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) = Zero(0/ M ). Now, Zero(F/w) ^ Zero(P/D) because (1,1) is contained in Zero(P/u) but not in Zew(P/D). This shows that the polynomial D cannot be abandoned during the projection. Keeping D, we have Zero(P/[u,D])=Zeio(P/D), so that for any u £ Zero(0/w) the system r - S = 0 ,
i-ii/0
has solutions for x, which can be computed form the above (triangularized) system. A method similar to TriSerP has been proposed by Wu [96] (see also [23]) via characteristic sets computation. The issue explained above is not correctly handled in [23], however. Keeping the polynomials like D in the above example makes the sets of inequation polynomials often very large, and it is not easy to simplify them. In contrast, the sets of inequation polynomials in regular systems are usually more compact. It is also not expensive to decompose a polynomial system into regular systems (compared to decomposition with projection or into simple systems, see Table 10). Therefore, we consider the use of regular systems a good alternative to deal with such computational problems as solving parametric systems, deriving locus equations, and implicitizing rational curves and surfaces, for which the projection property is required. Finally, we note that normalization is another alternative device for solving parametric systems. This may be seen from Examples 7.3.1 and 7.4.3. Normalization may simplify triangular sets in many case, but make them more complicated in many other cases. Another nontrivial parametric system is given by the four polynomial equations in Example 9.3.1 when x ~< y are regarded as parameters and u,v,w as unknowns. From the regular systems computed therein, we know for what values of x and y the given system has solutions for u,v,w, and such solutions may be determined for concrete values of x and y.
Chapter 8 Automated Theorem Proving and Discovering in Geometry
Automated theorem proving in geometry was a highlight of research, represented by the work of H. Gelernter [27] and his collaborators, in the early days of artificial intelligence. However, there had been no breakthrough until the later 1970s when W.-t. Wu introduced an algebraic method based on polynomial computation [85, 87]. After the surprising success of Wu's method, which has proved hundreds of nontrivial geometric theorems, automated deduction in geometry has witnessed two flourishing decades of development. Different algebraic methods based on characteristic and triangular sets, Grobner bases, variable and point elimination, and Clifford algebra as well as other synthetic approaches using heuristic search and term-rewriting have been proposed, investigated, implemented, and applied to prove and discover theorems in elementary and differential geometries (see [13, 71, 98, 99] and http://www-calfor.lip6.fr/~wang/GRBib). This chapter presents some of the coordinate-based methods together with a number of examples, which serve to demonstrate the power of elimination techniques for geometric reasoning and to illustrate the proving machines built into GEOTHER.
8.1
A Simple Algebraic Approach
A geometric theorem TT usually has a hypothesis HYP and a conclusion CON, where both HYP and CON are finite sets (or sequences) of geometric relations (or statements). The problem of proving the theorem TT is to determine whether every geometric relation in CON follows logically from the conjunction of relations in HYP. Using Wu's or other algebraic methods to prove TT, one needs to translate the involved geometric relations or statements into algebraic expressions, as we have
116
Chapter 8. Automated Theorem Proving and Discovering
seen in Example 1.2.1. To do so, a standard technique is to choose a coordinate system and denote the coordinates of points and other involved geometric entities by ^determinates x\,...,xn. Then the hypothesis and the conclusion of the theorem can be expressed, in most cases, as polynomial equations (=), inequations (^), and inequalities (<, <) inxi,... ,xn. The assignment of coordinates and the translation process may be done automatically, for example by the functions Coordinate and Algebraic in GEOTHER, which have implemented a dictionary for popular relations in plane Euclidean geometry. The algebraic expressions for most ordinary geometric relations like collinearity, perpendicularity, and congruence involve only polynomial equations. An important, large enough class of geometric theorems is of equality type, in which the algebraic formulation of any theorem involves only polynomial equations and inequations. The GEOTHER package is designed and implemented mainly for this class of theorems, and our presentation and examples in this chapter are also for theorems of equality type. It