/2, for which the range of y is positive only; in such cases both the positive and the negative range of yare plotted.
REFERENCES 1. Buck, R. c., Advanced Calculus, McGrawHill, New York, 1965, Chap. 5. 2. Poston, T., and I. Stewart, Catastrophe Theory and Its Applications, Pitman, New York, 1978. 3. Abramowitz, M., and I. A. Stegun, Eds., Handbook of Mathematical Functions, National Bureau of Standards, Department of Commerce, Washington, D.C., 1964. 4. Mandelbrot, B. B., The Fractal Geometry of Nature, W. H. Freeman, San Francisco, 1983. 5. Hayes, B., Computer recreations: Turning turtle gives one a view of geometry from the inside out, Sci. Am., 250, 14, 1984.
25
Chapter 2
ALGEBRAIC CURVES The curves of this chapter are mostly familiar equations found in tables of integrals. Many have acquired traditional or accepted names in the mathematical literature, and these are included wherever appropriate. The last section deals with curves more readily expressed in polar coordinates; this allows much easier computation of the curves than with the form y = I(x), especially when curves are multivalued in that form. For curves involving radicals, both the positive and negative branches are plotted to show the symmetry.
26
CRC Standard Curves and Surfaces
2.1. FUNCTIONS WITH x n / 2.1.1. y = cx n
1. 2. 3. 4. 5. 6.
c = c = c = c = c = c =
1, 1, 1, 1, 1, 1,
n
n n
n n n
y  cx n = 0 1 (linear) = 3 (cubic) = 5 (quintic) = 2 (quadratic, or simple parabola) = 4 (quartic) = 6 (sextic) =
= c/x n c = 0.01, n = c = 0.01, n = c = 0.01, n = c = 0.01, n = c = 0.01, n = c = 0.01, n =
yxn 
2.1.2. y
1. 2. 3. 4. 5. 6.
1 (hyperbola) 3 5 2 4 6
C
= 0
m
27
2.1.1
2.1.1
2.1.2
2.1.2
28
CRC Standard Curves and Surfaces
2.1.3. y = cx n / m y  cx n / m = 0 1. c = 1, n = 1, m = 4 2. c = 1, n = 1, m = 2 3. c = 1, n = 3, m = 4 4. c = 1, n = 5, m = 4 5. c = 1, n = 3, m = 2 (semicubical parabola) 6. c = 1, n = 7, m = 4 7. c = 1, n = 1, m = 3 8. c = 1, n = 2, m = 3 (cusp catastrophe) 9. c = 1, n = 4, m = 3 10. c = 1, n = 5, m = 3 2.1.4. y = c/x n / m 1. c = 0.01, n = 1, m = 4 2. c = 0.01, n = 1, m = 2 3. c = 0.01, n = 3, m = 4 4. c = 0.01, n = 5, m = 4 5. c = 0.01, n = 3, m = 2 6. c = 0.01, n = 7, m = 4 7. c = 0.01, n = 1, m = 3 8. c = 0.01, n = 2, m = 3 9. c = 0.01, n = 4, m = 3 10. c = 0.01, n = 5, m = 3
yx n / m  c
= 0
29
2.1.3
2.1.3
2.1.4
2.1.4
CRC Standard Curves and Surfaces
30
2.2. FUNCTIONS WITH xn AND (a 2.2.1. y = e(a + bx) 1. a = 0.5, b = 0.5, e = 1.0 2. a = 0.5, b = 1.0, e = 1.0 3. a = 0.5, b = 2.0, e = 1.0
y  bex  ae
+ bx)m
= 0
2.2.2. y = e(a + bX)2 1. a = 0.5, b = 0.5, e = 1.0 2. a = 0.5, b = 1.0, e = 1.0 3. a = 0.5, b = 2.0, e = 1.0 2.2.3. y = e(a
+ bX)3
y  b 3ex 3
a 3e = 1. 2.
3.
a a a
3ab 2ex 2  3a 2bex

0
= 0.5, b = 0.5, e = 1.0 = 0.5, b = 1.0, e = 1.0 = 0.5, b = 2.0, e = 1.0
2.2.4. y = ex(a + bx) 1. a = 0.5, b = 0.5, e = 1.0 2. a = 0.5, b = 1.0, e = 1.0 3. a = 0.5, b = 2.0, e = 1.0
y  bcx 2  aex
= 0
2.2.5. y = cx(a + bx)2 1. a = 0.5, b = 0.5, e = 1.0 2. a = 0.5, b = 1.0, e = 1.0 3. a = 0.5, b = 2.0, e = 1.0 2.2.6. y = cx(a 1. 2. 3.
a a a
+ bx)3
= 0.5, b = 0.5, e = 1.0 = 0.5, b = 1.0, e = 1.0 = 0.5, b = 2.0, e = 1.0
 3ab 2cx 3 a 3 ex = 0
y  b 3cx 4

3a 2bcx 2
31 3
3
2
2
2.2.2
2.2.1 3
2
3
3
2
2
3
2.2.3
2.2.4 2 3
3
2
2
3
2.2.5
2.2.6
2
32
CRC Standard Curves and Surfaces
2.2.7. 1. 2. 3.
Y = cx 2(a + bx) a = 0.5, b = 0.5, c = 1.0 a = 0.5, b = 1.0, c = 1.0 a = 0.5, b = 2.0, C = 1.0
2.2.8. 1. 2. 3.
cx 2(a + bX)2 a = 0.5, b = 0.5, c = 1.0 a = 0.5, b = 1.0, c = 1.0 a = 0.5, b = 2.0, c = 1.0 Y
=
2.2.9. Y
=
cx 2(a
y  bcx 3
y  b 3cX 5
+ bx?
acx 2


a 3 cx 2 = 1. 2. 3.
=
0
3ab 2cx 4

3a 2bcx 3

3a 2bcx 4
0
a = 0.5, b = 0.5, c = 1.0 a = 0.5, b = 1.0, c = 1.0 a = 0.5, b = 2.0, C = 1.0
2.2.10. Y = cx 3(a + bx) 1. a = 0.5, b = 0.5, C = 1.0 2. a = 0.5, b = 1.0, C = 1.0 3. a = 0.5, b = 2.0, C = 1.0
y  bcx 4
acx 3 = 0

2.2.11. Y = cx 3(a + bX)2 1. a = 0.5, b = 0.5, C = 1.0 2. a = 0.5, b = 1.0, C = 1.0 3. a = 0.5, b = 2.0, C = 1.0 2.2.12. y = cx3(a 1. 2. 3.
+ bX)3
a = 0.5, b = 0.5, a = 0.5, b = 1.0, a = 0.5, b = 2.0,
C
C C
y  b 3cX 6 a 3cx 3
= 1.0 = 1.0 = 1.0

3ab 2cx 5
=
0
33 3
2
1 3
3
2
3
2.2.7
2.2.8 3
2
2
3
2.2.10
2.2.9 3
2
3
3
2
2
3
2.2.11
2.2.12
2
2 1
CRC Standard Curves and Surfaces
34
2.2.13. y = e/(a + bx) 1. a = 1.0, b = 2.0, e = 0.02 2. a = 1.0, b = 3.0, e = 0.02 3. a = 1.0, b = 4.0, e = 0.02
ay
+ bxy
 e
°
=
2.2.14. y = e/(a + bX)2 1. a = 1.0, b = 2.0, e = 0.02 2. a = 1.0, b = 3.0, e = 0.02 3. a = 1.0, b = 4.0, e = 0.02 2.2.15. y
=
e/(a
+ bX)3
a 3y
+ 2a 2bxy + 2ab 2x 2y + b 3x 3y
=0 1. 2.
3.
a = 1.0, b = 2.0, e = 0.02 a = 1.0, b = 3.0, e = 0.02 a = 1.0, b = 4.0, e = 0.02
2.2.16. y = ex/(a + bx) 1. a = 1.0, b = 2.0, e = 0.1 2. a = 1.0, b = 3.0, e = 0.1 3. a = 1.0, b = 4.0, e = 0.1
ay
+ bxy
 ex
=
°
2.2.17. y = ex/(a + bX)2 1. a = 1.0, b = 2.0, e = 0.02 2. a = 1.0, b = 3.0, e = 0.02 3. a = 1.0, b = 4.0, e = 0.02 2.2.18. y = ex/(a
1. 2.
3.
+ bX)3
a = 1.0, b = 2.0, e = 0.01 a = 1.0, b = 3.0, e = 0.01 a = 1.0, b = 4.0, e = 0.01
a 3y + 3a 2bxy ex =
°
+ 3ab 2x 2y + b 3x 3 y
 e
35 23 1
23
23
2.2.13
2.2.14
1 23
2 3
2 3
1
23
2.2.15
2.2.16 2 3
1
1 1 2233
2.2.17
23
2.2.18
36
CRC Standard Curves and Surfaces
2.2.19. Y = ex 2/(a + bx) 1. a = 1.0, b = 2.0, e = 0.2 2. a = 1.0, b = 3.0, e = 0.2 3. a = 1.0, b = 4.0, e = 0.2
ay +bxy  ex 2
=
°
2.2.20. Y = ex 2/(a + bX)2 1. a = 1.0, b = 2.0, e = 0.1 2. a = 1.0, b = 3.0, e = 0.1 3. a = 1.0, b = 4.0, e = 0.1 a3 y
+ 3a 2bxy + 3ab 2x 2y + b 3 x 3 y
ex 2 1.
2. 3.
°
=
a = 1.0, b = 2.0, e = 0.02 a = 1.0, b = 3.0, e = 0.02 a = 1.0, b = 4.0, e = 0.02
2.2.22. Y = ex 3 /(a + bx) 1. a = 1.0, b = 2.0, e = 1.0 2. a = 1.0, b = 3.0, e = 1.0 3. a = 1.0, b = 4.0, e = 1.0
+ bxy
ay
 ex 3 =
°
2.2.23. Y = ex 3 /(a + bx)2 1. a = 1.0, b = 2.0, e = 0.2 2. a = 1.0, b = 3.0, e = 0.2 3. a = 1.0, b = 4.0, e = 0.2 a3y
+ 3a 2bxy + 3ab 2x 2y + b 3 x 3 y
ex 3 = 1. 2. 3.
a = 1.0, b = 2.0, e = 0.1 a = 1.0, b = 3.0, e = 0.1 a = 1.0, b = 4.0, e = 0.1
°
37 1
23
12233
2.2.20
2.2.19 1
23
1
2 3
1 23
2 3
2.2.22
2.2.21 2 3


7
I
1 23
12233
2.2.23
2.2.24
CRC Standard Curves and Surfaces
38
2.2.25. y = e(a + bx)lx 1. a = 1.0, b = 2.0, e = 0.04 2. a = 1.0, b = 4.0, e = 0.04 3. a = 1.0, b = 6.0, e = 0.04
xy  bex  ea
=
°
2.2.26. y = e(a + bxY Ix 1. a = 1.0, b = 2.0, e = 0.04 2. a = 1.0, b = 4.0, e = 0.04 3. a = 1.0, b = 6.0, e = 0.04 2.2.27. y = e(a
+ bX)3 Ix
xy  b 3cx 3  3ab 2cx 2  3a 2bcx  a 3e
=0 1. 2.
3.
a = 1.0, b = 2.0, e = 0.02 a = 1.0, b = 4.0, e = 0.02 a = 1.0, b = 6.0, e = 0.02
2.2.28. y = e(a + bx)lx 2 1. a = 1.0, b = 2.0, e = 0.04 2. a = 1.0, b = 4.0, e = 0.04 3. a = 1.0, b = 6.0, e = 0.04
x 2y  bex  ea =
°
2.2.29. y = e(a + bX)21x 2 1. a = 1.0, b = 2.0, e = 0.01 2. a = 1.0, b = 4.0, e = 0.01 3. a = 1.0, b = 6.0, e = 0.01 2.2.30. y = e(a
+
bX)31x 2
x 2y  b 3cx 3  3ab 2cx 2  3a 2bex
a 3e 1. 2.
3.
a a a
= 1.0, b = 2.0, e = 0.003 = 1.0, b = 4.0, e = 0.003 = 1.0, b = 6.0, e = 0.003
=
°
39
 3
3_ 2_
 2 1
1~~_
~~_
2
3
2.2.25
2.2.26 3
3
2
~:
1
2 3
2.2.27
2.2.28
~:.
,
=
~==:::::::::::::~_
1
~,
1====",~2
3
2.2.29
2.2.30
CRe Standard Curves and Surfaces
40
2.2.31. y = c(a + bx)jx 3 1. a = 1.0, b = 2.0, c = 0.02 2. a = 1.0, b = 4.0, c = 0.02 3. a = 1.0, b = 6.0, c = 0.02 2.2.32. y = c(a + bx )2jx 3 1. a = 1.0, b = 2.0, c = 0.01 2. a = 1.0, b = 4.0, c = 0.01 3. a = 1.0, b = 6.0, c = 0.01 2.2.33. y
= c(a + bX)3jx 3
x 3y  b 3cX 3  3ab 2cx 2  3a 2bcx
a 3c = 1.
2. 3.
a a a
= 1.0, b = 2.0, c = 0.002 = 1.0, b = 4.0, c = 0.002 = 1.0, b = 6.0, c = 0.002
a
41
~,
2
2.2.31
2.2.32
3
1====
2
2.2.33
CRC Standard Curves and Surfaces
42
2.3. FUNCTIONS WITH a 2 2.3.1. Y = cj(a 2 + x 2 ) Special case: c = a 3 gives witch
1. 2.
3.
a a a
= 0.2, c = 0.04
= 0.5, c = 0.04 = 0.8, c = 0.04
2.3.2. y = exj(a 2 Serpentine
1. 2. 3.
a a a
+ x 2)
= 0.2, c = 0.3 = 0.5, c = 0.3 = 0.8, c = 0.3
2.3.3. y = ex 2 j(a 2 + x 2 ) 1. a = 0.2, c = 1.0 2. a = 0.5, c = 1.0 3. a = 0.8, c = 1.0 2.3.4. y = ex 3 j(a 2 + x 2 ) 1. a = 0.2, c = 1.0 2. a = 0.5, c = 1.0 3. a = 0.8, c = 1.0
+ x 2 AND
a2 y + x 2y of Agnesi
C
= 0
xm
43
1
~~
2 3
2.3.1
2.3.2
2 3
2.3.3
2.3.4
44
CRC Standard Curves and Surfaces
2.3.5. y = c j[x(a 2 + x 2 )] 1. a = 0.2, c = 0.02 2. a = 0.5, c = 0.02 3. a = 0.8, c = 0.02 2.3.6. y = cj[x 2 (a 2 + x 2 )] 1. a = 0.2, c = 0.02 2. a = 0.5, c = 0.02 3. a = 0.8, c = 0.02 2.3.7. y = cx(a 2 + x 2 ) 1. a = 0.2, c = 1.0 2. a = 0.5, c = 1.0 3. a = 0.8, c = 1.0 2.3.8. y = cx 2 (a 2 + x 2 ) 1. a = 0.2, c = 1.0 2. a = 0.5, c = 1.0 3. a = 0.8, c = 1.0
4S 32
32 1
1
2.3.5
2.3.6 3
2.3.7
21
2.3.8
CRC Standard Curves and Surfaces
46
2.4. FUNCTIONS WITH a 2 2.4.1. Y = c/(a 2 1. a = 0.2, c = 2. a = 0.5, c = 3. a = 0.8, c = 2.4.2. 1. 2. 3.
x 2) 0.03 0.03 0.03
Y = cx/(a 2  x 2 ) a = 0.2, c = 0.1 a = 0.5, c = 0.1 a = 0.8, c = 0.1
2.4.3. Y = cx 2 /(a 2
1. 2. 3.
a a a
 x 2) = 0.2, c = 0.2 = 0.5, c = 0.2
= 0.8, c = 0.2
2.4.4. Y = cx 3 /(a 2
1. 2. 3.
 x 2) = 0.2, c = 0.2 = 0.5, c = 0.2 a = 0.8, c = 0.2
a a

x 2 AND xm
47 1
2
3
1
2
3
( 1
2.4.1
2
3
1
", 0
2
3
2.4.2
"!
~
\\\(7( 1
2.4.3
2
3
1
2.4.4
((
2
3
48
CRC Standard Curves and Surfaces
2.4.5. 1. 2. 3.
Y = c/[x(a 2  x 2 )] a = 0.2, c = 0.001 a = 0.5, c = 0.001 a = 0.8, c = 0.001
2.4.6. 1. 2. 3.
Y= a = a = a =
2.4.7. Y = 1. a = 2. a = 3. a = 2.4.8. 1. 2. 3.
c/[x 2 (a 2

x 2 )]
0.2, c = 0.0003 0.5, c = 0.0003 0.8, c = 0.0003 cx(a 2

x 2)
0.2, c = 1.0 0.5, c = 1.0 0.8, c = 1.0
Y = cx 2 (a 2 a = 0.2, c = a = 0.5, c = a = 0.8, c =
x2)
4.0 4.0 4.0
49 2
1
3
3
VV
U J
2
~
"'I
~
(
123
123
2.4.6
2.4.5
3
2 12
2.4.7
2.4.8
3
CRC Standard Curves and Surfaces
50
2.5. FUNCTIONS WITH a 3 2.5.1. 1. 2. 3.
Y
=
ej(a 3 + x 3 ) a = 0.2, e = 0.01 a = 0.3, e = 0.01 a = 0.4, e = 0.01
2.5.2. 1. 2. 3.
Y
=
2.5.3. 1. 2. 3. 2.5.4. 1. 2. 3.
a
=
a
=
a
=
exj(a 3 + x 3 ) 0.1, e = 0.01 0.3, e = 0.01 0.5, e = 0.01
Y = ex 2 j(a 3 + x 3 ) a = 0.1, e = 0.1
a
=
a
=
0.3, e 0.5, e
= =
0.1 0.1
Y = ex 3 j(a 3 + x 3 ) a = 0.1, e = 0.2
a a
= =
0.3, e 0.5, e
= =
0.2 0.2
2.5.5. 1. 2. 3.
Y = ej[x(a 3 + x 3 )] a = 0.5, e = 0.01
2.5.6. 1. 2. 3.
Y = ex(a 3 + x 3 ) a = 0.5, e = 1.0 a = 0.7, e = 1.0 a = 0.9, e = 1.0
a = 0.7, e = 0.01 a = 0.9, e = 0.01
+ x 3 AND
xm
51 3
3 2
2
3 21
2.5.1
2.5.2 3
2
J 3
2.5.3 3
3
2.5.4
2
2
3 21
n 1
2.5.5
2.5.6
52
CRC Standard Curves and Surfaces
2.6. FUNCTIONS WITH a 3 2.6.1. 1. 2. 3. 2.6.2. 1. 2. 3.
Y = e/(a 3 = 0.2, e = a = 0.3, e = a = 0.4, e = a
x 3) 0.01 0.01 0.01
Y = ex/(a 3  x 3 ) = 0.1, e = 0.01 = 0.3, e = 0.01 = 0.5, e = 0.01
a a a
2.6.3. y = ex 2 /(a 3  x 3 ) 1. a = 0.1, e = 0.1 2. a = 0.3, e = 0.1 3. a = 0.5, e = 0.1
2.6.4. Y = ex 3 /(a 3  x 3 ) 1. a = 0.1, e = 0.2 2. a = 0.3, e = 0.2 3. a = 0.5, e = 0.2 2.6.5. Y = e/[x(a 3  x 3 )] 1. a = 0.5, e = 0.01 2. a = 0.7, e = 0.01 3. a = 0.9, e = 0.01 2.6.6. 1. 2. 3.
Y = ex(a 3  x 3 ) a = 0.5, e = 1.0 a = 0.7, e = 1.0 a = 0.9, e = 1.0

x 3 AND xm
53 2
123
2.6.1
2.6.2
3
2.6.4
2.6.3
2
2.6.5
2
3
2.6.6
3
54
CRC Standard Curves and Surfaces
2.7. FUNCTIONS WITHa 4 2.7.1. Y = c/(a 4 + x 4 ) 1. a = 0.3, C = 0.007 2. a = 0.4, C = 0.007 3. a = 0.5, C = 0.007 Y = cx/(a 4 + x 4 ) = 0.2, C = 0.01 = 0.3, C = 0.01 = 0.4, C = 0.01
2.7.2. 1. 2. 3.
a a a
2.7.3. 1. 2. 3.
a = 0.3, c = 0.15 a = 0.4, c = 0.15 a = 0.5, c = 0.15
2.7.4. 1. 2. 3.
a a a
Y = cx 2 /(a 4
Y
+ x 4)
= cx 3 /(a 4 + x 4 ) = 0.2, c = 0.25 = 0.4, c = 0.25
= 0.6, c = 0.25
2.7.5. Y = cx 4 /(a 4 + x 4 ) 1. a = 0.2, c = 1.0 2. a = 0.5, c = 1.0 3. a = 0.8, c = 1.0 2.7.6. Y = cx(a 4 + x 4 ) 1. a = 0.5, c = 0.5 2. a = 1.0, c = 0.5 3. a = 1.2, c = 0.5
+ X4 AND xm
55
2.7.1
2.7.2
2.7.3
2.7.4 3
~1 3
2.7.5
2.7.6
2
56
CRC Standard Curves and Surfaces
2.8. FUNCTIONS WITH a 4 2.8.1. Y = c/(a 4  x 4 ) 1. a = 0.4, C = 0.01 2. a = 0.6, C = 0.01 3. a = 0.8, C = 0.01 2.8.2. Y = cx/(a 4  x 4 ) 1. a = 0.2, c = 0.01 2. a = 0.4, c = 0.01 3. a = 0.6, c = 0.01 2.8.3. y = ex 2/(a 4  x 4) 1. a = 0.2, c = 0.1 2. a = 0.4, c = 0.1 3. a = 0.6, c = 0.1 2.8.4. y = cx 3 /(a 4  x 4 ) 1. a = 0.2, c = 0.1 2. a = 0.4, c = 0.1 3. a = 0.6, c = 0.1 2.8.5. y = cx 4 /(a 4  x 4 ) 1. a = 0.2, c = 0.1 2. a = 0.5, c = 0.1 3. a = 0.8, c = 0.1 2.8.6. y = cx(a 4  x 4 ) 1. a = 0.4, c = 1.0 2. a = 0.8, c = 1.0 3. a = 1.0, c = 1.0