is usually implicit in the statement of a geometric theorem, as observed first by Wu [98], that the considered figures are in a generic position. For example, while speaking about a triangle, we mean a real triangle which does not degenerate into a line or a point. When formulating the theorem algebraically, one may rule out some of the degenerate cases a priori by adding inequations. Now consider a fixed geometry (25, a geometry-associated field HC of characteristic 0, and an appropriate coordinate system D under which a correspondence between statements in (25 and algebraic expressions over ^C m a v De established. Let the hypothesis of a given theorem TT in © be expressed under D as a finite system of polynomial equations and inequations
r«,w=o,...,«,w=o, (where each A = 0 corresponds usually to a predetermined degenerate case), and the conclusion be expressed as, without loss of generality, a single polynomial equation CON: G ( * ) = 0 . (8.1.2) All the polynomials H(,Dj and G are in the indeterminates x = (x\,...,xn) — which are coordinates of points and other geometric entities involved in the theorem — with coefficients in %^. Our problem of proving the theorem algebraically amounts to deciding (a) whether the logical formula (Vx) \Hx{x) = 0 A • • • AHs(x) = 0 A£»I(JC) ^ 0 A • • • A A W ^ 0 =^> G(x) = 0] (8.1.3)
8.1. Simple Algebraic Approach
117
is valid, where A and => denote the logical "and" and "imply" respectively. If (8.1.3) is determined invalid, we want to (b) find "appropriate" subsidiary conditions D\(x) ^ 0,... ,D*» (x) ^ 0 so that the formula (VJC) \R\{X) = 0 A • • • AHs(x) =0ADi(x) ^ 0 A • • • A A W ^ 0A D\ (x) ^ 0 A • • • AD*t, (x) ^ 0 = • G(x) = 0] becomes valid over %^ or some extension field of 3C. The additional inequations D* (x) ^ 0 are to be determined, ensuring that the configuration of the geometric hypotheses is in a generic position. In what follows, let . r={Hh...,Hs}, ®= {Di,...,Dt}. The considered geometric theorems of equality type thus have the algebraic form TT: P = 0AQ^0=^>G = 0. A simple method, which is essentially due to Wu [87, 98], consists of the following steps. The method either proves that the theorem TT is true under the subsidiary conditions SC, or reports that the hypothesis of TT is self-contradictory or TT is not confirmed. WO. [Preprocess.] Translate the geometric theorem into the algebraic form TT: P = 0 A Q ^ 0 = > G = 0. This can be done automatically by implementing a translator for some commonly used geometric relations. W l . [Transform the set P of hypothesis polynomials into a triangular set T.] Compute a (quasi-, weak-) medial set T of P over %_ by algorithm CharSetN or PriTriSys described in [78]. If T is contradictory or there is a polynomial in Q whose pseudo-remainder with respect to T is 0, then report that the hypothesis of TT is self-contradictory and the procedure terminates. W2. [Reduce G to 0 successively using the polynomials in T.] Compute the pseudo-remainder R of G with respect to T: /?:=prem(G,T). If 7? = 0, then let I\,... ,Ir be all the distinct irreducible factors of the polynomials in i n i s e t ( T ) which do not divide any D,-, set SC:=/i ^ 0 A - - - A / r ^ 0 , and output "TT is true under the subsidiary conditions SC." Otherwise, report that TT is not confirmed and the procedure terminates. W°o. [Postprocess]. Interpret the algebraic subsidiary conditions geometrically and determine which conditions are nondegeneracy ones. In most cases, the interpretation can be done easily and automatically (see, e.g., [13, 70]). Whether a subsidiary condition is a nondegeneracy condition may be seen from its geometric meaning or via dimension analysis, etc.
118
Chapter 8. Automated Theorem Proving and Discovering
This simple method is very efficient for confirming a large number of geometric theorems. The GEOTHER function Wprover is an implementation of the method with some technical considerations, where CharSetN and a top-down triangularization procedure are used for the computation of medial sets. The above Wl and W2 may be replaced by the following three alternative steps, in which Grobner bases are used. G l . Compute a Grobner basis G of P U {D\z\ — 1,... ,Dtzt — 1} over %^ with respect to the purely lexicographical term order determined by x\ < < xn -< z\ -<•••-< zt, where z\,...,zt are new indeterminates. If 1 e G, then report that the hypothesis of TV is self-contradictory and the algorithm terminates. G2. Compute the normalformR of G modulo G: /?:=normalf(G,G). IfR = 0, then output "the theorem IT is universally true" and the procedure terminates. G3. Take a quasi-basic set IB of G and compute R :=prem(/?,B). If R = 0, then let I\,... ,Ir be all the distinct irreducible factors of the polynomials in iniset(B) which do not divide any £),-, set SC:=/i ^ 0 A - - - A / r ^ 0 , and output "TT is true under the subsidiary conditions SC." Otherwise, report that TT is not confirmed and the procedure terminates. This alternative method has been implemented as GCprover in GEOTHER. See [98, 74, 78] for the correctness of the method and other details. Adding nondegeneracy conditions to the hypothesis is a good heuristic for geometric theorem proving using Grobner bases, so one should try to figure out such conditions in the way of formulating the theorem. In order to deal with subsidiary conditions effectively and to speak about genericness, one needs to separate the variables x into parameters u and geometric dependents y. The former are free variables, while the latter are constrained by the geometric conditions. When using the provers in GEOTHER, the user is advised to present only the dependent variables y as the third argument (list of ordered variables) in the Theorem specification. Whether or not the theorem is true in a degenerate case can be verified by using the same method, regarding the degeneracy condition as an additional hypothesis of the theorem. Unless explicitly stated, the Grobner bases mentioned in the examples of this chapter are always with respect to the purely lexicographical term order (plex) determined by the indicated variable ordering. For the sake of efficiency one can
119
8.1. Simple Algebraic Approach
also choose another elimination order instead. In some situation, the total degree term order is sufficient. An illustration of the above method can be found in Example 1.2.1 (the butterfly theorem). Here is another example which presents some more details about the proof of Simson's theorem (see Chap. 5 and also Examples 7.2.1 and 7.2.2 in [78]). Example 8.1.1. Let IP be the set of five hypothesis polynomials in the algebraic form of Simson's theorem shown in Sect. 5.2.2, and the variables xl,...,x9 be renamed xi,...,x$. Then, with respect to the ordering x\ -< • • • -< xg, a weak-characteristic set of P is hx\-x\(x\+x\-x\)x5+xix-i(x\-x\), hxd — /3X3X5 — /3X4 +
C=
x\x\,
I?,x7-x?,(x6+x\), 2
I4XS - /5X3X5 - IJX4 - X1X3,
I5x9-xi(xs-xl) where Ii=xix3,
I2
•A+ll
h=X2+X\,
h=x\+ll,
I5=X2-X\
are the initials of the five polynomials C\, • • • ,C$ in C respectively. Clearly, prem(/,-,C) ^ 0 for 1 < i < 5. Let G be the conclusion polynomial of Simson's theorem as in Sect. 5.2.2. It may be easily verified that prem(G,C) = 0, so the theorem is proved to be true under the subsidiary conditions /,• 7^ 0 for 1 < i < 5. The geometric meanings of the five conditions, interpreted automatically by GEOTHER, are as follows: • 1\ ^ 0 <^=> A,B,C are not collinear; • h 7^ 0 <S=> AC is nonisotropic; AC is not perpendicular to AS; • h ^ 0•
BC is nonisotropic; AB is not perpendicular to BC.
One can examine whether the theorem is true in each of the degenerate cases by taking /,• = 0 as a new hypothesis polynomial. Consider, e.g., the case h = 0 and let P ' ^ P U f f t } . A characteristic set of P* under the same
120
Chapter 8. Automated Theorem Proving and Discovering
variable ordering is
X7-X5,
(JC3 + 4X[) X8 + 2^1X3X5 — 4XjX4 — X1X3, _ {x\ + 4xj) X9 — X3X5 + 2x1X3X4 — 2xjx3 with some factors xi and X3 removed. Since prem(G,C*) = 0, the theorem is true as well in this case under the nondegeneracy condition x^ + Ax\ 7^ 0 (i.e., the line BC is nonisotropic). The other degenerate cases may be verified one by one in the same way. We shall see how this can be done systematically by computing a zero decomposition for P and then determining whether the conclusion holds true for each component. Such a systematic treatment will allow us to remove the two redundant nondegeneracy conditions h ^0 and I5 ^ 0. A plex Grobner basis G of P under the same variable ordering consists of 17 polynomials, and Groebner [normalf ] (G,G) = G ^ 0. Now G has a quasi-basic set identical to C (up to a sign for some polynomials). According to the above verifications, the theorem is true under the nondegeneracy conditions I\---I$ / 0. With respect to X5 -< • • • -< xg a Grobner basis of P is G* = [Ci/xi,C2,G3,C4,Gs], where G3 = hx-i - x\x5 - 73x3 (x4 + xi), G5 = h*9 ~ *\x5 - /5X3 (X4 - Xi), and C\,C2,C4,/i,...
,74 are as above. One can easily verify that Groebner [normalf] (G,
so the theorem is true under some nondegeneracy conditions. With different implementations [13,16,26, 30,70,87] of the above method and its variants, a large number of geometric theorems — including Steiner's theorem (generalized), Morley's trisector theorem and the recently confirmed conjecture of Thebault — have been proved; several interesting "new" theorems were also discovered (see, e.g., [87, 98, 13, 58]). Some of them will be presented in the following sections.