X4
AND xm
57 2
3
2
3
2.8.1
2
2.8.2 2
3
2
2.8.3
2.8.4 2
2
2.8.5
3
3
3
2.8.6
3
58
CRC Standard Curves and Surfaces
2.9. FUNCTIONS WITH (a 2.9.1. y = c(a
+ bX)lj2 AND
xm
+ bx)1/2
Parabola 1. 2. 3.
a a a
= 0.5, b = 0.5, c = 1.0 = 0.5, b = 1.0, c = 1.0
= 0.5, b = 2.0, c
2.9.2. y = cx(a
=
1.0
+ bX)1/2
Special case: c = 1/(3a) and b = "trisectrix of Catalan") 1. 2. 3.
a = 0.5, b = 0.5, c = 1.0 a = 0.5, b = 1.0, c = 1.0 a = 0.5, b = 2.0, c = 1.0
+ bx)1/2
2.9.3. 1. 2. 3.
y = cx 2(a a = 0.5, b a = 0.5, b a = 0.5, b
2.9.4. 1. 2. 3.
y a a a
2.9.5. 1. 2. 3.
= c(a + bX)1/2/X2 a = 0.5, b = 0.5, c = 0.1 a = 0.5, b = 1.0, c = 0.1 a = 0.5, b = 2.0, c = 0.1
2.9.6. 1. 2. 3.
y = c/(a + bX)1/2 a = 1.0, b = 1.0, c a = 1.0, b = 2.0, c a = 1.0, b = 4.0, c
= 0.5, c = 1.0 = 1.0, c = 1.0 = 2.0, c = 1.0
= c(a + bx)1/2/ X = 0.5, b = 0.5, c = 0.2 = 0.5, b = 1.0, c = 0.2 = 0.5, b = 2.0, c = 0.2
y
= 0.5 = 0.5 = 0.5
y2  bC 2X 3  ac 2x 2 = 0 1 gives Tschimhauser's cubic (also called
59
2.9.1
2.9.2
2.9.3
2.9.4
2.9.5
2.9.6
60
CRC Standard Curves and Surfaces
2.9.7. 1. 2. 3.
y = cx/(a + bX)1/2 a = 1.0, b = 1.0, c = 1.0
2.9.8. 1. 2. 3.
y = cx 2/(a + bX)1/2 a = 1.0, b = 1.0, c = 1.0
a = 1.0, b = 2.0, c = 1.0 a = 1.0, b = 4.0, c = 1.0
a = 1.0, b = 2.0, c = 1.0 a = 1.0, b = 4.0, c = 1.0
2.9.9. y = c/[x(a + bX)1/2] 1. a = 1.0, b = 0.8, c = 0.2 2. a = 1.0, b = 1.0, c = 0.2 3. a = 1.0, b = 1.2, c = 0.2 2.9.10. y = c/[x 2(a + bX)1/2] 1. a = 1.0, b = 0.8, c = 0.1 2. a = 1.0, b = 1.0, c = 0.1 3. a = 1.0, b = 1.2, c = 0.1 2.9.11. y = cx 1/ 2(a + bX)1/2 1. a = 2.0, b =  2.0, c = 1.0 2. a = 2.0, b =  3.0, c = 1.0 3. a = 2.0, b =  4.0, c = 1.0 4. a = 0.3, b = 1.0, c = 1.0 5. a = 0.5, b = 1.0, c = 1.0 6. a = 0.7, b = 1.0, c = 1.0
61
2 3
2
2
3
2.9.7
2.9.8
'((\
'7(\ 2
3
2
3
2.9.9
3
2.9.10 654
6 5 4
2.9.11
2.9.11
62
CRC Standard Curves and Surfaces
2.9.12. Y = cx 3/ 2(a Special case: b <
1. 2.
3. 4. 5. 6.
= = a = a = a = a = a a
4.0, 4.0, 4.0, 0.3, 0.5, 0.7,
b = b =
b b b b
= = = =
+
bX)1/2
a gives  4.0,  6.0,
y2  ac 2x 3  bC 2X 4
=
xy2  c 2bx  c 2a
=
a
x 3y  c 2bx  c 2a
=
a
piriform
= = 12.0, c = 1.0, c = 1.0, c = 1.0, C = C C
2.9.13. y = c(a + bX)1/2 j x l/2 1. a = 2.0, b = 4.0, c = 2. a = 2.0, b = 6.0, c = 3. a = 2.0, b = 8.0, c = 4. a = 2.0, b =  2.0, c = 5. a = 2.0, b =  3.0, c = 6. a = 2.0, b =  6.0, C = 2.9.14. y = c(a + bX)1/2jx 3/ 2 1. a = 2.0, b = 4.0, c = 2. a = 2.0, b = 6.0, C = 3. a = 2.0, b = 8.0, c = 4. a = 2.0, b =  2.0, C = 5. a = 2.0, b =  3.0, c = 6. a = 2.0, b =  6.0, c =
1.0 1.0 1.0 1.0 1.0 1.0
0.1 0.1 0.1 0.1 0.1 0.1
0.1 0.1 0.1 0.1 0.1 0.1
a
63
2.9.12
2.9.12
+*~4
2.9.13
2.9.13
2.9.14
2.9.14
CRC Standard Curves and Surfaces
64
2.9.15. Y = cx 1/ 2/(a 1. a = 1.0, b = 2. a = 1.0, b = 3. a = 1.0, b = 4. a = 4.0, b = 5. a = 4.0, b = 6. a = 4.0, b = 
+ bx)1/2 3.0, 5.0, 8.0, 2.0, 4.0, 8.0,
c = 1.0 c = 1.0 c = 1.0 c = 1.0 c = 1.0 c = 1.0
ay2 + bxy2  C2X 3 = 2.9.16. Y = cx 3/ 2/(a + bx)1/2 Special case: b =  a gives cissoid of Diodes
1. 2. 3. 4. 5. 6.
a a a a a a
= = = = = =
1.0, 1.0, 1.0, 1.0, 1.0, 1.0,
b b b b b
= = =
3.0, 4.0, 8.0, =  0.5, =  1.0, b =  2.0,
c = 1.0 c = 1.0 c = 1.0 c = 1.0 c = 1.0 c = 1.0
°
65
2J~i
6
5
t
~
2.9.15
2.9.15 6
~
;4 2.9.16
2.9.16
5
4
66
CRC Standard Curves and Surfaces
2.10. FUNCTIONS WITH (a 2

X 2 )1/2
AND xm
2.10.1. Y = c(a 2  X 2)1/2 Ellipse (c = 1 gives circle)
1. 2.
3. 4.
5. 6.
a = 1.00, c a = 0.75, c a = 0.50, c a = 0.5, c a = 0.5, c a = 0.5, c
= 1.0 =
1.0
= 1.0 =
1.20
= 1.60 = 2.00
y2  c 2a 2x 2 + C2X 4 = 0 2.10.2. Y = cx(a 2  x 2)1/2 Eight curve (also called lemniscate of Gerono)
1. 2. 3.
a = 0.6, c = 1.0 a = 0.8, c = 1.0 a = 1.0, c = 1.0
2.10.3. y = cx 2(a 2  X2)1/2 1. a = 0.6, c = 2.0 2. a = 0.8, c = 2.0 3. a = 1.0, c = 2.0 2.10.4. y = c(a 2  X 2)1/2 Ix 1. a = 0.50, c = 0.1 2. a = 0.75, c = 0.1 3. a = 1.00, c = 0.1 2.10.5. y = c(a 2  x 2)1/2Ix 2 1. a = 0.50, c = 0.1 2. a = 0.75, c = 0.1 3. a = 1.00, c = 0.1
67 6
2.10.1
2.10.1
2.10.2
2.10.3
3
2.10.4
2.10.5
68
CRe Standard Curves and Surfaces
2.10.6. Y
1. 2. 3.
=
c/(a 2
x 2 )1/2

a = 0.50, c = 0.1 a = 0.75, c = 0.1 a = 1.00, c = 0.1
2.10.7. Y = c/[x(a 2  X 2 )1/2] 1. a = 0.50, c = 0.1
2. 3.
a = 0.75, c = 0.1 a = 1.00, c = 0.1
2.10.8. Y = cx/(a 2

X 2 )1/2
Bulletnose curve 1. 2. 3.
a = 0.50, c = 0.4 a = 0.75, c = 0.4 a = 1.00, c = 0.4
2.10.9. Y
1. 2. 3.
=
cx 2 /(a 2
_ X 2 )1/2
a = 0.50, c = 0.4 a = 0.75, c = 0.4 a = 1.00, c = 0.4
69 1
2
3
1
2
3
LJu ~
~fJ
2.10.6
2.10.7
1
2.10.8
2
3
1
2.10.9
2
3
CRC Standard Curves and Surfaces
70
2.11. FUNCTIONS WITH (x 2 2.11.1. y = c(x 2  a 2 )1!2 Hyperbola
4. 5.
a = 0.1, c = 1.00 a = 0.3, c = 1.00 . a = 0.5, C = 1.00 a = 0.3, C = 0.75 a = 0.3, C = 1.00
6.
a
1. 2. 3.
= 0.3,
C
= 1.50
2.11.2. y = cx(x 2  a 2 )1/2 Kampyle of Eudoxus
1. 2.
3.
a = 0.1, a = 0.4, a = 0.7,
C = C C
1.0
= 1.0 = 1.0
2.11.3. y = CX 2(X 2 1. a = 0.1, C = 2. a = 0.4, C = 3. a = 0.7, C =
 a 2 )1/2
1.0 1.0 1.0
2.11.4. y = c(x 2 1. a = 0.1, C = 2. a = 0.3, C = 3. a = 0.5, C =
1.0 1.0 1.0
2.11.5. y = c(x 2 1. a = 0.1, C = 2. a = 0.2, C = 3. a = 0.3, C =
0.2 0.2 0.2
a 2 )1!2 Ix
a 2 )1/2Ix 2

a 2 )1/2 AND xm
71 6 5
4
2.11.1
2.11.1
2.11.2
2.11.3
2.11.4
2.11.5
CRC Standard Curves and Surfaces
72
2.11.6. Y = c/(x 2  a 2 )1/2 1. a = 0.1, c = 0.1 2. a = 0.3, c = 0.1 3. a = 0.5, c = 0.1 2.11.7. Y = c/[x(x 2  a 2 )1!2] 1. a = 0.1, c = 0.02 2. a = 0.3, c = 0.02 3. a = 0.5, c = 0.02 2.11.8. Y = cx/(x 2 Cross curve
1. 2.
3.
a a a

a 2 )1/2
= 0.1, c = 0.2
= 0.3, c = 0.2 = 0.5, c = 0.2
2.11.9. Y = CX 2 /(X 2 _ a 2 )1/2 1. a = 0.2, c = 0.5 2. a = 0.3, c = 0.5 3. a = 0.4, c = 0.5
73 1
2.11.6
2
3
1
2
3
2.11.7
J "l ~ ~ l rllfrr C;
'2.11.8
2.11.9
74
CRC Standard Curves and Surfaces
2.12. FUNCTIONS WITH (a 2 2.12.1. Y = c(a 2 + X 2)1/2 1. a = 0.1, c = 1.00 2. a = 0.3, c = 1.00 3. a = 0.5, c = 1.00 4. a = 0.1, c = 3.00 5. a = 0.3, c = 1.00 6. a = 0.5, c = 0.60 2.12.2. Y = cx(a 2 + X 2)1/2 1. a = 1.0, c = 0.5 2. a = 2.0, c = 0.5 3. a = 4.0, c = 0.5 2.12.3. Y = cx 2(a 2 + X 2)1/2 1. a = 1.0, c = 0.5 2. a = 2.0, c = 0.5 3. a = 4.0, c = 0.5 2.12.4. Y = c(a 2 + X 2)1/2 Ix 1. a = 0.2, c = 0.2 2. a = 0.5, c = 0.2 3. a = 0.8, c = 0.2 2.12.5. Y = c(a 2 + x 2)1/2Ix 2 1. a = 0.2, c = 0.2 2. a = 0.5, c = 0.2 3. a = 0.8, c = 0.2
+ X 2 )1/2 AND xm
75 321
4
5
6
2.12.1
2.12.1 3
2.12.2
2
3
2
2.12.3 1 2 3
123
~LJ)~
,r,r 2.12.4
2.12.5
76
CRC Standard Curves and Surfaces
2.12.6. Y = e /(a 2 + X 2)1/2 1. a = 0.2, e = 0.2 2. a = 0.5, e = 0.2 3. a = 0.8, e = 0.2 2.12.7. Y = e/[x(a 2 + 1. a = 1.0, e = 0.5 2. a = 2.0, e = 0.5 3. a = 4.0, e = 0.5
X 2 )1/2]
2.12.8. Y = ex/(a 2 + x 2 )1/z 1. a = 0.5, e = 1.0 2. a = 1.0, e = 1.0 3. a = 2.0, e = 1.0 2.12.9. Y = ex 2 /(a 2 + X 2 )1/2 1. a = 0.1, e = 1.0 2. a = 0.5, e = 1.0 3. a = 1.0, e = 1.0
2.12.6
2.12.8
2.12.9
CRC Standard Curves and Surfaces
78
2.13. MISCELLANEOUS ALGEBRAIC FUNCTIONS 2.13.1. y = e(e + x)/(b 1. a = 0.5, b = 0.3, 2. a = 0.5, b = 0.6, 3. a = 0.1, b = 0.6, 4. a =  0.5, b = 0.3, 5. a =  0.5, b = 0.6, 6. a =  0.1, b = 0.6, 2.13.2. y = 1. a = 2. a = 3. a = 4. a = 5. a = 6. a =
x) e = e = e = e = e = e =
by  xy  ex  ae = 0
0.1 0.1 0.1 0.1 0.1 0.1
c[(a + x)/(b  X)Jl/2 1.0, b = 0.8, e = 0.2 1.0, b = 0.3, e = 0.2 0.5, b = 0.3, e = 0.2  0.1, b = 0.5, e = 0.2  0.1, b = 0.8, e = 0.2  0.3, b = 0.8, e = 0.2
by2  xy2  e 2x  ae 2 = 0
2.13.3. y = ex[(a + x)/(b  x)]1/2 by2  xy2  ae 2x 2  e 2x 3 Special cases: a = b gives right strophoid a = 3b gives triseetrix of Maclaurin
1. 2. 3. 4. 5. 6.
a= 0.3, e = 1.0, b = 1.0, b = 0.8, e = a = a =  0.3, b = 0.8, e = 1.0, b = 0.1, e = a= a = 1.0, b = 0.3, e = 0.6, b = 0.3, e = a =
0.4 0.4 0.4 0.4 0.4 0.4
=
0
79 1
4
23
(f 1
(
32
4
2.13.1
2.13.1
I 2.13.2
2.13.2
4
1
2.13.3
56
2 3
2.13.3
56
80
CRC Standard Curves and Surfaces
2.13.4. y
(c/x)[(a
=
+
x)/(b  x)F/2
bx 2y2  x 3 y2 
c 2x 
ac 2
=0 1. 2. 3. 4.
5. 6.
a
=
a a a a a
= =

= = =
2.13.5. y
1.0, 1.0, 0.3, 1.0, 1.0, 0.6,
b b b
=
b
=
b b
=
= =
=
0.3, 0.8, 0.8, 0.1, 0.3, 0.3,
c c c c c c
= = = = = =
0.05 0.05 0.05 0.05 0.05 0.05
= c{[a 2x 2/(x  b)2]  x 2F/2
b 2y2  2bxy2 + x 2y2 + b 2x 2 a 2x 2  2bx 3 + X4 = 0
Conchoid of Nicomedes (also called "cochloid") 1. 2. 3.
a
=
a a
= =
2.13.6. y
1. 2. 3.
a a
=
a
=
= = =
0.50, c 0.25, C 0.50, C
= = =
1.0 1.0 1.0
= c(a 2 + x 2)/(b 2  x 2) 1.0, b 1.0, b 0.7, b
=
2.13.7. y
1. 2. 3.
0.50, b 0.50, b 0.25, b
=
= = =
0.7, 0.5, 0.5,
C
=
C
=
C
=
0.1 0.1 0.1
c[x  (x 2  a 2)1/2]
a = 0.3, a = 0.5, a = 0.7,
C
= 1.0
C
=
C =
1.0 1.0
2.13.8. y = c[x  (a 2 + X 2)1/2]
1. 2. 3.
a = 0.3, a = 0.5, a = 0.7,
C
C C
= 1.0 = 1.0 = 1.0
81
2.13.4
2.13.4 23
~ 2135
1
f!
12
3
213~
t"=====12 3
;J)
3
2 1
2.13.7
2.13.8
CRC Standard Curves and Surfaces
82
Bieorn 1. 2.
3.
a a a
= 0.50 =
0.75
= 1.00
2.13.10. y = e(1  Ix/alnjm)m/n yn/m + elx/al n / m  e = 0 Hyperellipse for n/m > 2, hypoellipse for n/m < 2 1. 2. 3. 4.
5. 6.
0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
y = = = a = a = a =
= e(1 + Ix/aln/m)m/n
2.13.11. 1. a 2. a 3. a 4.
5. 6.
0.2, 0.2, 0.2, 0.2, 0.2, 0.2,
n/m = n /m = n/m = n/m = n/m = n/m =
i, e =
a = a= a = a = a = a =
n/m n/m n/m n/m n/m n/m
= = = = = =
1.0
j, e = 1.0 1, 1, 2, 4,
e e e e
= = = =
1.0 1.0 1.0 1.0
i, e = 0.2
j, e = 0.2
1, 1, 2, 4,
e e e e
= = = =
0.2 0.2 0.2 0.2
yn/m _ elx/al n / m  e = 0
83 3
2.13.9
2.13.10
2.13.10 2
2.13.11
4
3
2.13.11
56
84
CRC Standard Curves and Surfaces
2.13.12. y
=
cx axn
= 10.0, 2. a = 10.0, 3. a = 10.0, 4. a = 2.0, 5. a = 5.0, 6. a = 8.0, 1.
a
c c c c c c
= = = = = =
1.0, 1.0, 1.0, 1.0, 1.0, 1.0,
n n
n
a = 0.75, c = 5.0 a = 1.00, c = 5.0 a = 1.50, c = 5.0
2.13.14. y = cx(1 + a 2x 2)2 1. a = 1.0, c = 0.1 2. a = 2.0, c = 0.1
3.
a
=
4.0, c
=
0.1
= 3
n = 2 n = 2
2.13.13. y = c(1  x2)a/x
1. 2. 3.
= 1
n = 2
= 2
85
2.13.12
2.13.12
2.13.13
2.13.14
CRC Standard Curves and Surfaces
86
2.13.15. 1. a 2. a 3. a
cx 2(1 = 1.0, c = = 2.0, c = = 4.0, c = y
=
2.13.16. xn 
nroll mill 1. 2.
n = 2 n = 3
+ a 2x 2)2 0.1 0.1 0.1
(~)xn2y2 + (~)xn4y4  ...
=
an
87
2.13.15 2
00 2.13.16
II 2.13.16
CRC Standard Curves and Surfaces
88
2.14. ALGEBRAIC FUNCTIONS EXPRESSIBLE IN POLAR COORDINATES 2.14.1. r = c(2a cos 8 + 1) Limaq,on of Pascal Domain: [0 < 8 < 21T] Special cases: a = ~ gives cardioid a = 1 gives trisectrix 1. 2. 3.
a a a
= 1.00, c = 0.25 = 0.50, c = 0.50 = 0.25, c = 0.50
2.14.2. r2 = c 2 cos 28 Lemniscate of Bernoulli Domain: [0 < 8 < 21T] 1.
c
=
1.0
2.14.3. r = c cot 8 Kappa curve Domain: [0 < 8 < 21T] 1.
c
=
0.6
2.14.4. r2 = c 2(1  a 2 sin 2 8) Hippopede curve Domain: [0 < 8 < 21T] 1. 2.
3.
a a a
= 0.70, c = 1.0 = 0.85, c = 1.0 = 1.00, c = 1.0
89
2.14.1
2.14.2
2.14.3
2.14.4
90
CRC Standard Curves and Surfaces
2.14.5.
r2
= a
2
. 2
8
b2
cos sin 2 8  cos 2 8
SIll

2
8
Devil's curve Domain: [0 < 8 < 27T] 1. 2.
3.
a = 0.2, b = 0.8 = 0.4, b = 0.8 = 0.6, b = 0.8
a a
2.14.6. r = (cos 8)(4a sin 2 8  b) Folium Domain: [0 < 8 < 7T] 1. 2. 3.
a a a
= 0.25, b = 1.0
= 0.50, b = 1.0 = 1.00, b = 1.0
2.14.7. r = c sin 8 cos 2 8 Bifolia Domain: [0 < 8 < 7T] 1. 2.14.8.
c = 3.0 r2
= (b 4
a 4 sin 2 28)1/ 2 +a 2 COS 28 
(x 2
+ y2 + a 2 )2
Cassinian oval Special case: a = b gives lemniscate of Bernoulli Domain: [0 < 8 < 27T] 1. 2. 3.
a = 0.45, b = 0.5 a = 0.50, b = 0.5 a = 0.55, b = 0.5
'k

4a 2 x 2

b1 =
0
91 3 2 1
3
2.14.5
2.14.6
2.14.7
2.14.8
CRC Standard Curves and Surfaces
92 2.14.9.
r = c[1 + 2 sinCe /2)]
(x 2 + y2)(X 2 + y2 + e 2  2ef [2e(x2 + y2)  2ex] = 0
Nephroid of Freeth Domain: [0 1.
< e < 4'7T]
e = 0.3
2.14.10. r
= e cos 3 (e /3)
4(x 2 + y2 _ ex)3 _ 27e2(x2
=0 Cayley's sextet Domain: [0 1.
< e < 3'7T]
e = 1.0 1acose
2.14.11. r
= e 1 + a cos e
Domain: [0 1.
2. 3. 4.
a a
< e < 2'7T]
= 0.50, e = 0.3 = 0.675, e = 0.3
a = 1.00, e = 0.3 a
= 10.0,
e = 0.3
+ y2 + ac.x:)Z (x 2 + y2)(e 
(x 2
ax)2
= 0
+ y2)2
93
2.14.9
2.14.10
2.14.11
2.14.11
3
4
4
CRC Standard Curves and Surfaces
94
2.15. ALGEBRAIC FUNCTIONS EXPRESSED PARAMETRICALLY c(8at 3 + 24t 5 ) y = c(  6at 2  15t 4 ) Butterfly catastrophe
2.15.1.
1. 2.
X =
a a
= 5.0, c = 0.03; 1.46 < t < 1.46 = 7.0, c = 0.02; 1.68 < t < 1.68
2.15.2. x = c(  2at  4t 3 ) y = c(at 2 + 3t 4 ) Swallowtail catastrophe
1. 2.
a a
= 1.0, c = 0.5; 1 < t < 1
=  2.0, c = 0.5;  2 < t < 2
95 2
2.15.1
2.15.1 2
2.15.2
2.15.2
97
Chapter 3
TRANSCENDENTAL FUNCTIONS This chapter treats the transcendental functions: trigonometric, logarithmic, and exponential. The equations found in this chapter can mostly be found in tables of integrals. Traditional or accepted names for certain curves are included wherever appropriate. A final section of the chapter comprises curves which are more easily expressed in the polar form r = fUJ) than in the Cartesian form y = f(x).
98
CRC Standard Curves and Surfaces
3.1. TRIGONOMETRIC FUNCTIONS WITH sinn(ax) AND cosm(bx) (n, m INTEGERS) 3.1.1. Y = sin(27Tx) 3.1.2. y = COS(27T x) 3.1.3. y = tan(27Tx) 3.1.4. y = COt(27TX) 3.1.5. y = 0.25 CSC(27TX) 3.1.6. Y = 0.25 sec(27Tx)
99
3.1.1
3.1.2
3.1.3
U U) ~ n n n n 3.1.5
3.1.6
100
CRC Standard Curves and Surfaces
3.1.9. y = sin(27Tax) sin(27Tbx) Modulated sine wave
1. 2. 3. 4.
a
=
a
=
0.5, 0.5, a = 0.5, a = 0.5,
b = 1.0 b = 1.5 b = 2.0 b = 2.5
3.1.10. y = cos(27Tax)cos(27Tbx) 1. a = 0.5, b = 1.0 2. a = 0.5, b = 1.5 3. a = 0.5, b = 2.0 4. a = 0.5, b = 2.5
101
3.1.7
3.1.8
3.1.9
3.1.9 2
3.1.10
3.1.10
102
CRC Standard Curves and Surfaces
3.1.11. y = sin(27Tax)cos(27Tbx) 1. a = 0.5, b = 1.0 2. a = 0.5, b = 1.5 3. a = 0.5, b = 2.0 4. a = 0.5, b = 2.5 3.1.12. Y = 2.0 sin(27Tx) cos 2 (27TX) 3.1.13. y
=
2.0 COS(27TX) sin 2 (27Tx)
103
3.1.11
3.1.11
3.1.12
3.1.13
104
CRC Standard Curves and Surfaces
3.1.14. Y
=
0.25 sin(27Tx)jcos 2 (27TX)
3.1.15. y = 0.25 sin 2 (27Tx)jcos(27TX) 3.1.16. y = 0.25 COS(27Tx)jsin 2 (27Tx) 3.1.17. y
=
0.25 cos 2 (27Tx)jsin(27Tx)
105
3.1.14
3.1.15
3.1.16
3.1.17
106
CRC Standard Curves and Surfaces
3.2. TRIGONOMETRIC FUNCTIONS WITH 1 ± sinn(ax) AND 1 3.2.1. y
=
0.5/[1 +
± cosm(bx)
COS(27T x)]
3.2.2. Y = 0.5/[1  COS(27TX)]
+ COS(27TX)]
3.2.3. y
=
0.5[sin(27Tx)]j[1
3.2.4. Y
=
0.5[sin(27Tx)]j[1  COS(27TX)]
3.2.5. Y
=
0.5[cos(27Tx)]/[1
+ COS(27TX)]
3.2.6. y = 0.5[cos(27Tx)]j[1  COS(27TX)]
107
~
)
uu
3.2.1
3.2.2
3.2.3
3.2.4
3.2.5
3.2.6
108
CRC Standard Curves and Surfaces
3.2.7. Y = 0.5/[1 3.2.8. Y
=
3.2.9. Y = 1. a = 2. a = 3. a =
+ COS(27TX)]l/2
0.5/[1  COS(27T x )]1/2 0.5/[a 2 + b 2 cos 2(27TX)] 0.0, b = 1.0 1.0, b = 1.0 2.0, b = 1.0
3.2.10. Y = 0.5/[a 2  b 2 COS 2(27TX)] 1. a = 0.0, b = 1.0 2. a = 1.0, b = 1.0 3. a = 2.0, b = 1.0 3.2.11. Y = sin(27Tx)/[1
+ cos 2(27TX)]
109
)
~
3.2.9
3.2.11
uu
\ n n(' 3.2.10
3.2.12
110
CRe Standard Curves and Surfaces
3.2.13. y = 2.0[sin(2'7Tx)]j[1
+ sin 2 (2'7Tx)]
3.2.14. y = 2.0[cos(2'7Tx)]j[1
+ cos 2 (21TX)]
3.2.15. y = [sin 2(2'7Tx)]j[1
+ cos 2(2'7Tx)]
3.2.16. y = [cos2(2'7Tx)]j[1
+ sin 2(2'7Tx)]
3.2.17. y = 2.0[sin 2 (2'7Tx)]j[1
+ sin2(2'7Tx)]
3.2.18. y = 2.0[cos 2(2'7Tx)}j[1
+ cos 2(2'7Tx)]
111
3.2.13
3.2.14
3.2.15
3.2.16
3.2.17
3.2.18
112
CRC Standard Curves and Surfaces
3.3. TRIGONOMETRIC FUNCTIONS WITH a sinn(cx) + b cosm(cx) 3.3.1. 1. 2. 3.
a COS(27TX) + b sin(27Tx) 0.4, b = 0.8 a = 0.6, b = 0.6 a = 0.8, b = 0.4
3.3.2. 1. 2. 3.
Y = l/[a COS(27TX) + b sin(27Tx)] a = 1.0, b = 1.0 a = 1.0, b = 2.0 a = 1.0, b = 4.0
3.3.3. 1. 2. 3.
Y = a 2 COS 2 (27TX) a = 0.25, b = 1.0 a = 0.50, b = 1.0 a = 0.75, b = 1.0
Y a
3.3.4. Y 1. a 2. a 3. a 3.3.5. 1. 2. 3.
=
=
= = = =
+ b 2 sin2 (27Tx)
1/[a 2 cos 2 (27TX) 1.0, b = 1.5 1.0, b = 2.0 1.0, b = 4.0
+ b 2 sin2 (27Tx)]
Y = [sin(27Tx)]j[a COS(27TX) + b sin(27Tx)] a = 1.0, b = 2.0 a = 1.0, b = 4.0 a = 1.0, b = 6.0
3.3.6. Y = 1. a = 2. a = 3. a =
[COS(27Tx)]j[a COS(27TX) 1.0, b = 2.0 1.0, b = 4.0 1.0, b = 6.0
+ b sin(27Tx)]
113 1 23
3 2
~ ~
1 23
3.3.1
3.3.2
3
3.3.4
3.3.3 123
123
3 1
3.3.5
3.3.6
114
CRC Standard Curves and Surfaces
3.4. TRIGONOMETRIC FUNCTIONS OF MORE COMPLICATED ARGUMENTS 3.4.1. y 1. a
=
3.4.2. y 1. a
=
=
=
sin(a1T Ix) 1.0
cos(a1T /x) 1.0
3.4.3. y = sin(a1Tlxl n1m ) 1. a = 4.0, n/m = t 2. a = 4.0, n/m = 2 3.4.4. y = cos(a1Tlxl n1m ) 1. a = 4.0, n/m = t 2. a = 4.0, n/m = 2
115
3A.1
3A.2 2
3A.3
3A.3 2
3AA
3AA
CRC Standard Curves and Surfaces
116
3.4.5. y = sin[ 7T cos(a7T x) /2] 1. a = 2.0 3.4.6. y = sin[ 7T sin(a7T x) /2] 1. a = 2.0 3.4.7. y = cos[7Tsin(a7Tx)/2] 1. a = 2.0 3.4.8. y = 1.
a
COS[7T
= 2.0
cos(a7Tx)/2]
117
3.4.5
3.4.6
3.4.7
3.4.8
118
CRC Standard Curves and Surfaces
3.5. INVERSE TRIGONOMETRIC FUNCTIONS 3.5.1. Y = (1/17 )arcsin x 3.5.2. y = (1/17 )arccos x 3.5.3. y = (l/17)arctan lOx 3.5.4. y = (1/17)arccot lOx 3.5.5. y = (1/17 )arcsec lOx 3.5.6. y = (1/17 )arccsc 10 x
119
3.5.1
3.5.2
3.5.3
3.5.4
3.5.5
3.5.6
CRC Standard Curves and Surfaces
120
3.6. LOGARITHMIC FUNCTIONS 3.6.1. y = 0.25 In lOx 3.6.2. y = 0.25 In[lj(lOx)] 3.6.3. y 3.6.4. y 1. 2.
3.
=
0.25 jln lOx
= 0.5 In[(x = 0.1 = 0.3 a = 0.5
a a
+ a)j(x
 a)]
121
3.6.1
3.6.2
3.6.3
3.6.4
CRC Standard Curves and Surfaces
122
3.6.5. 1. 2. 3.
y
=
a =
a
=
a
=
0.5In(x 2 0.5 1.0 2.0
+ a2)
3.6.6. Y = 0.25In[10(x 2 1. a = 0.1 2. a = 0.3 3. a = 0.5 3.6.7. y = 1. a = 2. a = 3. a =
O.5ln[x 0.1 0.3 0.5
3.6.8. y = O.5ln[x 1. a = 0.1 2. a = 0.3 3. a = 0.5

a 2 )]
+ (x 2 + a 2 )1/2]
+ (x 2 
a 2 )1/2]
123
3
123
3
3.6.5
3.6.6
3.6.7
3.6.8
2
CRC Standard Curves and Surfaces
124
3.7. EXPONENTIAL FUNCTIONS 3.7.1. 1. 2. 3. 4.
5. 6.
y a a a a a a
= ce ax = = =
= = =
1.0, c = 0.1 2.0, c = 0.1 3.0, c = 0.1  1.0, c = 1.0  2.0, c = 1.0  3.0, c = 1.0
3.7.2. y = 1/(a + be CX ) Special case: a = 1, b = 1 gives sigmoidal curve 1. 2. 3. 4. 5. 6.
3.7.3. 1. 2. 3. 4. 5. 6.
a a a a a a
= 1.0, b = 1.0, c = 2.0 = 1.0, b = 1.0, c = 4.0 = 1.0, b = 2.0, c = 4.0 =  2.0, b = 2.0, c = 2.0 =  2.0, b = 2.0, c = 4.0 =  2.0, b = 4.0, c = 4.0
y a a a a a a
= = = = = = =
ae bx + ce dx 1.0, b = 2.0, 1.0, b = 1.0, 1.0, b = 1.0, 0.1, b = 3.0, 0.1, b = 4.0, 0.1, b = 5.0,
c = 1.0, d = 3.0 c = 1.0, d = 3.0 c = 1.0, d = 5.0 c = 0.1, d = 3.0 c = 0.1, d = 2.0 c = 0.1, d = 1.0
125 3
3.7.1
3.7.1 6
5 4
3.7.2
3.7.2
654
3
2
3 2 1
3.7.3
3.7.3
CRC Standard Curves and Surfaces
126
3.7.4. 1. 2. 3. 4.
5. 6.
Y = l/(ae bX + ce dx ) a = 10.0, b = 2.0, c = 10.0, d = 3.0 a = 10.0, b = 1.0, c = 10.0, d = 3.0 a = 10.0, b = 1.0, c = 10.0, d = 5.0 a = 1.0, b = 3.0, c = 1.0, d =  3.0 a = 1.0, b = 4.0, c = 1.0, d =  2.0 a = 1.0, b = 5.0, c = 1.0, d = 1.0
3.7.5. y = c exp(ax 2 ) a < 1 gives Gaussian curve (also called normal curve)
1. 2.
a
= 1.0, c = 1.0
4.
a =  2.0, c = 1.0 a =  4.0, c = 1.0 a = 1.0, c = 0.3
5. 6.
a = a =
3.
2.0, c = 0.3 4.0, c = 0.3
3.7.6. y = c exp(l/ax) 1. a = 1.0, c = 0.2 2. a = 2.0, c = 0.2 3. a = 4.0, c = 0.2
127
3.7.4
3.7.4
4
3.7.5
3.7.5
3.7.6
128
CRC Standard Curves and Surfaces
3.7.7. y = c exp(1/ax 2 ) 1. a = 1.0, c = 0.1 2. a = 2.0, c = 0.1 3. a = 4.0, c = 0.1 3.7.8. y = c exp[l/(l  ax 2 )] 1. a = 2.0, c = 0.1 2. a = 4.0, C = 0.1 3. a = 8.0, C = 0.1 3.7.9. 1. 2. 3.
y = C exp[1/(l  alxl)] a = 2.0, C = 0.1 a = 4.0, C = 0.1 a = 8.0, C = 0.1
3.7.10. y = c[1 + exp(ax)]j[l  exp(bx)] 1. a = 1.0, b = 4.0, C = 0.02 2. a = 1.0, b = 1.0, C = 0.02 3. a = 4.0, b = 1.0, C = 0.02
129 3 2
3
3.7.7
3.7.8
32
3
3.7.9
1
2
3.7.10
1
2
130
CRC Standard Curves and Surfaces
3.8. HYPERBOLIC FUNCTIONS 3.8.1. y
=
0.1 sinh 5x
3.8.2. y
=
0.1 cosh 5x
Catenary
3.8.3. y
=
tanh 5x
3.8.4. y
=
0.1 coth 5x
3.8.5. y
=
sech 5x
3.8.6. y
=
0.1 csch5x
131
3.8.1
3.8.2
3.8.3
3.8.4
3.8.5
3.8.6
132
CRC Standard Curves and Surfaces
3.8.7. y = sinh 2 x 3.8.8. y
=
0.5 cosh 2 x
3.8.9. y
=
tanh 2 5x
133
3.8.8
3.8.7
3.8.9
134
CRC Standard Curves and Surfaces
3.8.10. y = 0.25/(sinh x cosh x) 3.8.11. y = sinh ax cosh bx 1. a = 0.5, b = 0.75 2. a = 1.0, b = 0.75 3. a = 1.0, b = 1.50 3.8.12. y = sinh ax sinh bx 1. a = 1.0, b = 1.5 2. a = 1.0, b = 2.0 3. a = 1.0, b = 3.0 3.8.13. y = 0.5 cosh ax cosh bx 1. a = 1.0, b = 1.25 2. a = 1.0, b = 2.00 3. a = 1.0, b = 4.00
135 3
3.8.11
3.8.10 3
3.8.12
2 1
3
3.8.13
2
2
136
CRC Standard Curves and Surfaces
3.9. INVERSE HYPERBOLIC FUNCTIONS 3.9.1. Y = 0.5 sinh 1 5x 3.9.2. y = 0.5 cosh 1 5x 3.9.3. y = 0.2tanh 1 x 3.9.4. y = coth 1 5x 3.9.5. y = 0.2sech 1 x 3.9.6. y = 0.2csch 1
X
137
3.9.1
3.9.2
3.9.3
3.9.4
3.9.5
3.9.6
138
CRC Standard Curves and Surfaces
3.10. TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS COMBINED 3.10.1. y = e ax sin(27Tbx) 1. a = 1.0, b = 4.0 2. a =  2.0, b = 4.0 3.10.2. y = e ax COS(27Tbx) 1. a = 1.0, b = 4.0 2. a =  2.0, b = 4.0
3.10.3. y = 0.5 e ax /sin(27Tbx) 1. a = 1.0, b = 4.0 2. a =  2.0, b = 4.0 3.10.4. y = O.5e ax /COS(27Tbx) 1. a =  1.0, b = 4.0 2. a =  2.0, b = 4.0
139
3.10.2
3.10.1
~
ij
~~
V
~
~~
2 1
A
~~
3.10.3
~~
, A
~
~~
3.10.4
1 2
140
CRC Standard Curves and Surfaces
3.11. TRIGONOMETRIC FUNCTIONS COMBINED WITH POWERS OF X 3.11.1. y = x sin(27Tax) 1. a = 4.0 3.11.2. y = x COS(27Tax) 1. a = 4.0 3.11.3. y = x/sin(27Tax) 1. a = 4.0 3.11.4. y = x/COS(27Tax) 1. a = 4.0 3.11.5. y = [sin(27Tax)]j27Tax sinc function 1. a = 4.0 3.11.6. y = [cos(27Tax)]j27Tax 1. a = 4.0
141
3.11.1
,
v~~
I
~~
3.11.3
3.11.5
3.11.2
~~v
~ ~,
v~~
A
~~v
~~ ~~ 3.11.4
3.11.6
n
142
CRC Standard Curves and Surfaces
3.11.7. Y = x sin 2 (2'7Tax) 1. a = 4.0 3.11.8. Y = x cos 2 (2'7Tax) 1. a = 4.0 3.11.9. Y = 0.025[sin(2'7Tax)]jx 2 1. a = 4.0 3.11.10. Y = 0.01[cos(2'7Tax)]jx 2 1. a = 4.0 3.11.11. Y = 0.5 x/sin 2 (2'7Tax) 1. a = 4.0 3.11.12. Y = 0.5 xlcos 2 (2'7Tax) 1. a = 4.0
143
3.11.7
3.11.8
3.11.9
3.11.10
3.11.11
3.11.12
144
CRC Standard Curves and Surfaces
3.11.13. y = 0.5 x/[l 1. a = 4.0
+ sin(27Tax)]
3.11.14. Y = 0.5 x/[l 1. a = 4.0
+ cos(27Tax)]
3.11.15. y = 0.5 x/[l  sin(27Tax)] 1. a = 4.0 3.11.16. y = 0.5 x/[l  cos(27Tax)] 1. a = 4.0
145
I
~\
3.11.13
~~
\~
3.11.14
~
3.11.16
l \~
3.11.15
CRC Standard Curves and Surfaces
146
3.12. LOGARITHMIC FUNCTIONS COMBINED WITH POWERS OF X 3.12.1. y = x In ax 1. a = 1.0 2. a = 2.0 3. a = 4.0 3.12.2. y = x 2 In ax 1.
2. 3.
a a
=
a
=
=
1.0 2.0 4.0
3.12.3. y = 0.05/(x In ax) 1. a = 1.0 2. a = 2.0 3. a = 4.0 3.12.4. y = 0.005/(x 2 In ax) 1. a = 1.0 2. a = 2.0 3. a = 4.0 3.12.5. y = O.lln(ax)/x 1. a = 1.0 2. a = 3.0 3. a = 9.0 3.12.6. y = 0.5 x/In(ax) 1. a = 1.0 2. a = 3.0 3. a = 9.0
147
3.12.1
~.12.2
3
3
2
2
2
3
3.12.3
3.12.4
3 2
3.12.5
3.12.6
(
148
CRC Standard Curves and Surfaces
3.12.7. Y = x In(ax + b) 1. a = 1.0, b = 2.0 2. a = 4.0, b = 2.0 3. a = 4.0, b = 4.0 4. a = 6.0, b =  1.0 5. a = 4.0, b =  1.0 6. a = 4.0, b =  2.0 3.12.8. Y = O.l[ln(ax + b )]jx 1. a = 1.0, b = 2.0 2. a = 4.0, b = 2.0 3. a = 4.0, b = 4.0 4. a = 6.0, b = 1.0 5. a = 4.0, b = 1.0 6. a = 4.0, b =  2.0 3.12.9. y = x In(x 2 1. a = 0.0 2. a = 0.5 3. a = 1.0
+ a2)
3.12.10. y = 0.5x In(x 2 1. a = 0.1 2. a = 0.3 3. a = 0.5

a2)
149 3
4
3 2
2
3.12.7
3.12.7
3.12.8
3.12.8
5
2
3
L4,~
3
3.12.9
3.12.10
150
CRC Standard Curves and Surfaces
3.13. EXPONENTIAL FUNCTIONS COMBINED WITH POWERS OF X 3.13.1. Y = 0.5 xe ax 1. a = 1.0 2. a = 2.0 3. a = 3.0 3.13.2. y = x 2 e ax 1. a = 1.0 2. a = 2.0 3. a = 3.0 3.13.3. Y = 4.0x 3e ax 1. a = 2.0 2. a = 4.0 3. a = 6.0 3.13.4. Y = 0.1 1. a = 1.0 2. a = 2.0 3. a = 3.0
eax Ix
3.13.5. Y = 0.03 e ax Ix 2 1. a = 2.0 2. a = 3.0 3. a = 4.0 3.13.6. Y = 0.01 e ax Ix 3 1. a = 3.0 2. a = 4.0 3. a = 5.0
151 3
2
1
3.13.1
3 2
3.13.2 321
3.12.3
~: 3.13.4
~:
I~: 3.13.5
1
3.13.6
152
CRC Standard Curves and Surfaces
3.13.7. y = ex exp(ax 2 ) 1. a =  1.0, e = 1.0 2. a =  2.0, e = 1.0 3. a =  3.0, e = 1.0 4. a = 1.0, e = 0.1 5. a = 2.0, e = 0.1 6. a = 3.0, e = 0.1 3.13.8. y = ex 2 exp(ax 2 ) 1. a =  1.0, e = 2.0 2. a =  2.0, e = 2.0 3. a =  3.0, e = 2.0 4. a = 1.0, e = 0.5 5. a = 2.0, e = 0.5 6. a = 3.0, e = 0.5
153 6
2
____~______~~3
3.13.7
3.13.7 65
2
3.13.8
3.13.8
4
154
CRC Standard Curves and Surfaces
3.14. HYPERBOLIC FUNCTIONS COMBINED WITH POWERS OF X 3.14.1. y
=
O.lx sinh5x
3.14.2. y = O.lx cosh 5x 3.14.3. y = x tanh 5x 3.14.4. Y = O.02(sinh5x)/x 3.14.5. Y = O.02(cosh5x)/x 3.14.6. Y = O.2(tanh5x)/x
")
155
3.14.1
3.14.2
3.14.3
3.14.4
3.14.5
3.14.6
156
CRC Standard Curves and Surfaces
3.15. COMBINATIONS OF TRIGONOMETRIC FUNCTIONS, EXPONENTIAL FUNCTIONS, AND POWERS OF X 3.15.1. y = 0.15xe aX sin(2'1Tbx) 1. a = 1.0, b = 4.0 2. a = 2.0, b = 4.0 3.15.2. y = 0.15xeaXcos(2'1Tbx) 1. a = 1.0, b = 4.0 2.
a
= 2.0, b = 4.0
3.15.3. Y = 0.1 e ax sin(2'1Tbx)/x 1. a = 1.0, b = 4.0 2. a = 2.0, b = 4.0 3.15.4. 1. 2.
y
= 0.1 e ax cos(2'1Tbx)/x
= 1.0, b = 4.0 a = 2.0, b = 4.0
a
157
2'~~~~~~~~~~~L4++~ 1
2 2
3.15.1
3.15.2
2
2
3.15.3
3.15.4
158
CRC Standard Curves and Surfaces
3.16. MISCELLANEOUS TRANSCENDENTAL FUNCTIONS 3.16.1. Y = a cosh~l(a/x)  (a 2 Tractrix
1. 2. 3.
a = 1.00 a = 0.75 a = 0.50
3.16.2. Y = x cot(rrx/2a) Quadratrix of Hippias
1. 2. 3.
a = 0.25 a = 0.35 a = 0.45
3.16.3. Y = 1  e ax Exponential Ramp
1. 2.
3.
2.0 4.0 a = 6.0 a
=
a
=
3.16.4. Y
1. 2.
3. 4.
5. 6.
=
cO 
2ax 2 )exp(ax 2 )
a =  3.0, c = 1.0 a =  6.0, c = LO a =  9.0, c = 1.0 a = 3.0, c = 0.1 a = 6.0, c = 0.1 a = 9.0, c = 0.1