8.2. Proving Theorems via Zero Decomposition
8.2
121
Proving Theorems via Zero Decomposition
The method described in the preceding section is elementary and effective for confirming many geometric theorems, but in a coarse fashion. Several refinements may be made to render the method more complete: 1. It should be verified, before determining the validity of (8.1.3), whether the hypothesis HYP is consistent (or self-contradictory). If HYP is inconsistent, then (8.1.3) is always a true formula. There are different ways to examine the consistency of a system ^)3 of polynomial equations and inequations (and thus of HYP), for instance, by computing a regular, simple, or irreducible triangular series of ty, or by means of projection or Grobner bases computation. However, complete examination of the consistency of ^3 is computationally expensive. 2. Formal definitions have to be given for what we call "appropriate" and "subsidiary conditions." Apparently, adding D*j ^ 0 to HYP should not exclude interesting cases of the theorem. In particular, every D*- = 0 should not be a formal consequence of HYP, i.e., the addition of D*: ^ 0 to HYP does not destroy the consistency. The purpose of finding nondegeneracy conditions is to rule out some degenerate cases in which the theorem becomes false or meaningless, so that a geometric theorem may be proved even if its algebraic formulation is not logically complete due to the missing of such conditions. The reader may find it difficult to predetermine all the degenerate cases. In fact, handling degeneracy is one of the crucial issues in geometric computation. 3. When geometric relations like trisection of angles and contact of circles are expressed as polynomial equations and inequations (without strict inequalities), some ambiguities corresponding to the reducibility of geometric configurations may occur naturally. In this case, the theorem is true usually only for some of the irreducible components and the method presented in the previous section does not work. Let us illustrate this last point by the following example from [98, 78]. Example 8.2.1. The bisectors of the three angles of an arbitrary triangle, three-to-three, intersect at four points. Let A,B,C be the three vertices of the triangle, the two bisectors of ZA and Z.B intersect at point D, and the bisector of Z.C meet line AS at point E. We need to prove that D lies on CE.
122
Chapter 8. Automated Theorem Proving and Discovering
D'
Fig. 17. Incenter and escenters As in [78] and without loss of generality, let the coordinates of the points be chosen as follows: A(xu0),
B(X2,0), C(0,x 3 ), D(x4,x5),
£(x 6 ,0).
The hypothesis of the theorem consists of three relations ' Hi = X3 [x\ - (x4 - X\ ) 2 ] - 2*1*5 (*4 - X\) = 0,
<—< DA is a bisector of Z.CAB
HYP • i
Hl = X3
4
X2
^ ~~ ^ ~ ^ ~
2X2X5
^ 4 ~ X2^ = °'
<—< DB is a bisector of ZABC H3 = X3 [(Xl - X6) (x\ + X2X6) + (X2 - X6) [x\ + X\X6)] = 0.
<—< EC is a bisector of ZBCA In this algebraic formulation, the equality of tangent of angles is used to express the equality of angles. We add the condition D\ = xs ^ 0,
«—< C does not lie on AB
to exclude the trivial degenerate case. The conclusion to be proved is CON:
G = X3X4 + X5X6 — X3X6 = 0.
«—< D lies on CE
A careful reader might observe that bisectors may be internal or external; both of them are represented by the same polynomial equations. Without using strict inequalities, the two kinds of bisectors cannot be distinguished from each other. If the bisector of one angle of AABC is external and those of the two others are internal, then the three bisectors are clearly not concurrent. Therefore, with the above formulation, the theorem would not be proved to be generically true. To deal with this situation, let us slightly modify the formulation (see [98, pp. 197— 199]).
8.2. Proving Theorems via Zero Decomposition
123
Example 8.2.2. Instead of the collinearity of D,C,E, we may prove that G* =
CON*:
[X\ (x5 - X3) + X3X4] [X3 (x5 - X3 ) - X2X4]
+ [X2 {x5 - X3) + X3X4] [x3 {x5 - x 3 ) - x\x4] = 0. <—< DC is a bisector of Z.BCA
Now point E need not be introduced, and the third relation H3 = 0 in Example 8.2.1 becomes redundant. This modified formulation allows us to exclude the four possibilities for which the three bisectors are not concurrent. Ambiguities of this kind also occur inherently in other geometric relations. They lead to the reducibility of the quasi-algebraic variety V defined by the hypothesis of the geometric theorem expressed as polynomial equations and inequations (without using strict inequalities). In a natural formulation of the theorem that does not take nondegeneracy conditions and ambiguities into account, the conclusion equation holds true usually only for some components of V. Those components for which the theorem is false have to be excluded. To solve this problem completely and systematically, we need to decompose 'U into irreducible components. Now we return to the algebraic problem of theorem proving and let (3, 3C and D be as in the previous section. Suppose that the hypothesis of a theorem TT in <3 has been expressed under D as a finite system (8.1.1) of polynomial equations and inequations, and the conclusion has been expressed as a single polynomial equation (8.1.2). Set P = {Hi,...,Hs} and Q = {DU...,D,}. The problem is to decide (c) whether Zero(IP/Q) = 0; and if not, (d) on which components of Zero(P/Q) the polynomial G vanishes (and thus TT is true). More precisely, let *F be an irreducible triangular series of [P, Q]. If *P = 0, then Zero(P/Q) = 0 and thus the hypothesis of the theorem TT is self-contradictory. Otherwise, problem (d) amounts to dividing *F into »F+ = {[T,U] e Y I prem(G,T) = 0},
V " = {[T,U] G V \ prem(G,T) ^ 0}.
y
TT is universally true if and only if ¥~ = 0. If *P+ = 0, then TT is universally false. Otherwise, TT is conditionally true. The subsidiary conditions are provided by excluding the triangular systems in *¥~ for which TT is false. Therefore, the refined method for proving TT works roughly as follows. Tl. Compute a characteristic series or triangular series *P of [P,Q] over %_ by algorithm CharSer, TriSer, or TriSerS described in [78]. If *P = 0 then the hypothesis of TT is self-contradictory and the procedure terminates.