X 2 )1/2
159 2
\ 3.16.1
1
2
65
4
3.16.2
====~1
3.16.3
3.16.4
3.16.4
3
160
CRC Standard Curves and Surfaces
3.16.5. Y = c arctan(e aX )  b Special case: a = 1, b = 1T12, c = 2 gives Gudermannian function
1. 2. 3.
a = 1.0, b = 1T14, c = 1.0 a = 3.0, b = 1T1 4, c = 1.0 a = 10.0, b = 1T14, c = 1.0
3.16.6. Y = c sin{ b
~da [( (b 
a)
~ +a
r
Sweep signal (linear) 1.
a = 5.0, b = 25.0, c = 0.5, d = 1.0
3.16.7. Y = sin(a1Tx) arcsin x 1. a = 1.0 2. a = 2.0 3. a = 3.0 3.16.8. y = c(bx)alnbx 1. a = 2.0, b = 2.0, c = 0.1 2. a = 2.0, b = 4.0, c = 0.1 3. a = 4.0, b = 4.0, c = 0.1 3.16.9. y = c/(bx)alnbx 1. a = 2.0, b = 2.0, c = 1.0 2. a = 2.0, b = 4.0, c = 1.0 3. a = 4.0, b = 4.0, c = 1.0
a 2 ]}
161
~
3.16.6
3.16.5
3
3
2
3.16.8
3.16.7
3.16.9
2
162
CRC Standard Curves and Surfaces
3.16.10. Y = Isin(a1Tx)IItan(a'ITx)l+l 1. a = 2.0 3.16.11. y = exp[b cos(a1Tx)  b] 1. a = 1.0, b = 2.0 2. a = 1.0, b = 10.0 3. a = 1.0, b = 50.0 3.16.12. y = signum[siIi(a1Tx)]lsin(a1Tx)1 1/ 1. a = 1.0, b = 10.0
b
3.16.13. y = sin(; sin(; sin( ... sin( a;x ) ... )))
Let n be the order of nesting of the sine function. 1.
a = 8, n = 5
163
3.16.11
3.16.10 r
,,....
,...
'
3.16.12
3.16.13
~
'
'
164
CRC Standard Curves and Surfaces
3.17. TRANSCENDENTAL FUNCTIONS EXPRESSIBLE IN POLAR COORDINATES 3.17.1. r = ce ao ~ In[(x 2 + y2)/c 2]  a arctan(y Ix) = 0 Logarithmic spiral (also called equiangular spiral or logistique)
1. 2.
a = 0.1, c = 0.10; 0 < 8 < 77T a = 0.2, c = 0.01; 0 < 8 < 77T
3.17.2. r = c8 1/ m (x 2 + y2)m/2  cmarctan(ylx) Archimedean spirals (m =1= 0)
1. 2. 3. 4.
c c
0.04, 0.20, c = 1.00, c = 0.50, = =
=
0
m = 1; 0 < 8 < 87T (Archimedes' spiral) m = 2; 0 < 8 < 87T (Fermat's spiral) m = 1; 1.00 < 8 < 67T (hyperbolic spiral) m = 2; 0.25 < 8 < 67T (lituus)
165 2
3.17.1
3.17.1 2
3.17.2
3.17.2
3
4
3.17.2
3.17.2
CRC Standard Curves and Surfaces
166
3.17.3. r = c cos me (x 2 + y2)1/ 2 Rhodonea (also called rose)
1. 2.
c = 1.0, m = 4.0; 0 < c = 1.0, m = 3.0; 0 <
C
cos[m arctan(y Ix)] = 0
e < 2'TT" e < 'TT"
3.17.4. r = c/cos me (x 2 + y2)1/2  c/cos[m arctan(y Ix)] = 0 Epispiral
1. 2.
c = 0.1, m = 4.0; 0 < c = 0.1, m = 3.0; 0 <
3.17.5. r = (4be)1/2 Parabolic spiral
1. 2.
a a
+a
e < 2'TT" e < 'TT"
[(x 2 + y2)1/ 2 
= 0.1, b = 0.01; 0 < e < 6'TT" = 0.1, b = 0.01; 0 < e < 6'TT"
aF 
4b arctan(y Ix)
=
0
167 2
3.17.3
3.17.3 2
3.17.4
3.17.4 2
3.17.5
3.17.5
168
CRC Standard Curves and Surfaces
3.17.6. r
=
c(sin aO)/O (x 2
+ y2)1/2
arctan(y Ix)  c sin[a arctan(y Ix)] = 0 Special case: a
1. 2.
=
1 gives cochleoid
a = 1.0, c = 1.0;  67T < 0 < 67T a = 2.0, c = 0.5;  37T < 0 < 37T
3.17.7. r = c /sinh aO (x 2 Spiral of Poinsot
1. 2.
1.

c /sinh[a arctan(y Ix)] = 0
a = 1.0; c = 0.5; 27T < 7T < 27T a = 0.5; c = 0.5,  47T < 0 < 47T
3.17.8. r = c/coshaO(x 2 Spiral of Poinsot
2.
+ y2)1/2
+ y2)1/2
a = 1.0, c = 1.0;  27T a = 0.5, c = 1.0;  47T

c/cosh[aarctan(y/x)]
< 0 < 27T < 0 < 47T
=
0
169 2
3.17.6
3.17.6 2
3.17.7
3.17.7 2
3.17.8
3.18.8
170
CRC Standard Curves and Surfaces
3.17.9. r = c(a 2 + 8 2 )1/2 (x 2 + y2)1/2  c{a 2 + [arctan(y /X)]2P/2 = 0 Special case: a = 1 gives involute of a circle 1. 2.
a = 1.0, c = 0.04; 0 < 8 < 6'lT a = 4.0, c = 0.04; 0 < 8 < 6'lT
3.17.10. r = c exp(a8 2 ) 1. a = 0.02, c = 0.04; 0 < 8 < 4'lT 2. a = 0.02, c = 1.00; 0 < 8 < 4'lT
\
171 2
3.17.9
3.17.9 2
3.17.10
3.17.10
172
CRC Standard Curves and Surfaces
3.18. PARAMETRIC FORMS 3.18.1. X = sin(at + b7T); y = sin t Lissajous curves (also called Bowditch curves) Let d = denominator of the parameter a
t
b = 0; 0 < t < 27Td
1.
a
=
2. 3. 4.
a
= }, b = 0; 0 < t < 27Td
a = a
=
L t,
b = 0; 0 < t < 27Td b = 0; 0 < t < 27Td
173 2
3.18.1
3.18.1
3
4
3.18.1
3.18.1
174
5. 6. 7. 8.
CRC Standard Curves and Surfaces
a=
t,
a = a =
%,
a
=
t i,
b = 0; 0 b = 0; 0 b = i;o b = i;o
3.18.2. x = cos t; Y Teardrop curve
1. 2.
3.
m = 1; 0 < m = 2; 0 < m = 3; 0 <
=
< t < 27rd < t < 27rd
< t < 27rd < t < 27rd sin t sin (t /2)m
t
< 27r < 27r
t
< 27r
t
175 5
6
3.18.1
3.18.1
7
8
3.18.1
3.18.1
3.18.2
176
CRC Standard Curves and Surfaces
3.18.3. x = at  b sin t; y = a  b cos t Cycloid 1. 2. 3.
<
a = 1/(41T), b = a;  lja t < 1/a (ordinary cycloid, a = b) a = 1/(41T), b = 2a; lja < t < 1/a (prolate cycloid, a < b) a = lj(41T), b = a/2; 1/a < t < 1/a (curtate cycloid, a > b)
177 2
3.18.3
3.18.3 3
3.18.3
178
CRC Standard Curves and Surfaces
3.18.4. x = d{(a  b)cos t + c cos[(a  b)t/b]} y = d{(a  b)sin t  c sin[(a  b)t/b]} Hypotrochoid
1.
a =
2. 3. 4. 5. 6.
a = 4.0, b = 1.0, c = 3.0, d = 0.15; a = 3.0, b = 1.0, c = 2.0, d = 0.25;
3.0, b = 1.0, c = 3.0, d = 0.15;
a = 4.0, b = 1.0, c = 2.0, d = 0.20; a = 3.0, b = 1.0, c = 1.0, d = 0.25; a = 4.0, b = 1.0, c = 1.0, d = 0.25;
°°<< °°<< °°<<
t t t t t t
< < < < < <
27T 27T 27T 27T 27T (deltoid) 27T (astroid)
179 2
3.18.4
3.18.4
3
4
3.18.4
3.18.4
5
6
3.18.4
3.18.4
CRC Standard Curves and Surfaces
180
3.18.5. x = d{(a y = d{(a
+ b )cos t + b)sin t 
n
c cos[(a + b)t /b c sin[(a + b)t/bn
Epitrochoid
1. 2. 3. 4. 5. 6.
a a a a
=
a a
=
= = = =
1.0, 2.0, 3.0, 3.0, 1.0, 2.0,
b b b b b b
= = = = = =
1.0, 1.0, 1.0, 1.0, 1.0, 1.0,
c c c c c c
= = = = = =
3.0, 3.0, 2.0, 5.0, 1.0, 1.0,
d d d d d d
= = = = = =
0.15; 0.15; 0.15; 0.10; 0.25; 0.25;
°°°<<< °°<< °<
t < 27T" t < 27T" t < 27T" t < 27T" t < 27T" (cardioid) t < 27T" (nephroid)
181 2
3.18.5
3.18.5
3
4
3.18.5
3.18.5
5
6
3.18.5
3.18.5
183
Chapter 4
POLYNOMIAL SETS The polynomial sets illustrated in this chapter are treated in detail in Abramowitz and Stegun1 and in Beyer. 2 Because efficient calculation of the curves is achieved by using the recurrence relations given in these references, the relations are repeated here for anyone who may wish to generate the curves for their own purposes.
184
CRC Standard Curves and Surfaces
4.1. ORTHOGONAL POLYNOMIALS 4.1.1. Legendre Polynomials Pn(x) Domain: [  1 < x < 1] Recurrence relation:
with Po(x) = 1 Pl(x)
o. 1. 2. 3. 4.
Po(x) Pix)
Pix) Pix) Pix)
5.
Ps(x)
6. 7.
P6 (x) Pix)
= X
185
4.1.1
4.1.1
186
CRC Standard Curves and Surfaces
4.1.2. Chebyshev Polynomials of the First Kind, Tn(x) Domain: [  1 < x < 1] Recurrence relation:
with To(x) = 1
T 1(x)
O. 1. 2. 3. 4.
5. 6.
7.
=X
To(x) Tlx) TzCx) Tix) Tix) Ts(x) T6(x) Tix)
4.1.3. Chebyshev Polynomials of the Second Kind, Un(x) Domain: [ 1 < x < 1] Recurrence relation:
with Uo(x) = 1
U1(x) =2x O. 1. 2.
3. 4.
5. 6. 7.
O.lUo(x) O.lU1(x) O.lUzCx) O.lUix) O.lUix) O.lUs(x) O.lU6(x) O.lU7 (x)
187
4
6
4.1.2
4.1.2
7
6 5 4
3
2 +~~~~~~~~~~~~~~O
4.1.3
4.1.3
188
CRC Standard Curves and Surfaces
4.1.4. Generalized Laguerre Polynomials L~(x) (a = 0 gives ordinary Laguerre polynomials) Domain: [x > 0] Recurrence relation:
L~'+1(X)=
(2n
+a +
1  X)L~,(X)  (n n+l
+ a)L~_l(x)
with L"o(x) = 1 L~(x)=Ix+a
10. 11. 12. 13. 14.
20. 21. 22. 23. 24.
0.IL1(10x) O.ILi(10x) 0.IL~(10x)
0.IL1(10x) 0.IL~(10x)
0.IL~(10x) 0.IL~(10x) 0.IL~(10x) 0.IL~(10x) 0.IL~(10x)
4.1.5. Laguerre Polynomials Ln(x) Domain: [x > 0] Recurrence relation: (2n L n + 1(x) =
+
1 x)Ln(x)  nLn_1(x)
n+1
with
La( x) = 1 L 1(x)=Ix
O. 1. 2. 3. 4. 5. 6. 7.
0.05L o(10x) O.05L 1(10x) 0.05LilOx) 0.05Lil0x) 0.05LilOx) 0.05L s(10x) 0.05L 6 (10x) 0.05L 7(10x)
189 22
12
1l.",H',~\10
11 14 13
24
4.1.4
4.1.4 5
2
__________~~~~===#o 6
37
4.1.5
4.1.5
190
CRC Standard Curves and Surfaces
4.1.6. Hermite Polynomials Hn(x) Domain: [x > 0] Recurrence relation:
with
Ho(x)
=
1
0.lHo(5x) 0.lH 1(5x) 0.lHi5x)/23 0.lH3(5x)/3 3 0.lHi5x)/4 3 0.lHs(5x)/5 3
O. 1. 2. 3. 4. 5.
4.1.7. Gegenbauer Polynomials C:(x) Domain: [ 1 < x < 1] Recurrence relation:
C:+1(x)
=
2(n + a)xC:  (n + 2a  1)C:_ 1 n + 1
with
CQ(x) = 1 Cf(x) = 2ax Special cases: a
=
a
=
O. 1. 2. 3. 4. 5. 6. 7.
1.0 gives Chebyshev polynomials of the second kind 0.5 gives Legendre polynomials
0.08C5(x) 0.08C~(x)
0.08Ci(x) 0.08Ci(x) 0.08Cl(x) 0.08C}(x) 0.08Cl(x) 0.08C?(x)
191 54 3
1""7''79.'H 0
4.1.6 46
4.1.7
4.1.7
192
CRC Standard Curves and Surfaces
4.1.8. Jacobi Polynomials p:,b(X) Domain: [  1 < x < 1] Recurrence relation: (2n
p:+~
+ a + b + 1)[(a 2 
b 2)
+
2(n
=
+ a + b + 2)(2n + a + b)x]Pna,b + a)(n + b)(2n + a + b + 2)P::....~
(2n
2~(~n+~I~)~(n+a~+b~+~I)~(~2n+~a+b~)~~
with PO,b
=
1
pa,b _ a  b 1
10. 11. 12. 13. 14. 15. 30. 31. 32. 33. 34. 35.

P o l/2,1/2(X)
+
(a
2
+ b + 2)x
P 3 l / 2, 1/2(X) P4 l / 2, 1/2(X) P5 l / 2, 1/2(X)
20. 21. 22. 23. 24. 25.
P6,1/2(X) Pl' 1/2(X) Pi' 1/2(X) pj' J/2(X) N' J/2(X) pJ' J/2(X)
40. 41. 42. 43. 44. 45.
P 1 l/2,1/2(X)
P:i 1/2, 1/2(X)
Po 1/2, lex)
Pl
l / 2,1(X)
Pi 1/2, lex) P j l/2,1(X) P4 l / 2,1(X) P 5 l / 2, l(X) P6,1/2(X) Pl,1/2(X) pi,1/2(X) pj,1/2(X) pl,1/2(x) pJ,1/2(X)
193 ~~__10
T~r_20
11
21
4.1.8
4.1.8 .r~_r_r~n__40
~~_r_._rr__30
35
41
4.1.8
4.1.8
194
CRC Standard Curves and Surfaces
4.2. NONORTHOGONAL POLYNOMIALS 4.2.1. Bernoulli Polynomials Bn(x) Domain: [00 < x < 00] Recurrence relation: none O. 1. 2.
3. 4.
5.
B o(2x) B 1(2x) Bi2x) Bi2x) B/2x) B 5 (2x)
4.2.2. Euler Polynomials En(x) Domain: [00 < x < 00] Recurrence relation: none O. 1.
E o(2x) E 1(2x)
2.
Ei2x) Ei2x) E/2x) E 5 (2x)
3. 4.
5.
195 ~~,~o
531
4.2.1 ~\T~,~~O
5 3 1
4.2.2
196
CRC Standard Curves and Surfaces
4.2.3. Neumann Polynomials 0n(x) Domain: [x > 0] Recurrence relation (for n > 1):
with I
Oo(x) =
X
Ol(X)
l 2 x
=
I
Oix) = 
x
O. 1. 2. 3. 4.
S.
4
+ 3" x
O.OSOo(Sx) O.OSOl(SX) O.OSOzCSx) O.OSOiSx) O.OSOiSx) O.OSOs(Sx)
4.2.4. Schlafli Polynomials Sn(x) Domain: [x > 0] Relation to Neumann polynomials: 2xOn( x)  2 cos 2 ( mT /2) S (x) = '..:...'.......:....''~ n n
Note: So = 0
1. 2. 3. 4. S.
O.OSSl(SX) O.OSSzCSx) O.OSSiSx) O.OSSiSx) O.OSSs(Sx)
197 012 3 4
5
4.2.3 1 2 3 4
4.2.4
5
198
CRC Standard Curves and Surfaces
REFERENCES 1. Abramowitz, M., and I. A. Stegun, Eds., Handbook of Mathematical Functions, National Bureau of Standards, U.S. Department of Commerce, Washington, D.C., 1964. 2. Beyer, W. H., Ed., Handbook of Mathematical Sciences, 6th Ed., CRC Press, Boca Raton, Florida, 1987.
199
Chapter 5
SPECIAL FUNCTIONS IN MATHEMATICAL PHYSICS The curves in this chapter are found in Abramowitz and Stegun, l and the names and notation used here conform with that reference. The approximations necessary to compute these curves are also given there; for purposes of illustrating the curves, the approximations were encoded into computer algorithms such that accuracy was attained to at least three significant figures for all plotted points of a curve. Such accuracy is sufficient for illustrative purposes and was efficiently achieved in all cases. The curves shown in this chapter are only representative, and the interested reader should, when necessary, consult the above reference, or similar ones such as Jahnke and Emde,2 Beyer,3 and Gradshteyn and Ryzhik,4 for a complete treatment of these curves. The reader should be aware that many of the functions are defined for a complex argument, while they are only plotted for a real argument in this chapter, thus showing only a vertical slice of the threedimensional surface over the complex plane.
200
CRC Standard Curves and Surfaces
5.1. EXPONENTIAL AND RELATED INTEGRALS 5.1.1. Exponential Integral En(x) = gee xl Itn) dt Domain: [x > 0] Recurrence relation: En+1(x) = (l/n)[eX  xEn(x)], n = 1,2,3, ... , with Eo(x) = e XIx O. 1. 2. 3. 4. 5.
6. 7.
Eo(x) E/x) EzCx) Eix) E/x) Es(x) E 6(x) Eix)
5.1.2. Exponential Integral Ei(x) Domain: [x > 0] 1.
=
 f~jellt)
dt
0.5Ei(x)
5.1.3. Alpha Integral an(x) = f~tneXl dt Domain: [x > 0] Recurrence relation: a n+l(x) = (llx)[e X + (n + l)a n(x)], n = 0,1,2, ... , with ao(x) = e xIx O. 1. 2. 3. 4. 5. 6. 7.
0.2a o(5x) 0.2al(5x) 0.2azC5x) 0.2ai5x) 0.2a/5x) 0.2a s(5x) O.2a6(5x) 0.2a 7 (5x)
201 2
o
~ 5.1.2
5.1.1
5.1.3
202
CRC Standard Curves and Surfaces
5.1.4. Beta Integral f3n(x) Domain: [x > 0] Recurrence relation:
= f~ltnext
dt
n=0,1,2, ... with f3 o(x) O. 1. 2. 3. 4. 5. 6. 7.
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
=
(2/x)sinh x
f3o(5x) f31(5x) f3zC5x) f3i5x) f3i5x) f3sC5x) f36(5x) f37(5x)
5.1.5. Logarithmic Integralli(x) Domain: [x > 0] 1.
0.005li(1000x)
203
1 357
5.1.4
5.1.5
204
CRC Standard Curves and Surfaces
5.2. SINE AND COSINE INTEGRALS 5.2.1. Sine Integral Six = !o[(sint)/t]dt Domain: [x > 0] 1.
0.5Si(20x)
5.2.2. Cosine Integral Ci x = 'Y Domain: [x > 0] 1.
+ In x + !o[(cos t
Ci(20x)
5.2.3. Sici Spiral x = c Ci t; Y = c[Si t  11"/2] 1.
c = 0.5, 0 < t < 611"
 l)/t] dt
205
5.2.1
5.2.2
5.2.3
206
CRC Standard Curves and Surfaces
5.3. GAMMA AND RELATED FUNCTIONS 5.3.1. Gamma Function f(x) = J;tx1e t dt Also called Euler's integral of the second kind Domain: [00 < x < 00] Recurrence relation: f(x + 1) = xf(x) 1.
0.2f(5x)
5.3.2. Complex Gamma Function f(x + iy) Domain: [00 < x < 00, 00 < y < 00] 1. 2.
Contours of the surface of If(x + iy)l;  5 < x < 5, 1 < y < 1 Contours of the surface of If(x + iy)l;  5 < x < 5,  5 < y < 5
5.3.3. Beta Function B(x, w) = JJt x  1(1  t)wl dt Also called Euler's integral of the first kind Domain: [00 < x < 00] Relation to gamma function: B(x, w) = f(x)f(w)/f(x + w) 1. 2. 3. 4.
0.5 0.5 0.5 0.5
B(5x, 1) B(5x,2) B(5x,3) B(5x,4)
5.3.4. Psi Function !/I(x) = [df(x)/dx]jf(x) Also called digamma function Domain: [00 < x < 00] 1. 0.2!/1(5x)
207
u 5.3.1 2
o
o
5
5.3.2 4
5
5.3.2
2
I
~
I
( 3
5.3.3
5.3.4
208
CRC Standard Curves and Surfaces
5.4. ERROR FUNCTIONS 5.4.1. Error Function Erf(x) Domain: [00 < x < 00]
=
(2/7T 1/ 2 ) It exp(  t 2) dt .
1. Erf(2x) 5.4.2. Complementary Error Function Erfc(x) = 1  Erf(x) Domain: [00 < x < 00] 1.
0.5Erfc(2x)
5.4.3. Derivatives of the Error Function Erf(n)(x) = dn[Erf(!)}jdXn Domain: [00 < x < 00] 1. 0.40 Erf(l)(2x) 2. 0.20 Erf(2)(2x) 3. 0.05 Erf(3)(2x)
209
5.4.1
5.4.2
3
~~~+~~
5.4.3
210
CRC Standard Curves and Surfaces
5.5. FRESNEL INTEGRALS 5.5.1. First Fresnel Integral Sex) = Domain: [00 < x < 00] 1.
S(5x)
5.5.2. Second Fresnel Integral C(x) Domain: [00 < x < 00] 1.
=
It COS(7Tt 2 j2) dt
C(5x)
5.5.3. Cornu's Spiral x 1.
It sin(7Tt 2 j2) dt
4 <
t
< 4
=
set), y
=
C(t)
211
5.5.1
5.5.2
5.5.3
CRC Standard Curves and Surfaces
212
5.6. LEGENDRE FUNCTIONS 5.6.1. Associated Legendre Functions of the First Kind, Pnm(x) Domain: [  1 < x < 1] Recurrence relations: n=I,2,3, ...
m
=
0,1,2, ...
with
pf =x Special case: p no = Legendre polynomials 10. 11. 12. 13. 14.
Pg(x) pf(x) pf(x) pf(x) P4O(x)
21. 22. 23. 24.
0.25 0.25 0.25 0.25
Pf(x) pi(x) Pj(x) Pj(x)
32. 33. 34.
0.10 P{(x) 0.10 pl(x) 0.10 pl(x)
43. 44.
0.025 pi(x) 0.025 pl(x)
213
5.6.1
5.6.1
34 44
5.6.1
5.6.1
214
CRC Standard Curves and Surfaces
5.6.2. Associated Legendre Functions of the Second Kind, Domain: [  1 < x < 1] Recurrence relations:
Q~n(x)
n=I,2,3, ... Q;;,+l(X)
=
(x 2  1) 1/2[ (n  m)xQ;;'(x)  (n
+ m)Qn_lm(X) ] , m = 0,1,2, ...
with
Q o = ~ In( 1 + X) 1 2 1 x 10. II. 12. 13. 14.
Qg(x) QP(x) Qg(x) Q~(x) Q~(x)
2L 22. 23. 24.
0.25 0.25 0.25 0.25
Q~(x) Q~(x) Q~(x) Q~(x)
32. 33. 34.
 1
0.05 Qi(x) 0.05 Q~(x) 0.05 Q~(x)
43. 44.
0.02 Q~(x) 0.02 Ql(x)
215
10
11
5.6.2
5.6.2 44
5.6.2
5.6.2
216
CRC Standard Curves and Surfaces
5.7. BESSEL FUNCTIONS 5.7.1. Bessel Functions of the First Kind, In(x) Also called simply Bessel functions Domain: [x > 0] Recurrence relation: In+1(x)
=
Symmetry: I _Jx)
=
O. 1. 2. 3. 4. 5.
2n xIn(x)  In_l(x), (
n=0,1,2, ...
J)nIn(x)
I o(20x) I 1(20x) Ii20x) I/20x) I/20x) I s(20x)
5.7.2. Bessel Functions of the Second Kind, Y,,(x) Also called Neumann functions or Weber functions Domain: [x > 0] Recurrence relation: 2n Y,,+I(X) = xY,,(x)  Y,,l(X),
Symmetry: Y_n(x) O. 1. 2. 3. 4. 5.
Y o(20x) Y I (20x) Y 2 (20x) Y/20x) Y/20x) Y s(20x)
=
(l) nyn (x)
n=0,1,2, ...
217 o
2
5.7.1
5.7.1
o
5.7.2
5.7.2
218
CRC Standard Curves and Surfaces
5.7.3. Hankel Functions HP)(x) and Domain: [x > 0] Relation to Bessel functions:
H~2)(X)
Recurrence relation: H(1,2)(X) n+l
o. 1. 2. 3. 4. 5.
=
2n H(l,2)(X)  H(1,2)(X) X
n
nl'
n = 0,1,2, ...
IHam)(20x)l, m = 1,2 IHi m)(20x)l, m = 1,2 IHi m)(20x)l, m = 1,2 IH~m)(20x)l,
m = 1,2
IHJm)(20x)l, m = 1,2 IHJm)(20x)l, m = 1,2
5.7.4. Complex Bessel Function fo(x + iy) Domain: [00 < x < 00, 00 < y < 00] 1.
Contours of the surface
IfoCx + iy)l; 0 .:::;; x.:::;;
10, 2.:::;; y .:::;;.:::;; 2
5.7.5. Bessel Function fn(x) versus Order and Argument Domain: [  00 < x < 00, n 2:: 0] 1.
Variableintensity plot of fn(x); 0 .:::;; x .:::;; 25, 0 .:::;; n .:::;; 20
219
2
4
5.7.4
5.7.3
5
10
15
5.7.5
20
25
6
8
10
220
CRC Standard Curves and Surfaces
5.8. MODIFIED BESSEL FUNCTIONS 5.8.1. Modified Bessel Function In(x) Domain: [x > 0] Recurrence relation:
2n In+1(x) = In1(x)  x1n(x)
n
=
0,1,2, ...
Symmetry: Ln(x) = In(x) O. 1. 2.
3. 4. 5.
0.1 0.1 0.1 0.1 0.1 0.1
I o(10x) Ij(lOx) Ii10x) 13(lOx) I/lOx) Is(lOx)
5.8.2. Modified Bessel Function Kn(x) Domain: [x > 0] Recurrence relation:
n = 0,1,2, ...
O. 1. 2. 3. 4.
5.
0.1 0.1 0.1 0.1 0.1 0.1
K o(5x) K 1(5x)
KzC5x) Ki5x) K/5x) K s(5x)
221 012345
5.8.1 12 3 4
5.8.2
5
222
CRC Standard Curves and Surfaces
5.9. KELVIN FUNCTIONS 5.9.1. Kelvin Function bern(x) Domain: [x > 0] Recurrence relation: 2l/2n
.
bern+l(x) =  x [bern(x)  beln(x)]  bern_leX),
n=1,2,3, ...
Symmetry: becn(x) = (_l)n bern(x) O. 1. 2. 3. 4. 5.
0.1 beroC8x) 0.1 ber l(8x) 0.1 beri8x) 0.1 beri8x) 0.1 beri8x) 0.1 bers(8x)
5.9.2. Kelvin Function bein(x) Domain: [x > 0] Recurrence relation: .
2l/2n.
.
beln+l(x) =  x [beln(x) + bern(x)]  beln_l(x), Symmetry: beLn(x) = (_l)n bein(x) O. 0.1 bei o(8x) 1. 0.1 bei l(8x) 2. 0.1 bei z(8x) 3. 0.1 bei 3 (8x) 4. 0.1 beii8x) 5. 0.1 beisC8x)
n=1,2,3, ...
223 1
40
3 2
5.9.1
o 5.9.2
3
224
CRC Standard Curves and Surfaces
5.9.3. Kelvin Function kern(x) Domain: [x > 0] Recurrence relation: 2l/2n
kern+l(x)
= 
.
x[kern(x)  keln(x)]  kern_leX),
Symmetry: kecn(x)
=
n=1,2,3, ...
(_l)n kern(x)
O. ker o(8x) 1. keri8x) 2. keri8x) 3. keri8x) 4. keri8x) 5. kers(8x) 5.9.4. Kelvin Function kein(x) Domain: [x > 0] Recurrence relation:
n=1,2,3, ...
O. 1. 2. 3. 4. 5.
kei o(8x) kei l (8x) kei 2 (8x) keii8x) keii8x) kei s(8x)
225 035
5.9.3 245
3
5.9.4
226
CRC Standard Curves and Surfaces
5.10. SPHERICAL BESSEL FUNCTIONS 5.10.1. Spherical Bessel Functions of the First Kind, jix) Domain: [x > 0] Relation to Bessel Function: jn(x) = (7T /2x)1/z In+1/ix) Recurrence relation: .
( ) = 2n x+ 1 in . (X)  Inl . (X) ,
In+l x
o. 1. 2. 3.
n = 0,1,2, ...
jo(20x) jl(20x) ji20x) M20x)
5.10.2. Spherical Bessel Functions of the Second Kind, yix) Domain: [x > 0] Relation to Bessel Function: yix) = (7T /2x)1/ZYn+l/ix) Recurrence relation: Yn+l(x) =
2n
+1
X
yn(x)  Ynl(X),
n = 0,1,2, ...
Symmetry: yn(x) = (_l)lnjn_l(X) O. 1. 2. 3.
yaC20x) Yl(20x) Yz(20x) yi20x)
5.10.3. Spherical Bessel Functions of the Third Kind, h~l)(X) and M!')(x) Domain: [x > 0] Relation to Hankel Function: h~'Z)(x) = (7T/2x)1/zH~~~jix) Recurrence relation: h(l,Z)(X) n+l
=
2n
+ 1 h(l,Z)(X)
X
O. 1.
Ihbm)(20x)l, m = 1,2 Ihim)(20x)l, m = 1,2
2. 3.
Ih~m)(20x)l, Ih~m)(20x)l,
m = 1,2 m = 1,2
n
 h(l,Z)(X)
nl,
n = 0,1,2, ...
227
o
5.10.2
5.10.1 13
02
5.10.3
228
CRC Standard Curves and Surfaces
5.11. MODIFIED SPHERICAL BESSEL FUNCTIONS 5.11.1. Modified Spherical Bessel Functions of the First Kind, (rr /2X)1/2 In + 1/2(X) Domain: [x > 0]
Recurrence relation: n=0,1,2, ...
O. 1. 2. 3.
4. 5.
0.1 0.1 0.1 0.1 0.1 0.1
(rr /20x )1/2 (rr /20x )1/2 (rr/20x)1/2 (rr/20x)1/2 (rr /20X)1/2 (rr/20x)1/2
I 1/ 2(10x) I3/zC10x) Is/zC10x) I7/i10x) I9/zClOx) I l1 / 2(10x)
5.11.2. Modified Spherical Bessel Functions of the Second Kind, (rr /2X)1/2Ln_l/ix) Domain: [x > 0]
Recurrence relation: L n 3j2(X)
O. 1. 2.
3. 4. 5.
0.1 0.1 0.1 0.1 0.1 0.1
=
Ln+1/2(X) 
(rr/20x)l/2 (rr /20X)1/2 (rr /20x )1/2 (rr /20X)1/2 (rr/20x)1/2 (rr/20x)1/2
2n
+1
x
L n l/ 2(x),
n
=
0,1,2, ...
L 1/ 2(10X) L3/ilOx) Ls/i10x) L 7/ 2(lOx) L 9/ 2(10X) Ll1/ilOx)
5.11.3. Modified Spherical Bessel Function of the Third Kind, (rr /2X)1/2Kn+1/ix) Domain: [x > 0]
Recurrence relation: n=0,1,2, ...
O. 1. 2. 3. 4. 5.
(rr/20x)1/2 (rr/20x)1/2 (rr/20x)1/2 (rr/20x)1/2 (rr/20x)1/2 (rr /20X)1/2
K 1/ 2(10X) K 3 / 2(lOx) Ks/zC10x) K7/zC10x) K9/ilOx) K l1 / 2(10x)
229 024 012345
1 3 5
5.11.2
5.11.1 012345
5.11.3
230
CRC Standard Curves and Surfaces
5.12. AIRY FUNCTIONS 5.12.1. Airy Function Ai(x) Domain: [00 < x < 00] 1.
Ai(lOx)
5.12.2. Airy Function Bi(x) Domain: [00 < x < 00] 1.
Bi(lOx)
5.13. RIEMANN FUNCTIONS 5.13.1. Zeta Function (x) Domain: [00 < x < 00] 1.
0.21(5x)1
5.13.2. Zeta Function I(t + iy)1 The line x = t in the complex plane is the critical line of the zeta function. Domain: [00 < y < 00] 1.
0.21(1/2
+ i50y)1
5.13.3. Complex Zeta Function (x + iy) Domain: [00 < x < 00, 00 < y < 00] 1.
Contours of the surface I(x
+ iy)l; 4 < x < 4, 0 < y < 15
5.14. PARABOLIC CYLINDER FUNCTIONS 5.14.1. HalfInteger Orders Solving y"  (x 2 /4 Domain: [x > 0] 1.
0.5y(5x)
+ a)y
=
0
231
5.12.1
5.12.2
5.13.1
5.13.2
5.14.1
232
CRC Standard Curves and Surfaces
5.15. ELLIPTIC INTEGRALS 5.15.1. Elliptic Integral of the First Kind, F(
Contours of F(
5.15.2. Complete Elliptic Integral of the First Kind, K(m) Domain: [0 < m < 1] 1. 0.2K(m) 5.15.3. Elliptic Integral of the Second Kind, E(
Contours of E(
5.15.4. Complete Elliptic Integral of the Second Kind, E(m) Domain: [0 < m < 1] 1. (2/'Tr )E(m) 5.15.5. Elliptic Integral of the Third Kind, lIen;
Contours of lIen;
=
t (
5.15.6. Complete Elliptic Integral of the Third Kind, II(n; m) Domain: [0 < n < 1, 0 < m < 1] 1. 2.
3. 4.
5.
0.1 0.1 0.1 0.1 0.1
II(0.1; m) II(0.3; m) II(0.5; m) 1I(0.7; m) 1I(O.9; m)
233
5.15.1
5.15.2
5.15.3
5.15.4
5.15.5
5.15.6
234
CRC Standard Curves and Surfaces
5.16. JACOBI ELLIPTIC FUNCTIONS In the plots of the Jacobi elliptic functions, the independent variable u (abscissa) varies linearly from 0 to 4K(m), where K is the complete elliptic integral of the first kind. 5.16.1. sn u, cn u, dn U 11. 12. 13.
sn u; m = 0.25 cn u; m = 0.25 dn u; m = 0.25
31. 32. 33.
sn u; m = 0.75 cn u; m = 0.75 dn u; m = 0.75
21. 22. 23.
sn u; m = 0.50 cn u; m = 0.50 dn u; m = 0.50
235 3
2
2
5.16.1
5.16.1
2
5.16.1
236
CRC Standard Curves and Surfaces
5.16.2. sd u, cd u, nd
U
11. 0.5 sd u; m = 0.25 12. 0.5 cd u; m = 0.25 13. 0.5 nd u; m = 0.25 31. 32. 33.
0.5 sd u; m = 0.75 0.5 cd u; m = 0.75 0.5 nd u; m = 0.75
21. 22. 23.
0.5 sd u; m = 0.50 0.5 cd u; m = 0.50 0.5 nd u; m = 0.50
237
5.16.2
5.16.2
5.16.2
238
CRC Standard Curves and Surfaces
5.16.3. se u, de u, ne
U
11. 12. 13.
0.5 se u; m = 0.25 0.5 de u; m = 0.25 0.5 ne u; m = 0.25
31. 32. 33.
0.5 se u; m = 0.75 0.5 de u; m = 0.75 0.5 ne u; m = 0.75
21. 22. 23.
0.5 se u; m = 0.50 0.5 de u; m = 0.50 0.5 ne u; m = 0.50
239
12
3
5.16.3
5.16.3
12
5.16.3
3
240
CRC Standard Curves and Surfaces
5.16.4. cs u, ds u, ns
U
11. 0.5 cs u; m = 0.25 12. 0.5 ds u; m = 0.25 13. 0.5 ns u; m = 0.25 31. 32. 33.
0.5 cs u; m = 0.75 0.5 ds u; m = 0.75 0.5 ns u; m = 0.75
21. 0.5 cs u; m = 0.50 22. 0.5 ds u; m = 0.50 23. 0.5 ns u; m = 0.50
241
5.16.4
5.16.4
5.16.4
242
CRC Standard Curves and Surfaces
REFERENCES 1. Abramowitz, M., and I. A. Stegun, Eds., Handbook of Mathematical Functions, National Bureau of Standards, U.S. Department of Commerce, Washington, D.C., 1964. 2. Jahnke, E., and F. Emde, Tables of Functions, Dover Publications, Inc., New York, 1945. 3. Beyer, W. H., Ed., Handbook of Mathematical Sciences, 6th Ed., CRC Press, Boca Raton, Florida, 1987. 4. Gradshteyn, I. S., and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, Orlando, Florida, 1973.
243
Chapter 6
SPECIAL FUNCTIONS IN PROBABILITY AND STATISTICS 6.1. DISCRETE PROBABILITY DENSITIES The following discrete densities are plotted with the variable m on the x axis. Although a continuous line is plotted, the functions must be understood as discrete, having values only at integer m, the domain of which is listed in each case. The vertical scale is arbitrary, chosen to plot the density so that its maximum value is of uniform height for all plots. Thus, the scale may change among a series of plots for a given density function. A property common to all discrete densities is that the sum over all possible m must equal unity: m2
L
P(m) = 1
where m 1 and m 2 are the minimum and maximum possible values of m.
244
CRC Standard Curves and Surfaces
6.1.1. Binomial, P(mln, p) Here
=
(::Z )pln(l 
p)nm
m = number of given outcomes in n trials
1. n = 25, 2. n = 25, 3. n = 25, 4. n = 10, 5. n = 10, 6. n = 10,
n
=
total number of trials
p
=
probability of a given outcome in a single trial
p = 0.25; p = 0.50; p = 0.75; p = 0.25; p = 0.50; p = 0.75;
m = 0, 1,2, ... , n m = 0, 1,2, ... , n m = 0, 1,2, ... , n m = 0, 1,2, ... , n m = 0, 1,2, ... , n m = 0,1,2, ... , n
6.1.2. Geometric, P(mlp) = pO  p)ml Here m = number of events (m
> 0)
p = probability of a given event
1. p
=
2. p
=
3. p
=
0.25; m 0.50; m 0.75; m
= =
=
1,2,3, ... , 10 1,2,3, ... , 10 1,2,3, ... ,10
6.1.3. Hypergeometric,P(mln,N,p) =
(~)(N~l_,:))/(~)
Here
m = number of items of a given type in a sample, with the upper bound given by m = min(n, Np) n = sample size N = total number of items available (N > n) p = probability of a given item type in total number N 1. 2. 3. 4. 5. 6.
n = 10, N = 40, p = 0.25; n = 10, N = 40, p = 0.50; n = 10, N = 40, p = 0.75; n = 20, N = 40, p = 0.25; n = 20, N = 40, p = 0.50; n = 20, N = 40, p = 0.75;
m = 0,1,2, ... , 10 m = 0, 1, 2, ... , 10 m = 0, 1,2, ... , 10 m = 0, 1,2, ... , 10 m = 0, 1,2, ... ,20 m = 0,1,2, ... ,20
245 0.5
0.25
3 4
6
6.1.1
6.1.1 0.75
6.1.2 0.5
0.5
3
4
6.1.3
6
6.1.3
246
CRC Standard Curves and Surfaces