124
Chapter 8. Automated Theorem Proving and Discovering
T2. Let *F = {[Ti,Ui],..., [Te,Ve]} and compute /?,-:=prem(G,T,-),
1 < / < e.
If there exists an index i (I < i < e) such that Rt ^ 0, then go to step T3. Otherwise, determine whether there exists a k (1 < k < e) such that Zero(Ti/U^) ^ 0. If so, then TT is universally true; otherwise, the hypothesis of TT is self-contradictory. The procedure terminates in both cases. T3. For each i with Rt ^ 0, compute an irreducible triangular series *?,• of [T,-, U,-] over % by algorithm IrrTriSer, IrrCharSer, or IrrCharSerE presented in [78]. Set
If Y ^ 0, then go to step T4. If Rt ^ 0 for all 1 < i < e or (J
RiS0
Zero(T,/U ( ) = 0,
(8.2.1)
1<;<e
then the hypothesis of TT is self-contradictory. Otherwise, TT is universally true. In any case, the procedure terminates. T4. Let 4* = {[f i, tfi],..., [f/,,©/,]} and compute /? 7 -:=prem(G,f / ),
\<j
If Rj = 0 for all 1 < j < h, then TT is universally true and the procedure terminates. If /?, ^ 0 for all 1 < / < e or (8.2.1) holds, and Rj ^ 0 for all 1 < j
Vi-Vm): (i) prem(V, fj) = 0 for all; with Rj ^ 0; (ii) if R{ = 0 and prem(V,T,) = 0, then Zero(Ti/Ui) does not represent an interesting case of TT; (iii) if Rj = 0 and prem(V, fy) = 0, then Zero(f j/tlj•) does not represent an interesting case of TT; (iv) the number of cases for i in (ii) and for j in (iii) is kept as small as possible; (v) the index m is as small as possible.
8.3. Illustration with Examples
125
Then the theorem IT is true under the subsidiary conditions V\ ^ 0 A • • • A Vm ^ 0 (or V ^ 0). Some of the conditions may be interpreted as nondegeneracy conditions according to W°°. The GEOTHER function Tprover is an implementation of this method, where triangular series and irreducible triangular series are computed by t r i s y s [ t r i ser] and t r i s y s [its] respectively. A slightly different presentation of the method is given as algorithm ProverB in [78] with formal proof. For practical efficiency some redundant triangular systems, for example those [T,U] for which |T| > |F|, should be removed from *P and *F,-. The proposed method starts by computing a triangular series, not an irreducible one, mainly for avoiding unnecessary (algebraic) polynomial factorization. The triangular series may also be computed over ^C(M) when the parameters u are correctly identified from the variables x and the theorem is considered only for the nondegenerate cases. Correct identification of parameters and geometric dependents allows one to speak about generic truth more precisely and to determine simpler nondegeneracy conditions in the algebraic sense using projection and other devices. To confirm theorems, one may also follow a refutation approach by verifying the inconsistency of the hypothesis relations with the negation of the conclusion equation. The verification can be carried out by computing (regular, simple, or irreducible) triangular series or Grobner bases. We refer to the algorithms ProverC and ProverD and relevant discussions in [78] for other details.
8.3
Illustration with Examples
A straightforward formulation of Steiner's theorem (see [78, pp. 200-203]) may have to use square of distance, where the reducibility problem may occur because the side of a line on which an equilateral triangle is drawn cannot be easily distinguished. Using vector rotation in which orientation is taken into account, we can give a simple formulation of Steiner's theorem in a generalized form as shown in the following example. With this formulation, the machine proof becomes quite trivial. Example 8.3.1 (Steiner theorem generalized [74]). Let ABC', BCA' and CAB1 be three similar isosceles triangles drawn all inward or all outward on the three sides of an arbitrary triangle ABC. Then the three lines AA', BB' and CC' are concurrent. As AABC", ABC A' and ACAB' are similar, their altitudes are proportional to the lengths of the corresponding bases \AB\, \BC\ and \CA\. Let the ratio
126
Chapter 8. Automated Theorem Proving and Discovering
\
\
\
/ / / A
Fig. 18. Steiner theorem generalized be a a n d t h e six points be located as A(0,0), B(JCI,0), C(x 2 ,x 3 ), A'(x4,xs),
B'(x6,Xl),
C'(x 8 ,x 9 ).
To avoid the problem of reducibility, we consider t h e point A' as the end of t h e vector starting from t h e midpoint of B and C with length equal t o a|BC| a n d t h e same direction as the vector obtained by rotating BC 90° anticlockwise. Similarly, t h e points B' and C" are so constructed. Then the hypothesis of t h e theorem m a y be expressed as ' Hi = 2x4 - (xi +x2) +200:3 = 0, H2 = 2 x 5 - X 3 + 2 a ( x i -X2) = 0 , # 3 = 2x6 —X2 — 20CX3 = 0, < H4 = 2X7 — X3 + 20UC2 = 0, H5 = 2 x 8 - x i = 0 , ^Hb=x<) — ooci = 0. T h e polynomial set T = [Hi,...,H^\ is already a triangular set and a plex Grobner basis with respect t o xi -< X2 -< X3 -< a -< X4 -< • • • -< xg. T h e conclusion of the theorem is G = [(X2 - Xi) X4XJ - X2X5 (x6-Xi)]xg
+ [{xiX5 -X3X4)x7
+X3X5
(x6-Xi)]x&
— xi (X2X5 — X3X4) xj = 0.
It is easy t o verify t h a t prem(G,T) = Groebner [normalf ] (G,T) = 0, and 1 is contained in the reduced Grobner basis of T u { G z — 1}. So the theorem is proved t o be true universally.
127
8.3. Illustration with Examples
The above generalized version of Steiner's theorem was discovered by the author in January 1994. We do not know where the generalized theorem can be found. It is the centroid theorem when a = 0, the orthocenter theorem when a = °°, and Steiner's theorem when a = A / 3 / 2 . To show the power of the methods explained in Sects. 8.1 and 8.2, we present a few more geometric theorems and their machine proofs. These theorems are well known and may be proved automatically in a matter of seconds. For some of them, polynomial factorization over algebraic extension fields is required. Let us first recall one of the most surprising and beautiful theorems in elementary geometry that was discovered by F. Morley around 1899. The first automated proof of Morley's theorem in the generalized form stated below is attributed to Wu [87], who worked out a tricky and elegant algebraic formulation. Since then, several simplified machine proofs of the theorem have been given by other researchers [13,67]. Example 8.3.2 (Morley theorem [13, 67, 87]). The neighboring trisectors of the three angles of an arbitrary triangle intersect to form 27 triangles in all, of which 18 are equilateral.
B
C Fig. 19. Morley theorem
Following Wu [87], the hypothesis of the theorem consists of ZABC = 3ZPBC, ZABR = ZPBC,
ZACB = 3ZPCB, ZACQ = ZPCB,
tan2 0 = 3,
ZBAR = ZQAC,
ZCBP + ZPCB + ZBAR = 6 mod 2n, and the conclusion to be proved is ZQPR =
ZRQP=*.
Let X(, = tan 9 and take the coordinates of the points as A(X4,X5),
B(xi,0),
C(x2,0),
P(0,X3),
Q{xW,Xg),
R(X8,X7).
128
Chapter 8. Automated Theorem Proving and Discovering
We use an index triple [t v d] to characterize an arbitrary polynomial P, where t is the number of terms in P, v the leading variable of P, and d the degree of P in v. Then, by taking tangent for the equalities of angles both the hypothesis and the conclusion of Morley's theorem can be expressed as polynomial equations with index triples [6 x5 1], [6 x5 1], [2 x6 2], [9 x8 1], [9,xw 1], [41 x10 I], [40 x8 1] and [9xio 1], [lOxio 1] with respect to the variable ordering *!-<•••-< xio- Some of the occurring polynomials are rather large, so we list their index triples instead of reproducing them here. The theorem can be easily proved by using the elementary method described in Sect. 8.1. For instance, a plex Grobner basis of the hypothesis-polynomial set under X4 -< • • • -< xio consists of 7 polynomials with index triples [7X4 1], [9X5 1], [2X6 2], [10X7 1], [13X8 1], [10X9 1], [13X10 1]. The normal forms of the conclusion polynomials modulo this Grobner basis are both 0. Therefore, the theorem is proved to be true under some possible nondegeneracy conditions which are not explicitly provided. Without using Wu's trick, let us consider a natural formulation of the theorem, where the hypothesis consists of ZABC = 3ZPBC,
ZACB = 3ZPCB, ZCAB = 3ZRAB,
ZABR = ZPBC,
ZACQ = ZPCB,
ZBAR = ZQAC,
and the conclusion to be proved is \PQ\ = \PR\,
\PQ\ = \QR\.
Let the coordinates of the points be chosen as A(yi,yi),
B(uu0),
C(ii2,0), P(0,1), Q(y6,y5),
Rfays).