6.1.4. Negative Binomial, P(mln, p) Here m
=
=
(n
+ :  1 )pn(1
_ p)m
number of failures prior to nth success
n = number of successes p
1. n = 10, p = = = = = =
2. n = 10, p 3. n = 10, p 4. n = 25, p 5. n = 25, p 6. n = 25, p
= probability of a given event 0.25; 0.50; 0.75; 0.25; 0.50; 0.75;
m m m m m m
= = = = = =
0, 1,2, ... ,50 0, 1, 2, ... , 50 0,1,2, ... , 50 0, 1,2, ... , 100 0,1,2, ... , 100 0,1,2, ... , 100
6.1.5. Poisson, P(mlr) = err m 1m! Here m = number of events occurring per sample of unit time r = mean rate (number of events per unit time)
1. r = 2; m = 0, 1, 2, ... , 25 2. r = 6; m = 0, 1,2, ... , 25 3. r = 10; m = 0, 1,2, ... ,25
247 0.25 0.125
6
3
5
2
4
6.1.4
6.1.4 0.5
6.1.5
248
CRC Standard Curves and Surfaces
6.2. CONTINUOUS PROBABILITY DENSITIES The following probability densities are continuous functions, plotted so that the xaxis limits are  1 to + 1 (with the actual domain of x given by the argument as listed). The range of y is arbitrary, selected only to plot the function in an easily viewable manner. As for the discrete densities, the scale may change among plots for a given function. A property common to all continuous probability densities is that the integral equals unity:
tp(x) dx = 1 a
where a and b are the limits of the particular density function. 6.2.1. Beta, P(x) = [l/B(a, b)]al(l  X)bl Domain: [0 < x < 1] 1. O.4P(x): a = O.4P(x): a = O.4P(x): a = O.3P(x): a = O.3P(x): a =
2. 3. 4. 5. 6.
2, b = 2, b = 2, b = 3, b = 3, b = O.3P(x): a = 3, b =
1 2 4 2 3 6
6.2.2. Cauchy, P(x) = (1/7Tb){1 + [(x  a)/b]2}l Domain: [  00 < x < 00] 1. 2.0P(5x): a 2. 2.0P(5x): a 3. 2.0P(5x): a
= 0, b = 1 = 0, b = 2 = 0, b = 3
6.2.3. ChiSquare, P(x) Domain: [0 < x < 00]
= [2 n/ 2r(n/2)]lx(n2)j2 e x/2
1. 5.0P(50x): n = 5 2. 5.0P(50x): n = 15 3. 5.0P(50x): n = 25 6.2.4. Exponential, P(x) = (l/b)exp[ (x  a)/b] Domain: [a < x < 00] 1. P(5x): a = 0, b = 1 2. P(5x): a = 0, b = 2 3. P(5x): a = 0, b = 3
249 3
6
6.2.1
6.2.1
6.2.2
6.2.3
6.2.4
250
CRC Standard Curves and Surfaces
6.2.5. Extreme Value, P(x) Domain: [00 < x < 00]
=
(l/b)exp{ lex  a)/bl  exp[ I(x  a)/bl])
1. P(lOx): a = 0, b = 1 2. P(lOx): a = 0, b = 2 3. P(lOx): a = 0, b = 3 6.2.6. Gamma, P(x) = [l/f(a)balxaIex/b (a, b Domain: [x > 0] 1. P(lOx): 2. P(lOx): 3. P(lOx): 4. P(lOx): 5. P(lOx): 6. P(lOx):
a a a a a a
= 2, b = 0.5 = 2, b = 1.0 = 2, b = 2.0 =
3, b
=
0.5
= 3, b = 1.0
= 3, b = 2.0
6.2.7. Laplace, P(x) = [1/(2b)]exp[ I(x  a)l!b] Domain: [00 < x < 00] 1. 2.0P(5x): a = 0, b = 1 2. 2.0P(5x): a = 0, b = 2 3. 2.0P(5x): a = 0, b = 3 . . 1 6.2.8. LOgIStiC, P(x) = Ii Domain: [0
< x < 00]
exp [(x  a) /b ]
{I + exp[(x
1. 4.0P(lOx): a = 0, b = 1 2. 4.0P(lOx): a = 0, b = 2 3. 4.0P(10x): a = 0, b = 3
 a)/b]}
2
> 0)
251
6.2.5
4
6
6.2.6
6.2.6
6.2.7
6.2.8
252
CRC Standard Curves and Surfaces
6.2.9. Lognormal, P(x)
{ [ (In x  a) /b ]2 } exp 2
1
=
1/2
(217)
b
Domain: [x > 0] 1. P(lOx): a = 0, b = 0.5 2. P(lOx): a = 0, b = 1.0 3. P(lOx): a = 0, b = 2.0 6.2.10. Maxwell, P(x) = (4/171/2a 3 )x 2 exp( x 2/a 2) Domain: [x > 0] 1. P(lOx): a = 1 2. P(10x): a = 2 3. P(lOx): a = 3 6.2.11. Normal (Gaussian), P(x)
=
1 (217)1/2b
Domain: [00
<x < 00]
1. P(lOx): a = 0, b = 1 2. P(10x): a = 0, b = 2 3. P(lOx): a = 0, b = 3 6.2.12. Pareto, P(x) = (x/b)(l Domain: [x > 0] 1. 5.0P(10x): a = 2, b = 1 2. 3. 4. 5. 6.
5.0P(10x): a = 2, b = 2 5.0P(lOx): a = 2, b = 3 10.0P(lOx): a = 4, b = 1 10.0P(10x): a = 4, b = 2 10.0P(lOx): a = 4, b = 3
+ x/b)a1
exp { _ [( x  a) /b ]2 } 2
253
6.2.10
6.2.9
6.2.11
123
6.2.12
6.2.12
254
CRC Standard Curves and Surfaces
6.2.13. Rayleigh, P(x) = a12x exp[ _
(x~a)2]
Domain: [x > 0] 1. P(lOx): a 2. P(10x): a 3. P(lOx): a
=
=
1 2
=3
6.2.14. Snedecor's F, P(x)
( mm/2nn/2)
=
Domain: [x > 0] 1. 2. 3. 4. 5. 6.
P(5x): P(5x): P(5x): P(5x): P(5x): P(5x):
m m m m m m
= = = = = =
5, n = 10 15, n = 10 50, n = 10 5, n = 20 15, n = 20 50, n = 20
B( m12, n12) x(m2)/2(n + mx)(m+n)/2
255
6.2.13
6.2.14
6.2.14
256
CRC Standard Curves and Surfaces
6.2.15. Student's t, P(x) Domain: [00 < x < 00]
1 =
BG,
n 1/ 2
(. x 2 )
1. P(5x): n = 2 2. P(5x): n = 5 3. P(5x): n = 25 6.2.16. Weibull, P(x) = (b/ab)xbl exp[ (x/a)b] Domain: [x > 0] 1. 2. 3. 4. 5. 6.
P(5x): P(5x): P(5x): P(5x): P(5x): P(5x):
a a a a a a
= 1, b = 1, b = 1, b = 2, b = 2, b =
2, b
= 1 = 2 = 3 = 2 = 3 =
4
257
3
6.2.15 3 6
6.2.16
6.2.16
258
CRC Standard Curves and Surfaces
6.3. SAMPLING DISTRIBUTIONS The following sampling distributions are expressed as integrals of a density function. By definition, at the upper limit the integral equals unity; therefore, the distributions are plotted so that the maximum is always unity. The actual domain of the sampling variable is as listed. 6.3.1. Normal Distribution P(X) =
11/2
(27T)
b
X
I x exp { 
=
[n 1/ 2BG, n/2)]1
_
[(Xa)/b]2} 2
00
dx
< X < 00]
Domain: [00
1. P(SX): a = 0, b = O.S 2. P(SX): a = 0, b = 1.0 3. P(SX): a = 0, b = 2.0 6.3.2. Student t Distribution P(tln)
Domain: [00
< t < 00]
1. pest): n
2
=
+ (x 2/n)](n+l)/2 dx
X
foo[1
=
[2 n/ 2f(n/2)]1
2. pest): n = S 3. peSt): n = 99
6.3.3. ChiSquare Distribution P(x 2In) Domain: X 2 >
X
°
jX x (n2)/2 e X/2 dx 2
o
1. P(2SX2): n = 2 2. P(2SX 2): n = 6 3. P(2Sx 2): n = 16 m/2 n/2
· ·b· P(FI )m n 634 . . . FD lstn utlOn m, n  B(m/2, n/2)
Domain: F> 1. P(SF): 2. P(SF): 3. P(SF): 4. P(SF): S. P(SF): 6. P(SF):
m m m m m m
°
(n
= 2, n = 10 = 6, n = 10 =
20, n
=
10
= 2, n = 20 = 6, n = 20 = 20, n = 20
j0Fx (m2)/2
+ mx)(m+n)/2 dx
259
6.3.2
6.3.1
6.3.3
6.3.4
. 6.3.4
261
Chapter 7
THREEDIMENSIONAL CURVES As opposed to curves which lie wholly in a plane (called "plane curves"), certain curves occupy three dimensions (called "skew curves"). All threedimensional curves must necessarily be expressed in parametric form: x =
f(t)
y =
get)
z=h(t) Because there are innumerable variations of the functions f, g, and h, threedimensional curves can assume a wide variety in appearance. Only those curves having some accepted significance and use are illustrated here. Many interesting and useful threedimensional curves can be generated simply by adding a z variation of some sort to the curves given in the previous chapters, after they are put into parametric form. The curves in this chapter are plotted as points (x p' yp) projected on a plane which is normal to the vector between the origin (0,0,0) and the viewpoint and which passes through the origin. The projection used is the perspective one (see Foley and VanDam l for a full treatment of projections). If the viewing point is at large distance relative to the projected points (as is the case in this chapter), then the view approaches the parallel one, which is given by the transformations xp = x sin e + y cos e
yp = 
x cos
e cos 4>  y sin e cos 4> + z sin 4>
where (x, y, z) are the coordinates of the point on the curve prior to projection and (e, 4» are the angles in spherical coordinates (see Section 1.3) of the vector normal to the projection plane. The three axes are plotted with solid lines between the limits of 1.0 and + 1.0, with the positive z axis up.
262
CRC Standard Curves and Surfaces
7.1. HELICAL CURVES 7.1.1. Circular Helix This is also called the right helicoid: x
=
a sin t
y = a cos t
z 1.
a = 0.5,
C
=
t/(21TC)
= 5.0; 0 < t < 101T; viewpoint = (40,  50,20)
7.1.2. Elliptical Helix x
=
a sin t
y
=
b cos
t
z = t/(21TC) 1.
a = 0.3, b = 1.0,
C
= 5.0; 0 < t < 101T; viewpoint = (40,  50,20)
7.1.3. Conical Helix
x = (at/21Tc)sint y = (at /21TC )cos t z = t/(21TC)
1.
a = 0.5,
C
= 5.0; 0 < t < 101T; viewpoint = (40,  50,20)
263
7.1.1
7.1.2
7.1.3
264
CRC Standard Curves and Surfaces
7.1.4. Spherical Helix x = sin[tj(2c)]cos t y
= sin[tj(2c)]sin t
z = cos[tj(2c)] 1.
c = 5.0;
°<
t < 1O'lT; viewpoint = (40,  50,20)
7.1.5. nHelix
x = a cos(t + 2'lTijn) ,
i = 1, ... , n
+ 2'lTijn),
i = 1, ... ,n
y
= a sin(t
z = tj(2'lTc) 1.
a = 0.3, c = 3.0, n = 2;
°< t < 6'lT; viewpoint = (40,  50,20)
265
7.1.4
7.1.5
266
CRC Standard Curves and Surfaces
7.2. SINE WAVES IN THREE DIMENSIONS 7.2.1. Sine Wave on Cylinder x = b cos t y = b sin t z
1.
a
= 10.0, b = 1.0,
C
=
c cos at
= 0.5; 0 < t < 27T; viewpoint = (40,  50,20)
7.2.2. Sine Wave on Sphere
1.
a
1/2
X =
(b 2

c 2 cos 2 at)
y =
(b 2

c 2 cos 2 at) 1/2 sin t
z
c cos at
=
cos t
= 10.0, b = 1.0, c = 0.30; 0 < t < 27T; viewpoint = (40,  50,20)
7.2.3. Sine Wave on Hyperboloid of One Sheet
1.
a
1/2
X =
(b 2 + c 2 cos 2 at)
y
=
(b 2 + c 2 cos 2 at) 1/2 sin t
z
=
c cos at
cos t
= 10.0, b = 1.0, c = 0.30; 0 < t < 27T; viewpoint = (40, 50,20)
267
7.2.1
7.2.2
7.2.3
268
CRC Standard Curves and Surfaces
7.2.4. Sine Wave on Cone x = b(l
+ cos at)cos t
y = b(l
+ cos at)sin t
z 1.
a = 10.0, b = 0.5,
C
= C (1
+ cos at)
= 0.4; 0 <
t
< 2rr; viewpoint = (40,  50,20)
7.2.5. Rotating Sine Wave x = sin at cos bt
1. 2.
a = 3.0, b = 1.00, a = 3.0, b = 0.25,
C C
y
=
sin at sin bt
z
=
ct/(2rr)
= 1.0;  2rr < t < 2rr; viewpoint = (40,  50,20) = 1.0;  2rr < t < 2rr; viewpoint = (40,  50,20)
269
7.2.4 2
7.2.5
7.2.5
CRC Standard Curves and Surfaces
270
7.3. MISCELLANEOUS SPIRALS 7.3.1. Sici Spiral x = a Ci(t) y
= a SiC t)
z = tic where Si and Ci are the sine and cosine integrals 1. a = 0.5, c = 20.0; 0.2 < t < 20; viewpoint = (40,  50,30) 7.3.2. Fresnel Integral Spiral This is also called Cornu's spiral:
x = C(t) Y = S(t)
z
=
tic
where Sand C are the first and second Fresnel integrals 1. c = 5.0; 0 < t < 5; viewpoint = (40,  50,30) 7.3.3. Toroidal Spiral x = (a sin ct y
+ b )cos t
= (a sin ct + b) sin t
z = a cos ct 1.
a
= 0.2, b = 0.8, c = 20.0; 0 < t < 27T; viewpoint
7.3.4. Intersection of Sphere and Cylinder
x = (1 + cos t) 12 y = (sin t)/2
z 1.
0
=
sin(tI2)
< t < 27T; viewpoint = (40,  50,20)
=
(40,  50,20)
271
7.3.1
7.3.2
7.3.3
7.3.4
272
CRC Standard Curves and Surfaces
REFERENCES 1. Foley, J. D., and A. VanDam, Fundamentals of Interactive Computer Graphics, AddisonWes
ley, Reading, Massachusetts, 1983.
.
273
Chapter 8
ALGEBRAJCSURFACES The following forms are plotted in the perspective projection described at the beginning of Chapter 7. There are numerous, different ways to represent 3D surfaces. The method chosen here is a shadedrelief type of illustration, with a grid superimposed to show lines of constant x and y on the surface. Artificial light sources are used to enhance the features of each surface, and so the shading is not a uniform gray. The viewpoint of the observer is given as (x, y, z) coordinates relative to the origin of the projection plane, which is coincident with the origin of the figure and normal to the line connecting the origin with the observer. The surfaces are shown in their true aspect ratios, and a bounding box is placed about the surface in each case. This box extends from  1 to + 1 on both the x and y axes. The z range is also 1 to + 1, with a few exceptions as indicated. The surfaces sometimes intersect the· top or bottom of the bounding box; at these intersections, the surface is not represented accurately by the method used here.
274
CRC Standard Curves and Surfaces
8.1. FUNCTIONS WITH ax 8.1.1. z = ax Plane
1. a 2. a
= =
+ by
0.5, b 0.1, b
= =
8.1.2. z = 1/(ax
1. a 2. a
= =
5.0, b 2.0, b
= =
ax
+ by + z =
0.5; viewpoint 0.3; viewpoint
+ by)
axz
= =
+ byz
5.0; viewpoint 4.0; viewpoint
= =
0
(5,  6, 4) (5,  6,4)
 1= 0 (6,  4,3) (6,  4,3)
+ by
275
8.1.1 8.1.1 2
8.1.2 8.1.2
276
CRC Standard Curves and Surfaces
8.2.1. z = c(x 2ja 2 + y2 jb 2) x 2ja 2 + y2 jb 2  zjc = 0 Elliptic paraboloid
1. a = 0.5, b = 1.0, c = 1.0; viewpoint = (5,  6,4) 2. a = 1.0, b = 1.0, c =  2.0; viewpoint = (5,  6,4) 8.2.2. z = c(x 2ja 2  y2jb 2) x 2ja 2  y2jb 2  zjc = 0 Hyperbolic paraboloid (commonly called saddle)
1. a = 0.50, b = 0.5, c = 1.0; viewpoint = (4,  6, 4) 2. a = 1.00, b = 0.5, c = 1.0; viewpoint = (4,  6, 4)
8.2.3. 1 = x 2ja 2 + y2jb 2 x 2ja 2 + y2jb 2  1 = 0 Elliptic cylinder 1. a = 1.0, b = 1.0; viewpoint = (4,  5,2) 8.2.4. 1 = x 2ja 2  y2jb 2 x 2ja 2  y2jb 2  1 = 0 Hyperbolic cylinder 1. a = 1.0, b = 1.0; viewpoint = (4,  6,3)
277 2
8.2.1
8.2.1 2
8.2.2
8.2.2
8.2.3
8.2.4
278
CRC Standard Curves and Surfaces
8.3.1. z = (1  x 2 
y2)1/2
x2
+ y2 + Z2

1=
°
Sphere
1. Viewpoint 8.3.2.
Z
=
(4, 5,2)
= cO  x 2ja 2  y2jb 2)1/2 x 2ja 2 + y2jb 2 + z2jc 2  1 =
Ellipsoid
Special cases: a a
= =
b > c gives oblate spheroid b < c gives prolate spheroid
1. a = 1.00, b = 1.00, c = 0.5; viewpoint = (4,  5,2) 2. a = 0.50, b = 0.50, c = 1.0; viewpoint = (4, 5,2) 8.3.3. z = (x 2
+ y2)l/2
x2
Cone
1. Viewpoint = (4,  5,2)
+ y2

Z2
=
°
°
279
8.3.1 2
8.3.2
8.3.2
8.3.3
280
CRC Standard Curves and Surfaces
8.3.4. z = c(x 2/a 2 + y2/b 2)1/2 x 2/a 2 + y2/b 2  Z2/C2 = 0 Elliptic cone (circular cone if a = b) 1. a = 0.5, b = 0.5, c = 1.00; viewpoint = (4, 5,2) 2. a = 1.0, b = 1.0, c = 0.50; viewpoint = (4,  5,2) 8.3.5. z = c(x 2/a 2 + y2/b 2  1)1/2 x 2/a 2 + y2/b 2  Z2/C2  1 Hyperboloid of one sheet
=
0
=
0
1. a = 0.1, b = 0.1, c = 0.2; ±z = cm; viewpoint = (4,  5,2) 2. a = 0.2, b = 0.2, c = 0.2; ±z = cm; viewpoint = (4,  5,2) 8.3.6. Z = c(x 2/a 2 + y2/b 2 + 1)1/2 x 2/a 2 + y2/b 2  Z2/C2 + 1 Hyperboloid of two sheets
1. a = 0.125, b = 0.125, c = 0.2; ±z = cm; viewpoint = (4,  5,2) cm; viewpoint = (4, 5,2)
2. a = 0.25, b = 0.25, c = 0.2; ±z =
281
2
8.3.4
8.3.4
2
8.3.5
8.3.5
2
8.3.6
8.3.6
282
CRC Standard Curves and Surfaces
1. a = 1.0, b = 1.0, 2. a = 1.0, b = 1.0,
1. a = 1.0, b = 1.0, 2. a = 1.0, b = 1.0,
C C
C
C
= 0.5; viewpoint = (1,  5,2) = 2.0; viewpoint = (1,  5,2)
= 0.5; viewpoint = (1,  5,2) = 2.0; viewpoint = (1,  5,2)
283
2
8.4.1 8.4.1 2
8.4.2 8.4.2
284
1. a 2. a
1. a 2. a
CRC Standard Curves and Surfaces
= =
= =
1.0, b 1.0, b
=
1.0, b 1.0, b
=
=
=
1.0, c 1.0, c
=
1.0, c 1.0, c
=
=
=
0.5; viewpoint 2.0; viewpoint
=
0.5; viewpoint 2.0; viewpoint
= (
=
(2,  4, 1) (2,  4, 1)
= (
4,3,2) 4,3, 2)
285 2
8.5.1
8.5.1 2
8.5.2
8.5.2
286
CRC Standard Curves and Surfaces
8.6. MISCELLANEOUS FUNCTIONS 8.6.1. z Torus
=
{a 2  [(x 2
+ y2)1/2
b ]2}1/2

1. a = 0.2, b = 0.8; ±z = 2a; viewpoint = (4,  5,2) 8.6.2. z = c(x 3 Monkey saddle
x3
3xy2)

3xy2 
1. c = 1.0; viewpoint = (5,3,3) 8.6.3. z =
Cxy2
Cxy2 
Z
= 0
1. c = 1.0; viewpoint = (5,3,3) 8.6.4. z = cx 2 y2 cx 2 y2 Crossed trough

Z
= 0
1. c = 4.0; viewpoint = (2,  5,3)
zjc = 0
8.6.1
8.6.2
8.6.3
8.6,4
288
CRC Standard Curves and Surfaces
8.6.5. z = (ax 2 + by2)j(X 2 + y2) Conoid of Plucker or cylindroid
ax 2 + by2  ZX2  zy2 = 0
1. a =  0.5, b = 0.8; viewpoint = (5,  3,2)
1. a = 1.5, b = 1.0; viewpoint = ( 3,  4,2) 2. a = 1.5, b = 3.0; viewpoint = (  3,  4,2)
289
8.6.5 2
8.6.6
8.6.6
291
Chapter 9
TRANSCENDENTAL SURFACES The perspective projection described at the beginning of Chapter 7 is used for the figures in this chapter. The surface representation is as described at the beginning of Chapter 8. The bounding box, shown with most figures of this chapter, has limits of  1 to + 1 for all three axes, unless noted otherwise. The true aspect ratio of all figures is preserved. Wherever the surfaces intersect the top or bottom of the bounding box, they are not accurately represented by the method used here.
292
CRC Standard Curves and Surfaces
9.1. TRIGONOMETRIC FUNCTIONS 9.1.1. z 1.
=
=
=
C
C
C
= 0.25; viewpoint = (5,  4,4)
cos(27Taxy) C
= 0.25; viewpoint = (5,  4,4)
sin(27Tax)sin(27Tby)
a = 2.0, b = 1.0,
9.1.6. z = 1.
c sin(21Taxy)
a = 3.0,
9.1.5. z 1.
c cos[21Ta(x 2 + y2)1/2]
a = 3.0,
9.1.4. z = 1.
+ y2)1/2]
a = 3.0, c = 0.25; viewpoint = (5,  4,4)
9.1.3. z 1.
c sin[21Ta(x 2
a = 3.0, c = 0.25; viewpoint = (5,  4,4)
9.1.2. z 1.
=
C
C
= 0.25; viewpoint = (5,  4,4)
cos(27Tax)cos(21Tby)
a = 2.0, b = 1.0,
C
= 0.25; viewpoint = (5,  4, 4)
293
~1.1
~12
~13
~lA
~lS ~1.6
294
CRC Standard Curves and Surfaces
9.2. LOGARITHMIC FUNCTIONS 9.2.1. z = c InCalxl 1.
a
=
1.0, b
=
+ blyl) 2.0, c
=
0.5; viewpoint
=
(4, 4,6)
=
0.2; viewpoint
=
(4,  4,5)
9.2.2. z = c In(ax 2 + by2) 1.
a
9.2.3. z 1.
=
=
1.0, b
=
2.0, c
c In(lxyl)
c = 0.2; viewpoint = (4,  4, 5)
295
9.2.2
9.2.1
9.2.3
CRC Standard Curves and Surfaces
296
9.3. EXPONENTIAL FUNCTIONS 9.3.1. z = c exp(ax
1.
a
+ by)
= 2.0, b = 2.0, c = 0.25; viewpoint = (  3,  5,3)
9.3.2. z = c exp(ax 2 + by2)
1. 2. 3.
a = 1.0, b = 0.5, c = 0.25; viewpoint = (5,  3,3) a = 1.0, b =  0.5, c = 0.25; viewpoint = (5,  3,3) a =  2.0, b =  1.0, c = 1.00; viewpoint = (5,  3, 3)
9.3.3. z = c exp(axy)
1.
a = 1.0, c = 0.35; viewpoint = (5,  3, 2)
297
9.3.2
9.3.1
3
2
9.3.2
9.3.2
9.3.3
CRC Standard Curves and Surfaces
298
9.4. TRIGONOMETRIC AND EXPONENTIAL
FUNCTIONS COMBINED
1.
a
= 3.0, b = 2.0, c = 1.0; viewpoint = (3,2,3)
1.
a
= 3.0, b = 2.0, c = 1.0; viewpoint = (3,2,3)
9.4.3. z 1.
a
9.4.4. z 1.
a
=
c COS(27Tbx) eay
= 1.0, b = 1.0, c = 0.4; viewpoint = (3,  4,3) =
c sin(27Tbx) eay
= 1.0, b = 1.0, c = 0.4; viewpoint = (3,  4,3)
299
9.4.1 9.4.2
9.4.3 9.4.4
300
CRC Standard Curves and Surfaces
9.5. SURFACE SPHERICAL HARMONICS For the surface harmonics, onehalf of the surface is "cut away" along the (x, z) plane in order to more completely illustrate the shape of these
surfaces. The three orthogonal axes, all from 1 to + 1, are added to also clarify the illustration. 9.5.1. r = 1 + cPnO(cos cp) Zonal harmonicsPno is the Legendre polynomial