The hypothesis and the conclusion can both be expressed as polynomial equations with index triples . HYP: [6y 2 1], [6 v2 1], [191 y4 3], [9 y4 1], [9,y6 1], [41 y6 1]; . CON: [6y 6 2], [6 y6 1]
129
8.3. Illustration with Examples
with respect to the variable ordering y\ -< • • • -< y§. The set H of hypothesis polynomials can be decomposed over Q{u\,u2) into two irreducible triangular sets T=
[ri,r2,r3,r4,r5,r6], T = [TUT2,T3* ,T4J5,T6],
where 7\=/yi-ap\ T2 = P (y2 - u2) + u2 (u\ - 3)yi, Ti = Iy23 - 4MI («ip + 4u2)y3 + 4 M 2 p, T4 = 2wiv4 + {u\ - l)y 3 - 2w2, r 5 = {[ai4 + «?P+(7niM2 + 3)(M2 + Ill)]y3-2Ml(M?+l)P}3'5 — 2au2{u\ + l)y3, T(, = (y2 + u2yi - u2) (y6 - u2) - {u2y2
-y\-u^)ys,
T
3 = ^ 3 + 2«1 («2 - «l) Pi I = au2 + Suiu2-ul
+ 3,
a = 3wi-l,
P = 3M2~1.
In computing the zero decomposition, no algebraic factorization is needed. The pseudo-remainders of the conclusion polynomials are both 0 with respect to T, but not 0 with respect to T*. Therefore, under some nondegeneracy conditions the algebraic form of the theorem is true for one component and false for the other. In the tricky formulation of Wu, the constraint Z.CBP + ZPCB + Z3AR = 9 mod 2% with tan2 0 = 3 is imposed. After the addition of this constraint to HI the component T* is then excluded, so that only T remains. Therefore, we may arrive at the same conclusion as Wu without using his trick in the formulation. Note that T-?, is of degree 2 and T3* of degree 1 in V3. This can be explained roughly as follows. After the trisectors are fixed for two angles of the triangle, the trisectors for the third angle would have three possibilities in forming the triangle PQR. T3 corresponds to two of these possibilities for which APQR is equilateral, and r3* corresponds to the third possibility for which APQR is not equilateral in general. To see the former more clearly, let us introduce a new variable yo and add TQ — y2, — 3 to EL Then 73 can be factorized, over Q_(uiiu2,yo) with yo having adjoining polynomial 7b, as (B.4) so {TQ} UT can be further decomposed into two irreducible triangular sets T' = [7b,r1,72,r3',r4,r5,76], T" = [T0JX,T2JI:,TA,T5,T6].
130
Chapter 8. Automated Theorem Proving and Discovering
Instead of \PQ\ = \PR\ and \PQ\ = \QR\, we may prove the conclusion relations tan2 ZQPR = 3 and tan2 ZPQR = 3, which can be written as (tan ZQPR +y0) (tanZQPR-y0)
= 0,
(tan ZPQR+y0) (tan ZPQR - y0) = 0. It is easy to verify that tan ZQPR + yo = 0 and tan ZPQR — yo = 0 are true for T', and so are tanZQPR-y0 = 0 and tanZPQR + y0 = 0 for T". That is, for both of the components that correspond to the two possibilities of 73 in forming APQR the theorem is true. By means of polynomial factorization, H can also be decomposed over Qiu\,U2) into two plex Grobner bases Gi and G2 such that Zero(H) = Zero(
^1,^2,73,74, uicys + au2y3-2u\U2{u2 + ui), 2u\cy(, — adys + 2ui (u\d — 2u2) 'Ti,G2,Tf,
G2 =
Iy4 — 3&«2 — 2u\c — lu\u2 — U2, Iy5-2au2(u2-ui), Iyd — ?>u\d — l U1U2 — 2au2 — u\ G2 =Iy2~8uiU2(u2
+ ui);
• - „ ?