1. 2. 3.
n = 1, c = 1.0; viewpoint = (2,  3, 2) n = 2, c = 1.0; viewpoint = (2,  3,2) n = 3, c = 1.0; viewpoint = (2,  3,2)
9.5.2. r = 1 + cPnn(cos cp)cos nO Sectoral harmonicsPnn is the associated Legendre function of the first
kind 1. 2.
3.
n = 1, c = 1; viewpoint = (2,  3,2) viewpoint = (3,  2,2) n = 3, c = is; viewpoint = (3,  2,2)
n = 2, c =
·L
9.5.3. r = 1 + c~;n(cos cp) cos me Tesseral harmonicsPnm is the associated Legendre function of the first kind 1. 2. 3.
n = 2, m = 1, c = n = 3, m = 1, c = n = 3, m = 2, c =
t; viewpoint = t; viewpoint =
i;
(2, 3,2) (2, 3,2) viewpoint = (2, 3,2)
301 2 3
9.5.1
9.5.1
9.5.1
9.5.2
9.5.2
9.5.2
9.5.3
9.5.3
9.5.3
302
CRC Standard Curves and Surfaces
9.6. MISCELLANEOUS TRANSCENDENTAL FUNCTIONS 9.6.1. Catenoid Parametrically:
x
=
u cos u
y=usinu z 1.
=
arccosh u
1 < u < 5; 0 < u < 27T; viewpoint = (4,  5, 1); box limits: 5 < x < 5; 5 < y < 5; arccosh5 < z < arccosh5
9.6.2. Right Helicoid Parametrically:
x = u cos u y =
U Slll U
z = cu 1.
2. 3.
c = box c = box c = box
1/(27T);  0.5 < u < 0.5,  27T < u < 27T; viewpoint = (4,  5,1); limits: 0.5 < x < 0.5, 0.5 < y < 0.5, 1 < z < 1 1/(27T); 0.0 < u < 0.5,  27T < u < 27T; viewpoint = (4,  5,1); limits: 0.5 < x < 0.5, 0.5 < y < 0.5, 1 < z < 1 1/(27T); 0.25 < u < 0.5,  27T < u < 27T; viewpoint = (4,  5,1); limits:  0.5 < x < 0.5,  0.5 < y < 0.5, 1 < z < 1
9.6.3. Conocuneus of Wallis Also called conical wedge Parametrically:
x
=
u cos u
y
=
u sin u
z=c(12cos 2 u) 1.
c = cos(7T/4); 0 < u < 1, 7T/4 < u < 37T/4; viewpoint (4,3,1); box limits: c <x < c, 0
303
9.6.1
9.6.2
2
3
9.6.2
9.6.2
9.6.3
304
CRC Standard Curves and Surfaces
9.6.4. Fresnel's Elasticity Surface
Parametrically:
x' = a sin u cos v
1.
y'
=
b sin u sin v
z'
=
c cos u
a = 1.0, b = 2.0, c = 1.0; 0 < u < 7T, 0 < v < 2 7T; viewpoint = (4, 3, 3);box limits: 1 < x < 1, 2 < y < 2, ':"1 < z <1
9.6.5. Cornucopia Parametrically:
x
=
e bu cos v + e GU cos u cos u
y = e bv sin u
+ e au cos u sin u
z = e av sin u 1.
a = 0.3, b = 0.5; 0 < u < 27T,  3 < u < 3; viewpoint (  4,  4,2); box limits:  1 < x < 1,  1 < Y < 1,  1 < z < 1
9.6.6. Sherk's Surface z 1.
c In [ ( cos 2 7T Y ) / ( cos 2 7T x) ]
=
c = 1.0; viewpoint 4 < z < 4
=
(2,  4,3); box limits: 1 < x < 1, 1 < y < 1,
305
9.6.5
9.6.4
9.6.6
306
CRC Standard Curves and Surfaces
9.6.7. Cylindrical Spiral Parametrically:
1.
2.
x
=
a cos nv (1 + cos u) + c cos nv
y
=
a sin nv (1 + cos u) + c sin nv
z
=
bv/(27T) + a sin u
a = 0.1, b = 1.0, c = 0.5, n viewpoint = (2,  4, 2); box limits: a = 0.3, b = 0.9, C = 0.5, n viewpoint = (2, 4,2); box limits:
= 3; 0 < u < 27T, 1 < x < 1, 1 < y < = 3; 0 < u < 27T, 1 < x < 1, 1 < y <
0 1, 0 1,
< V < 27T; 1 < z < 1 < V < 27T; 1 < z < 1
9.6.8. Conical Spiral Parametrically:
x
=
a[l  V/(27T)] cos nv [1
+ cos u] + C cos nv
y
=
a[l  V/(27T)] sin nv [1
+ cos u] + c sin nv
z = bv/(27T) + a[l  v/(27T)]sin u 1.
2.
a = 0.2, viewpoint 0.2 < z a = 0.3, viewpoint 0.3 < z
b = 1.0, c (2,  4,2); < 1.0 b = 0.3, C = (2,  4,2); < 0.3 =
= 0.1, n = 2; 0 < u < 27T, 0 < V < 27T; box limits:  0.5 < x < 0.5,  0.5 < y < 0.5, = 0.1, n = 2; 0 < u < 27T, 0 < V < 27T; box limits:  0.7 < x < 0.7,  0.7 < y < 0.7,
307
9.6.7
9.6.8
9.6.7
9.6.8
309
Chapter 10
COMPLEXVARIABLE SURFACES The functions of this chapter are given by w = fez) where z is the complex number x + iy. To illustrate the functions, both abs(w) and arg(w) are plotted, where 2
2 1/2
abs( w) = (u + u )
arg( w) = arctan( u ju) u
=
real(w)
u = imaginary( w ) The perspective projection described at the beginning of Chapter 7 is used for the figures in this chapter. The surface representation is as described at the beginning of Chapter 8. The bounding box shown with all abs(w) figures of this chapter has limits of  1 to + 1 for x and y axes, but limits of 0 to + 1 for the z axis. The true aspect ratio of all abs( w) figures is preserved. For arg( w) plots, the bounding box also has limits of  1 to + 1 for x and y axes but limits of  7T" to + 7T" for the z axis. Many of the arg( w) plots exhibit jumps of 27T" in the complex plane, along branch cuts of the particular function.
310
CRC Standard Curves and Surfaces
10.1. ALGEBRAIC FUNCTIONS 10.1.1. w = cz
1. 1.
abs( w); c = arg(w); c =
1/ Ii; viewpoint = (1,  4,2) II Ii; viewpoint = (1, 4,3)
10.1.2. w = cz 2
1. 2.
abs(w); c = 0.5; viewpoint = (1, 4,2) arg( w); c = 0.5; viewpoint = ( 1,  4, 3)
10.1.3. w = clz
1. 2.
abs(w); c = 0.1; viewpoint = (5,3,2) arg( w); c = 0.1; viewpoint = (5, 3, 2)
1. 2.
abs(w); c = 0.01; viewpoint = (4,3,3) arg( w); c = 0.01; viewpoint = (4,3,4)
311
10.1.1
10.1.1
10.1.2
10.1.2 2
10.1.3
10.1.3 2
10.1.4
10.1.4
312
CRC Standard Curves and Surfaces
10.1.5. w = az
1. 2.
+b
abs( w); a arg(w); a
= =
0.5, b 0.5, b
= =
0.25; viewpoint 0.25; viewpoint
= =
(3,  4, 4) (3, 4,4)
10.1.6. w = c/(z  a)
1. 2.
abs(w); a arg(w); a
= =
0.5, c 0.5, c
= =
0.1; viewpoint 0.1; viewpoint
(1,4,3) (1,4,3)
= =
10.1.7. w = c/(z  a)2
1. 2.
abs(w); a arg(w); a
10.1.8. w
1. 2.
=
= =
0.5, c 0.5, c
= =
0.01; viewpoint 0.01; viewpoint
=
0.25; viewpoint 0.25; viewpoint
=
=
(1,4,3) (1,4,3)
cz/(z  a)
abs(w); a arg( w); a
= =
0.5, c 0.5, c
= =
1,4,3) (1,4,3)
= (
313 2
10.1.5
10.1.5
10.1.6
10.1.6 2
10.1.7
10.1.7 2
10.1.8
10.1.8
314
1. 2.
CRC Standard Curves and Surfaces
abs(w); a arg(w); a
= =
0.05; viewpoint 0.05; viewpoint
= =
(1,4,3) (1,4,3)
abs(w); c = 0.1; viewpoint = (4,2,3) arg(w); c = 0.1; viewpoint = (4,2,3)
+ b)/(cz + d)
10.1.11. w = (az
1. 2.
= =
+ liz)
10.1.10. w = c(z
1. 2.
0.5, c 0.5, c
abs(w); a arg(w); a
=

=

0.2, b 0.2, b
= =
0.1, c 0.1, c
= =
0.6, d 0.6, d
= =
0.3; viewpoint 0.3; viewpoint
= =
(4,2,3) (4,2,3)
10.1 .9
10.1.9 2
10.1 .10 2
10.1 .11
316
CRC Standard Curves and Surfaces
10.2. TRANSCENDENTAL FUNCTIONS 10.2.1. w = ce Z 1.
2.
abs( w); c = 0.3; viewpoint = (  4,  2,3) arg(w); c = 0.3; viewpoint = (4, 2,3)
10.2.2. w = c In z 1.
2.
abs(w); c = 0.25; viewpoint = (4,  2,3) arg(w); c = 0.25; viewpoint = (4, 2,3)
10.2.3. w = c sin z 1.
2.
abs( w); c = 0.1; viewpoint = (4,  2, 2) arg(w); c = 0.1; viewpoint = (4, 2,3)
:2
10.2.1
10.2.1 :2
10.2.2
10.2.2 :2
10.2.3
10.2.3
318
CRC Standard Curves and Surfaces
10.2.4. w = c cos z 1.
2.
abs(w); c = 0.1; viewpoint = (4, 2,2) arg(w); c = 0.1; viewpoint = (4, 2,3)
10.2.5. w 1.
2.
=
c tan z
abs(w); c = 0.1; viewpoint = (4, 2,3) arg(w); c = 0.1; viewpoint = (4, 2,3)
10.2.6. w = c sinh z 1.
2.
absCW); c = 0.5; viewpoint = ( 2,  4,1) arg(w); c = 0.5; viewpoint = (2, 4,3)
319 2
10.2.4
10.2.4 2
10.2.5
10.2.5 2
10.2.6
10.2.6
320
CRC Standard Curves and Surfaces
10.2.7. w = c cosh z
1. 2.
abs(w); c = 0.5; viewpoint = (2, 4,1) arg(w); c = 0.5; viewpoint = (  2,  4,3)
10.2.8. w = c tanh
1. 2.
abs(w); a arg(w); a
= =
az 5.0, c 5.0, c
= =
0.5; viewpoint 0.5; viewpoint
= =
(2, 4,3) (2,  4,3)
321 2
10.2.7 10.2.7 2
10.2.8 10.2.8
I
323
Chapter 11
NONDIFFERENTIABLE AND DISCONTINUOUS FUNCTIONS In the equations of this chapter, the symbol H is used for the unit step function and the symbol 8 for the unit impulse function. The function 8 is defined only over an infinitesimal interval of x such that its integral over the infinitesimal interval is unity. This requires 8 to have an infinite amplitude, and the amplitude is truncated here at y = 1 for purposes of illustration. The function H is defined such that H(a) is zero for x < a and H(a) = 1 for x > a. Therefore, H(a) is the integral of 8(a).
324
CRC Standard Curves and Surfaces
11.1. FUNCTIONS WITH A FINITE NUMBER OF DISCONTINUITIES 11.1.1. Y = S(x  a) Delta function 1.
a
= 0.5
11.1.2. Y = 8' (x  a) Doublet function 1.
a
= 0.5
11.1.3. y = c[H(x  a)] Step function 1.
a
= 0.5, c = 0.5
11.1.4. y = c[H(x  a)  H(x  b)] Boxcar function 1.
a = 0.25, b = 0.75, c = 0.50
11.1.5. y = c[H(x  a)  2H(x  b) Double boxcar function 1.
a = 0.25, b = 0.50, c = 0.50
+ H(x
 2b
+ a)]
325
11.1.1
11.1.2
11.1.3
11.1.4

11.1.5
326
CRC Standard Curves and Surfaces
11.2. FUNCTIONS WITH AN INFINITE NUMBER OF DISCONTINUITIES 11.2.1. Y = cr,~_oH(x  na)
1.
a
= 0.2, c = 0.1
1.
a a
= 0.2, c = 0.5
2.
= 0.2, c =  0.5
11.2.3. y = c[ 1 + Square sine wave
1.
2r,~_
oo( 1)nH(x  na)]
a = 0.1, c = 0.5
11.2.4. y = c{ 1 + 2r,~_ oo( 1)nH[x  (n  ~)a]) Square cosine wave
1.
a
= 0.1, c = 0.5
327
11.2.1 2
11.2.2

~
'
~
~
'
~
I
L....
11.2.2
.
.
.
'
11.2.3

~
.
r
.
.
r
'
r
'
11.2.4
,....
r
'
328
CRC Standard Curves and Surfaces
2
11.2.5.
y
00
= c ( aX n~oo {H[x  na]  H[x  (n + 1)a])
 2n
~
00
n {H [x  na]  H [x  (n + 1) a]}  1 )
Sawtooth wave 1.
a = 0.2, c = 0.5
11.2.6. y = c( 
2 :
f
00
n
00
{H [x  na]  H [x  (n + 1) a ]}
+ 2 n ~ (n + 00
1) {H [x  na]  H [x  (n
+
1) a]}  1)
Sawtooth wave 1.
a = 0.2, c = 0.5
11.2.7. Y = cr,~~
00
{H[x  n(a + b)]  H[x  n(a + b) 
Comb function 1.
a = 0.05, b = 0.07, c = 0.50
an
329
11.2.6
11.2.5
11.2.7
CRC Standard Curves and Surfaces
330
11.3. FUNCTIONS WITH A FINITE NUMBER OF DISCONTINUITIES IN THE FIRST DERIVATIVE 11.3.1. y = [cj(b  a)][(x  a)H(x  a)  (x  b)H(x  b)] Ramp function 1.
a
=  0.5, b = 0.5, c = 0.5
11.3.2. y = c(l  !xl!a)[H(x Triangular function 1.
+ a)
 H(x  a)]
a = 0.5, c = 0.5
11.3.3. y = c(1  x 2ja 2)1/2[H(x + a)  H(x  a)] Semiellipse (semicircle for a = c) 1.
a
= 0.75, c = 0.50
11.3.4. y = c(1  eaX)H(x) Exponential ramp 1.
a
= 5.0, c = 0.5
331
11.3.1
11.3.2
11.3.3
11.3.4
332
CRC Standard Curves and Surfaces
11.4. FUNCTIONS WITH AN INFINITE NUMBER OF DISCONTINUITIES IN THE FIRST DERIVATIVE 11.4.1. y = 2c n~oo [H( (2n ; l)a ) _ H( (2n ; 3)a )]
+ n~oo (2~X  4nc)[ H( (2n ; l)a ) _ 2H( (2n ; l)a )
+H( (2n; 3)a)] Triangular sine wave a = 0.25, c = 0.5
1.
11.4.2. y
=
L
c
n=
+
{H[(2n  l)a]  H[(2n + l)a]}
00
f.
n=
(2~x
 4nc ){H[(2n  l)a]  2H[2na] + H[(2n + l)a]}
00
Triangular cosine wave 1.
a = 0.25, c = 0.5
11.4.3. y
=
f. n=
(2cn 
C; ){H[(2n 
00
Rectified triangular sine wave 1.
a = 0.25, c = 0.5
l)a]  2H[2na] + H[(2n + 1)a]}
333
11.4.2
11.4.1
11.4.3
CRC Standard Curves and Surfaces
334
11.4.4. Y
=
i2 (c;  2cn ){H[(2n n=
1)a]  2H[2na]
co 00
+c
L n=
{H[(2n  1)a]  H[(2n + 1)an
00
Rectified triangular cosine wave
1.
a
= 0.25, c = 0.5
11.4.5. Y = clsin(27Tax)1 Rectified sine wave
1.
a
= 2.0, c = 0.5
11.4.6. Y = clcos(27Tax)1 Rectified cosine wave 1.
a
= 2.0, c = 0.5
+ H[(2n + 1)an
335
11.4.4
11.4.5
11.4.6
337
Chapter 12
POLYGONS The familiar shapes of twodimensional geometry are shown in the first four sections of this chapter. Scale is entirely relative for these figures. The last three sections show how triangles, squares, and hexagons can be combined into more complicated shapes; these can serve as building blocks for even larger patterns, and some are capable of tiling the plane.
338
CRC Standard Curves and Surfaces
12.1. REGULAR POLYGONS 1. n 2. 3. 4. 5. 6. 7. 8. 9. 10.
n n n n n n n n n
= 3 sides (triangle) = 4 sides (square) = 5 sides (pentagon) = 6 sides (hexagon) = 7 sides (heptagon) = 8 sides (octagon) = 9 sides (nonagon) = 10 sides (decagon) = 11 sides (undecagon) = 12 sides (dodecagon)
12.2. STAR POLYGONS 3 points 4 points 3. n = 5 points 4. n = 6 points 5. n = 7 points 6. n = 8 points 7. n = 9 points 8. n = 10 points 9. n = 11 points 10. n = 12 points 1. n
=
2. n
=
12.3. IRREGULAR TRIANGLES 12.3.1. Right Triangle One angle = 90° 12.3.2. Isosceles Triangle Two angles equal 12.3.3. Acute Triangle All angles < 90° 12.3.4. Obtuse Triangle One angle > 90°
339
[>00 ·000 1
4
2
3
5
6
000
o 7
8
9
10
12.2.1
12.2.2
12.3.1
12.3.2
12.3.3
12.3.4
340
CRC Standard Curves and Surfaces
12.4. IRREGULAR QUADRILATERALS 12.4.1. Rectangle a = band c = d; all angles
=
90°
12.4.2. Parallelogram a = band c = d; all angles
=1=
90°
12.4.3. Rhombus a = b = c = d; all angles
90°
=1=
12.4.4. Trapezoid a = b; c and d parallel; all angles
=1=
90°
12.4.5. Deltoid a = band c = d; two angles are equal
341
12.4.1
12.4.2
12.4.3
12.4.4
12.4.5
342
CRC Standard Curves and Surfaces
12.5. POLYIAMONDS 12.5.1. Triamonds Three connected equilateral triangles 12.5.2. Tetriamonds Four connected equilateral triangles 12.5.3. Pentiamonds Five connected equilateral triangles 12.5.4. Hexiamonds Six connected equilateral traingles
12.6. POLYOMINOES 12.6.1. Trominoes Three connected squares 12.6.2. Tetrominoes Four connected squares 12.6.3. Pentominoes Five connected squares
12.7. POLYHEXES 12.7.1. Trihexes Three connected regular hexagons 12.7.2. Tetrahexes Four connected regular hexagons
343
\li/ 12.5.1
12.5.2
12.5.3
vvV\~
00
~
vs&~
4
~
J~~& 12.5.4
IIII 12.6.1
IIIIIBTI
cEPEB 12.6.2
db
cB IIIIII~ 8:::B EEb B:a~rr8:JEtrn
qf cdF
% CW
12.6.3
c8=> c¢YQ?
J3 OSQ}D 12.7.2
db
345
Chapter 13
POLYHEDRA AND OTHER CLOSED SURFACES WITH EDGES Smooth, closed surfaces form the boundary of a volume, or solid, and the outward normal of the surface at any point is everywhere continuous in all directions about that point (for example, a sphere or torus). However, some closed surfaces contain edges, which are defined to be where the derivative of the surface (and therefore the normal) is discontinuous. Most of these surfaces with edges are classified as polyhedra, which are closed, 3D figures made up of polygons. The known polyhedra are numerous (see, for instance, the comprehensive survey of Williams l ), and only the simpler and more common ones are presented here. The projection used here is the perspective one (see start of Chapter 7), with the projection plane normal to the line joining the origin of the axes and the viewing point. The viewing point is not given for the figures here because it is not meaningful in this context; however, it is at a large distance from the figure in all cases, so that the projection is effectively a parallel one.
346
CRC Standard Curves and Surfaces
13.1. REGULAR POLYHEDRA 13.1.1. Tetrahedron (n
=
4)
13.1.2. Hexahedron (n = 6) 13.1.3. Octahedron (n= 8) 13.1.4. Dodecahedron (n = 12) 13.1.5. Icosahedron (n = 20)
347
13.1.1
13.1.2
13.1.3
13.1.4
13.1.5
348
CRC Standard Curves and Surfaces
13.2. STELLATED (STAR) POLYHEDRA Let a be the distance to the star vertices divided by the distance to the regular vertices, both measured from the center of the figure. 13.2.1. Tetrahedron (n = 4, a = 2) 13.2.2. Hexahedron (n = 6, a = 3) 13.2.3. Octahedron (n = 8, a = 3) 13.2.4. Dodecahedron (n = 12, a = 2) 13.2.5. Icosahedron (n = 20, a = 3)
349
13.2.2
13.2.1
13.2.4
13.2.3
13.2.5
350
CRC Standard Curves and Surfaces
13.3. IRREGULAR POLYHEDRA 13.3.1. Prism 1. Triangular 2. Hexagonal 13.3.2. Star Prism 1. Triangular 2. Hexagonal 13.3.3. Antiprism 1. Triangular 2. Hexagonal
351 2
13.3.1
13.3.1
13.3.2
13.3.2 2
13.3.3
13.3.3
352
CRC Standard Curves and Surfaces
13.3.4. Prismoid 1. Triangular 2. Hexagonal 13.3.5. Prismatoid 1. Triangularhexagonal 2. Squareoctagonal 13.3.6. Parallelepiped 1. Oblique 2. Right
353 2
13.3.4 13.3.4
13.3.5 13.3.5 2
13.3.6 13.3.6
354
CRC Standard Curves and Surfaces
13.3.7. Pyramid 1. Triangular 2. Hexagonal 13.3.8. Dipyramid 1. Triangular 2. Hexagonal 13.3.9. Trapezohedron 1. Triangular 2. Hexagonal 13.3.10. Irregular Dodecahedron
355 2
13.3.7 13.3.7 2
13.3.8 13.3.8 2
13.3.9 13.3.9
13.3.10
356
CRC Standard Curves and Surfaces
13.4. MISCELLANEOUS CLOSED SURFACES WITH EDGES 13.4.1. Cylinder 1. Right circular 2. Oblique circular 3. Right circular (disk) 13.4.2. Cone 1. Right circular 2. Oblique circular 13.4.3. Frustrum of a Cone 13.4.4. Hemisphere 13.4.5. Rectangular Torus
357 2
13.4.1
13.4.1 3
13.4.1 2
13.4.2
13.4.2
13.4.4
13.4.3
13.4.5
358
CRC Standard Curves and Surfaces
REFERENCES 1. Williams, R, The Geometrical Foundation of Natural Structure, Dover Publications, New York, 1979.
359
Chapter 14
RANDOM PROCESSES A large number of classes of random processes exist, and many variations are recognized within each class. Extensive treatments of random processes, from differing perspectives, can be found in Mandelbrot,1 Box and Jenkins, 2 and Parzen. 3 This chapter only attempts to show those onedimensional processes which are of simple form or are in common usage. There is a large and complex suite of random processes called Markov processes; they are of such variety that no representative examples are even given here. The random processes of this chapter are plotted as time evolutions, with the time axis being horizontal. For each process, three realizations are shown, each independent of the others, so that the reader can appreciate the variability of each process. For each realization, 200 (sometimes 100 or 50) points or increments are plotted. The plots are done with arbitrary scaling of the amplitude, such that the plotted range of the amplitude is always in constant proportion to the plotted length of the time axis.
360
CRC Standard Curves and Surfaces
14.1. ELEMENTARY RANDOM PROCESSES 14.1.1. White Noise 14.1.2. TwoValued Process 14.1.3. Unit Random Walk This is the integral of the Poisson wave.
361
14.1.1
14.1.2
~
•
nn~ D
8fOl\DVJDDnr 0
VD D
14.1.3
362
CRC Standard Curves and Surfaces
14.2. GENERAL LINEAR PROCESSES The discrete, general linear (ARMA) process is described by the equation:
y(i) = aly(i  1) + a 2y(i  2) + ... +any(i  n) + blr(i 1) + b 2 r(i  2) + ... +bmr(i  m) + rei) where the ai and bi are constant coefficients and r is a random variable. If all bi = 0, the process is called an autoregressive (AR) process of order n. If all a i = 0, theprocess is called a movingaverage (MA) process of order m. If at least one aj and at least one bj are nonzero, the process is called a mixed (ARMA) process. 14.2.1. FirstOrder Autoregressive 1. a l = 0.2 = 0.8 3. a l = 0.2 4. a l = 0.8
2. a l
363 2
IYv~~ J;J,~~Jv/ ~~ 14.2.1 3
~~jr1~
14.2.1 4
fi~~
~ ~'~Itt+
up JiL~!~1~'~ "I
~ · T "." 'II .t·+~f~"t1
14.2.1
14.2.1
. H
364
CRC Standard Curves and Surfaces
14.2.2. SecondOrder Autoregressive 1. 2. 3. 4.
at = 0.4, a 2 = 0.4 at = 0.4, a 2 = 0.4 a t =0.4,a 2 = 0.4 at = 0.4, a 2 = 0.4
365 2
~
~~
~~~ 14.2.2
14.2.2
3
4
~~A'\ulAAb In~AnM""JA&~MAtr. ~ ~V '~vyr~, "~ V , .,V V'lr~
Ii
V
tIJrI,A\J.'1 ~l.! .1!1I "'v V~ VYVf VV""VY
r'oIil4/ ~I \r.~ IV' ~'~ 'i
"U~! &nAiII4)~+;tllirtl Vf' 'V r V' T' ~ W,IV R
'J
"
14.2.2
14.2.2
366
CRC Standard Curves and Surfaces
14.2.3. FirstOrder Moving Average 1. b i 2. b i 3. b i 4. b i
= = = =
0.2 0.8 0.2 0.8
367 2
14.2.3
14.2.3
3
4
r' rmY'r"j n"1 ' ~~ ~~~.I·IIlf+t··+ ~~~ ~~~~ I
14.2.3
r
,
~
i
14.2.3
368
CRC Standard Curves and Surfaces
14.2.4. SecondOrder Moving Average 1. 2. 3. 4.
b i = 0.4, b2 = b i = 0.4, b 2 b i = 0.4, b 2 = b i = 0.4, b 2
0.4 = 0.4 0.4 = 0.4
369 2
14.2.4
14.2.4 3
4
1fI1~1~ .!I'flil11IIft' U'd~jjV' vl'V'~~fAd~ IPVp vnvy
~.~~. ~T"U IAoWl~AlI ~J.r.~~LI~~ftM Ii' ~V'l V In rprr ~ vii IV
r ,11( PI" '1"plY~ r tr~ I'V I~~"fl~ ~
14.2.4
~ r IV
T
14.2.4
370
CRC Standard Curves and Surfaces
14.2.5. FirstOrder Mixed 1. a l = 0.4, hI = 0.4 = 0.4, hI = 0.4 3. a l = 0.4, hI = 0.4 4. a l = 0.4, hI = 0.4
2. a l
14.2.6. HighOrder Moving Average 1. m = 4 2. m = 8
371 2
14.2.5 3
14.2.5
14.2.5 4
14.2.5 2
14.2.6
14.2.6
372
CRC Standard Curves and Surfaces
14.3. INTEGRATED PROCESSES The discrete, general integrated (ARIMA) process y(i) is described by the equation
Vdy(i) = alVdy(i  1) + a 2V dy(i  2) + ... +anVdY(i  n) + b1r(i  1)
+ b 2 r(i  2) + ... +bmr(i 
m)
+ rei)
where the aj and bj are constant coefficients, r is a random variable, and d is the order of the differential. For example, if d = 1, then y is the first integral of an ordinary ARMA process (see the explanation of a general linear process above). If all bj = 0, the process is called an integrated autoregressive (ARI) process of order n. If all aj = 0, the process is called a integrated movingaverage (IMA) process of order m. If at least one aj and at least one bj are nonzero, the process is called an integrated mixed (ARlMA) process. 14.3.1. FirstOrder Autoregressive 1. a l = 0.2 2. al = 0.8 3. a l = 0.2 4. a l = 0.8
373 2
14.3.1
14.3.1
3
4
.~.~
.~\;\
~\N'I.A
AAC'
~,VV'~
r
.~
W'
•V •
~I~""
rei
A.~ M1tJ\,~
~v
u
14.3.1
V~\}
14.3.1
374
CRC Standard Curves and Surfaces
14.3.2. FirstOrder Moving Average 1. hI = 0.2 2. hI = 0.8 3. hI = 0.2 4. b i = 0.8
375 2
14.3.2
14.3.2
3
4
~'1• yl..M~y~
j~,~ ~~'''~I~ ,"V m VI
14.3.2
14.3.2
376
CRC Standard Curves and Surfaces
14.3.3. FirstOrder Mixed 1. a l = 0.4, hl = 0.4 2. a l = 0.4, hl = 0.4
3. a l = 0.4, hl = 0.4 = 0.4, hl = 0.4
4. a l
377
14.3.3
14.3.3 4
14.3.3
14.3.3
378
CRC Standard Curves and Surfaces
14.4. FRACTAL PROCESSES The general fractal process (fractallinetoline function) is a process whose increments are distributed according to the Gaussian (or normal) density law. If yet  at) and yet + at) are two values separated by two increments, then the correlation of the two values is given by r = 2 2H 
1 
1
When H = ~ exactly, r = 0 and the process is the classical onedimensional Brownian motion because each new value is incremented by an independent random variable from the last value. Processes for which H > ~ (r > 0) are called persistent processes because they have longwavelength components. Processes for which H < ~ (r < 0) are called antipersistent processes because they are dominated by short wavelengths. H is in the range of 0 to 1; thus r is in the range of  ~ to 1. The general fractal process is also called a fractional Brownian process. The method of construction is an approximate method taken from K. Falconer 4 (Equation 16.13). 14.4.1. Brown Function The Brown function is also called Bachelier or Wiener or Levy function. 1. H = 0.5 14.4.2. Persistent Fractal Process 1. H
=
0.8
14.4.3. Antipersistent Fractal Process 1. H = 0.2
379
14.4.2
14.4.1
14.4.3
380
CRC Standard Curves and Surfaces
14.5. POISSON PROCESSES In the Poisson process, events occur over time at a mean rate of a. Let N(t) be the number of events which have occurred since t = O. Then the probability that N(t) = m is given by:
P[N(t)
=
m]
=
eat(at)m 1m!
The interevent time T is exponentially distributed thus:
peT) = ae aT The above is called the ordinary or homogeneous Poisson process. When the mean rate of events, a, varies with time, it is a nonhomogeneous process. Another variation is not to track the cumulative number of events, but rather to track the accumulation of a random variable at the times given by an ordinary Poisson process; such processes are called compound Poisson processes. 14.5.1. Homogeneous Poisson Process 1. a
=
1.0
14.5.2. Nonhomogeneous Poisson Process 1. a
=
1.0 + 0.5t
14.5.3. Compound Poisson Process The random variable is from a normal distribution with mean zero. 1. a
=
1.0
14.5.4. Poisson Wave (Telegraph Signal) This process takes one of the values lor times corresponding to a Poisson process. 1. a
=
1.0
+ 1, with equal probability, at
381
~~ ~~ ~~ 14.5.1
14.5.2
1~53
1~5A
382
CRC Standard Curves and Surfaces
REFERENCES 1. Mandelbrot, B. B., The Fractal Geometry of Nature, W. H. Freeman, San Francisco, 1983. 2. Box, G. E. P., and G. M. Jenkins, Time Series Analysis: Forecasting and Control, HoldenDay, San Francisco, 1976. 3. Parzen, E., Stochastic Processes, HoldenDay, San Francisco, 1962. 4. Falconer, K., Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, New York, 1990.
INDEX A acute triangle, 338 Agnesi (witch of), 42 Airy functions, 230 algebraic curves, 14, 2594, see also specific types algebraic surfaces, 273288, see also specific types alpha integral, 200 antiprism, 350 antisymmetric, 6 Archimedean spirals, 164 Archimedes' spiral, 164 ARlMA process, 372376 ARMA process, 362370 associated Legendre functions of the first kind, 212 of the second kind, 214 astroid, 178 asymptotes, 7 autoregressive process, 362364, 372 axial symmetry, 7
B Bachelier function, 378 Bernoulli (lemniscate of), 88 Bernoulli polynomials, 194 Bessel functions of the first kind, 216 modified, 220 modified spherical, 228 of the second kind, 216 spherical, 226 beta function, 206 beta integral, 202 beta probability density, 248 bicorn,82 bifolia, 90 binomial probability density, 244 Bowditch curves, 172174 boxcar function, 324 Brown function, 378 bullet nose curve, 68 butterfly catastrophe, 94
c cardioid, 88, 180 Cartesian coordinates, 23
Cassinian oval, 90 Catalan (trisectrix of), 58 catastrophe curves, 28, 94 catenary, 130 catenoid, 302 Cauchy probability density, 248 Cayley's sextet, 92 Chebyshev polynomials of the first kind, 186 of the second kind, 186 chisquare distribution, 258 probability density, 248 circle, 66 circular cone, 356 circular cylinder, 356 circular helix, 262 cissoid of Diocles, 64 classification of curves, 1314 closed surfaces, 345 cochleoid, 168 cochloid, 80 comb function, 328 complementary error function, 208 complex variable surfaces, 309320 conchoid of Nicomedes, 80 cone, 278 elliptic, 280 oblique circular, 356 right circular, 356 conical helix, 262 conical spiral, 306 conical wedge, 302 conjugate (isolated) points, 10 conocuneus of Wallis, 302 conoid of Plucker, 288 continuity, 9 continuous probability densities, 248256 cornucopia, 304 Cornu's spiral, 210 in three dimensions, 270 cosine integral, 204 critical points, 1011 cross curve, 72 crossed trough, 286 cubic curve, 26 curtate cycloid, 176 curvature, 13 cusp catastrophe, 28 cusp points, 10
383
384
CRC Standard Curves and Surfaces
cycloid curtate, 176 ordinary, 176 prolate, 176 cylinder elliptic, 276 hyperbolic, 276 oblique circular, 356 right circular, 356 cylindrical coordinates, 4 cylindrical spiral, 306 cylindroid, 288
D decagon, 338 degenerate point, 1011 delta function, 324 deltoid, 178, 340 derivative, 56 determinate curve, 1314 devil's curve, 90 differentiable, 9 digamma function, 206 dimension, 1 Diocles (cissoid of), 64 dipyramid, 354 discontinuities, 9 finite number of, 324 infinite number of, 326330 discontinuous functions, 323334, see also specific types discrete probability densities, 243246 disk, 356 dodecagon, 338 dodecahedron, 346, 348 irregular, 354 domain, 7 double boxcar function, 324 double cusp point, 10 doublet function, 324
E eight curve, 66 ellipse, 66 ellipsoid, 278 elliptical helix, 262 elliptic cone, 280 elliptic cylinder, 276 elliptic integrals, 232 elliptic paraboloid, 276 epispiral, 166
epitrochoid, 180 equiangular spiral, 164 error function, 208 essential discontinuity, 9 Euclidean space, 1 Eudoxus (kamplyle of), 70 Euler polynomials, 194 Euler's integral of the first kind, 206 of the second kind, 206 even function, 12 explicit form of curve, 1 exponential functions, 124128, 296 powers of x, combined with, 150152 and trigonometric functions combined, 138 exponential integral, 200 exponential probability density, 248 exponential ramp, 158, 330 extremevalue probability density, 250
F F distribution, 258 Fermat's spiral, 164 finite discontinuity, 9 folium, 90 fractal processes, 378 Freeth (nephroid of), 92 Fresnel integral first, 210 second, 210 Fresnel integral spiral, 210 in three dimensions, 270 Fresnel's elasticity surface, 304 frustrum of a cone, 356 fully bounded, 7
G gamma function, 206 gamma probability density, 250 Gaussian curve, 126 distribution, 258 probability density, 252 second derivative, 158 Gegenbauer polynomials, 190 generalized Laguerre polynomials, 188 general linear processes, 362370 geometric probability density, 244 Gerono (lemniscate of), 66 Gudermannian function, 160
385
H Hankel function, 218 helical curves, 262264 helix, 262264 hemisphere, 356 heptagon, 338 Hermite polynomials, 190 hexagon, 338 hexahedron, 346, 348 hexiamonds, 342 Hippias (quadratrix 00, 158 hippopede curve, 88 homomorphic operations, 17 hyperbola, 26, 70 hyperbolic cylinder, 276 hyperbolic functions, 130132 inverse, 136 powers of x, combined with, 154 hyperbolic paraboloid, 276 hyperbolic spiral, 164 hyperboloid of one sheet, 280 of two sheets, 280 hyperellipse, 82 hypergeometric probability density, 244 hypoellipse, 82 hypotrochoid, 178
I icosahedron, 346, 348 implicit form of curve, 1 improper integral, 12 infinite discontinuity, 9 inflection point, 10 integrability, 1112 integral curve, 1516 integrated processes, 372376 inverse hyperbolic functions, 136 inverse trigonometric functions, 118 involute of a circle, 170 irrational function, 14 irregular polygons, 338340 irregular polyhedra, 350354 irregular quadrilaterals, 340 irregular triangles, 338 isolated point, 10 isosceles triangle, 338
J Jacobi elliptic functions, 234240 Jacobi polynomials, 192
K kampyle of Eudoxus, 70 kappa curve, 88 Kelvin functions, 222224
L Laguerre polynomials, 188 Laplace probability density, 250 Legendre functions of the first kind, 212 of the second kind, 214 Legendre polynomials, 184 lemniscate of Bernoulli, 88, 90 of Gerono, 66 Levy function, 378 limacon of Pascal, 88 linear curve, 26 Lissajous curves, 172174 lituus, 164 logarithmic functions, 120122, 294 powers of x, combined with, 146148 logarithmic integral, 202 logarithmic spiral, 164 logistic probability density, 250 logistique, 164 lognormal probability density, 252 loop, 10
M Maclaurin (trisectrix of), 78 mathematical physics, special functions in, 199242 matrix methods for transformation, 2021 maximum points, 1011 Maxwell probability density, 252 minimum points, 1011 mixed process, 370, 374 modified Bessel functions, 220 modified spherical Bessel functions of the first kind, 228 of the second kind, 228 of the third kind, 228 modulated sine wave, 100 monkey saddle, 286 movingaverage process, 366370, 374 multiple values, 1213
386
CRC Standard Curves and Surfaces
N negative binomial probability density, 246 nephroid, 180 nephroid of Freeth, 92 Neumann function, 216 Neumann polynomials, 196 nhelix, 264 Nicomedes (conchoid of), 80 node point, 10 nonagon, 338 non degenerate point, 1011 nondifferentiable functions, 323334, see also specific types nonlinear scaling, 19 nonorthogonal polynomials, 194196 normal curve, 126 distribution, 258 probability density, 252 nroll mill, 86
o oblique circular cone, 356 oblique circular cylinder, 356 obtuse triangle, 338 octagon, 338 octahedron, 346, 348 odd function, 12 ordinary cycloid, 176 orthogonal, 16 orthogonal polynomials, 184192 osculation, 10 oval of Cassini, 90
piecewise continuous functions, 16 piriform, 62 plane, 274 plane symmetry, 7 Poinsot (spiral of), 168 point symmetry, 6 Poisson probability density, 246 Poisson processes, 380 Poisson wave, 380 polar coordinates, 3 polygons, 337342, see also specific types irregular, 338340 regular, 338 star, 338 polyhedra, 345358, see also specific types irregular, 350354 regular, 346 stella ted (star), 348 polyhexes, 342 polyiamonds, 342 polynomial sets, 183198, see also specific types polyominoes, 342 prism, 350 prismatoid, 352 prismoid, 352 probability densities continuous, 248256 discrete, 244246 probablility distributions, 258 prolate cycloid, 176 psi function, 206 pure shear, 19 pyramid, 354
Q
p parabola, 26, 58 parabolic cylinder functions, 230 parabolic spiral, 166 parallelepiped, 352 parallelogram, 340 parallel projection, 261 parametric representation, 1 Pareto probability density, 252 partial derivative, 6 Pascal (Iimacon of), 88 pentagon, 338 pentiamonds, 342 pentominoes, 342 periodicity, 8 perspective projection, 261
quadratic curve, 26 quadratrix of Hippias, 158 quadrilateral, 340 quartic curve, 26 quintic curve, 26
R radial translation, 1819 radius of curvature, 13 ramp function, 330 random processes, 359382 random walk, 360 range, 7 rational function, 14 Rayleigh probability density, 254
387 rectangle, 340 rectangular coordinates, 3 rectified wave cosine, 334 sine, 334 reflection, 18 regular polygons, 338 regular polyhedra, 346 removable discontinuity, 9 rhodonea, 166 rhombus, 340 Riemann functions, 230 right circular cone, 356 right circular cylinder, 356 right helicoid, 262, 302 right strophoid, 78 right triangle, 338 roll mill, 86 roots, 11 rose, 166 rotating sine wave, 268 rotation, 17 rotational scaling, 18
s saddle, 276 saddle point, 10 sampling distributions, 258 sawtooth wave, 328 scaling, 1718 Schlafli polynomials, 196 sectoral harmonics, 300 semibounded, 7 semicircle, 330 semicubical parabola, 28 semiellipse, 330 serpentine, 42 sextet (Cayley's), 92 sextic curve, 26 shear (simple), 1920 Sherk's surface, 304 sici spiral, 204 in three dimensions, 270 _sigmoidal curve, 124 sinc function, 140 sine integral, 204 sine wave on cone, 268 on hyperboloid, 266 modulated, 100 rectified, 334 rotating, 268
on sphere, 266 square, 326 in three dimensions; 266268 triangular, 332 singlevalued curve, 1213 singular point, 910 skew curve, 261 smoothness criterion, 1 Snedecor's F distribution, 258 probability density, 254 space curves, 261272, see also specific types sphere, 278 spherical Bessel functions of the first kind, 226 modified, 228 of the second kind, 226 of the third kind, 226 spherical coordinates, 4 spherical harmonics, 300 spherical helix, 264 spheroid oblate, 278 prolate, 278 spiral of Poinsot, 168 square, 338 square wave cosine, 326 sine, 326 star polygons, 338 star (stellated) polyhedra, 348 star prism, 350 statistics, special functions of, 243258 step function, 324 strophoid (right), 78 Student's t distribution, 258 probability density, 256 surface spherical harmonics, 300 swallowtail catastrophe, 94 sweep signal, 160 symmetry, 67
T teardrop curve, 174 telegraph signal, 380 tesseral harmonics, 300 tetrahedron, 346, 348 tetrahexes, 342 tetriamonds, 342 tetrominoes, 342
388
CRC Standard Curves and Surfaces
threedimensional curves, 261272 toroidal spiral, 270 torus, 286 rectangular, 356 tractrix, 158 transcendental curves, 15, 97180, see also specific types transcendental surfaces, 291306, see also specific types transformations, 1721 transient function, 12 translation, 17 trapezohedron, 354 trapezoid, 340 triamonds, 342 triangle, 338 triangular function, 330 triangular wave cosine, 332 rectified cosine, 334 rectified sine, 332 sine, 332 trigonometric functions, 98116, 292 and exponential functions combined, 138 inverse, 118 powers of x, combined with, 140144 trihexes, 342 trisectrix, 88 of Catalan, 58
of Maclaurin, 78 trochoids, 178180 trominoes, 342 Tschirnhauser's cubic, 58 turtletrack curve, 16 twovalued process, 360
u unbounded, 7 un decagon, 338
w Wallis (conocuneus of), 302 Weber function, 216 Weibull probability density, 256 weighting, 19 white noise, 360 Wiener function, 378 witch of Agnesi, 42
z zeros, 11 zeta function, 230 zonal harmonics, 300