«2 + 1, d = U2 — 1. 1, b = u\—\, It may be easily verified that the normal forms of the two conclusion polynomials are both 0 modulo GQ , but not 0 modulo
[ 1 2 x 6 l ] , [13 x7 1]
131
8.3. Illustration with Examples
and there is only one conclusion polynomial G with index triple [11 xj 1] in the variables u\ -< ui -< uj, -< x\ -< ••• -< xq. H can be decomposed over Q(u\,U2,U3,) into four irreducible triangular sets Ti,...,T4 with T 1 = [r 1 ,72,r 3 ,r 4 ,...,77],
T2 =
[T1,T2,TI,T4,---,T7},
T3 = [TUT2\T3,T4,...,T7],
T4 =
[TI,T2^T^---M
T\ =4u\u\x\ — {2u\ui-yJr2u\u^){2u\u\u2,
+ u\j-2u^),
72 = 2u2CldX2 + 2lliU2(x(xi + M3) + 8, 73 = 2u2adx3 — 2u\U2<x(xi — K3) + 8, 74 = U1112X4 — ab, T5 = M1M2X5 — cd,
T6 = 2u\[u\ (x5 +xA) - p]x6 - u\y(x5+X4) + (3{u\ + 1), Tj — abcdxj + \2u\u\x*, + cd {u\u\ +\)]xf, — u\yxs + u\ — u\, T[ = 2 U2bcx2 — 2 W1M2OC (xi +113) + 8, T3' = 2 ii2bcxT, + 2 W1M2OC (x\ — M3) + 8; a = «lM2+l,
b = U\U2— 1, -
C = Ul+U2,
d = Ui — U2,
=M
a = ul-l, P = M2 1, Y 2 + 1> 8 = («i + l ) a 2 . The pseudo-remainder of G is 0 with respect to Ti, but not 0 with respect to T2,T3 and T4. Hence, the algebraic form of the theorem is true for one component and false for all the others. The largest polynomial occurring in the proof reductions has 168 terms. More than half of the computing time was spent for the two algebraic factorizations (B.7) and (B.8) given in Appendix B. The above examples demonstrate the significance of algebraic factorization in geometric theorem proving. See (B.9) and (B.10) in Appendix B for algebraic factorizations that may have to be computed for the proof of Steiner-Lehmus' theorem [100] (see Example 9.1.1). For these examples, the polynomials to be factorized as well as the adjoining polynomials are all quadratic. The degree is low mainly because the geometric theorems considered so far only involve figures like triangles and circles whose algebraic character is no more than quadratic and the algebraic formulations are often made carefully to simplify the computation and to avoid polynomials of high degree. If one does not take good care of algebraic formulation or geometric figures with algebraic character of high order are considered, polynomials may have to be factorized over algebraic extension fields with adjoining polynomials of degree greater than 2. This can be seen from the factorizations (B.ll), (B.5), (B.6), and Example 8.4.4 (Feuerbach's theorem) in [75].
132
Chapter 8. Automated Theorem Proving and Discovering
The theorem in the following example is the second of the five killers that were sent to the author by S. Markelov [43] from the Moscow Center for Continuous Mathematics Education in February 1998 to challenge the power of our theorem provers in the GEOTHER package. These theorems can be proved in geometric ways, and are called killers to analytic ways of geometric problem-solving because no analytic proof could be found even by expert geometers. In [32] we have presented several machine proofs together with a human proof of the second killer, which provides a beautiful representation of the area of an arbitrary quadrilateral in terms of its four sides and four internal angles. Later Markelov informed the author that this theorem was first discovered by the German mathematician W. Blaschke, first appeared in [5]. Here let us recall the theorem with one machine proof. Example 8.3.4 (Russian killer no. 2 [32]). Let ABCD be an arbitrary quadrilateral with sides \AB\ = k, \BC\ = I, \CD\ = m, \DA\ = n and internal angles 2a,2b,2c,2d at vertices A,B,C,D respectively, and let 5 be the area of the quadrilateral. Then (k + l + m + n)2 cota + cotb + cote + cotd
(l + n-k-m)2 tana + tanfe + tanc + tan J
Fig. 20. Russian killer According to the geometric hypotheses, we have the following relations: H\ = 2S — knsm2a — lmsm2c = 0, H2 = 2S-klsm2b-mnsm2d = 0, 2 2 2 2 H3=k + n - 2kncos2a -l -m + 2lmcos2c = 0, 2 2 2 2 H4 = k + l - 2klcos2b -m -n + 2mncos2d = 0, H5 = sin(a + b + c + d) = 0.
133
8.3. Illustration with Examples
Let 0 — {a,b,c,d}
and xe = sin8, ye = cos0,
0 € 0.
By expanding the sine and cosine of double angles and sum of angles with simple substitution, the above H\,...,Hs will become expressions in xa,.-.,Xd,ya,---,yd &nd S,k,l,m,n. Clearly,
HQ=4+yl-1 = 0, e e e . Thus Hi=0,...,H5=0,Ha=0,...,Hd
=0
(8.3.1)
constitute the hypothesis of the theorem, and the conclusion to be proved is
G = s - r 2 / y ^ + / 2 / X - = o8G0
*e
eeeye
The statement of the theorem implies that the denominators do not vanish (e.g., XQ ^ 0 and ye ^ 0 for every 0 € 0 ) . So we only need to prove that (8.3.1) implies that the numerator G* of G is 0. Without loss of generality, take n = 1. Then G* becomes a polynomial consisting of 91 terms. With respect to the variable ordering yb < *b < yc •< xc -< yd •< Xd -< ya -< xa < I -< k -< m •< S,
a quasi-characteristic set C of {Hi,... ,H^,Ha,... ,Hd} consists of the following 8 polynomials: C\ = Hb, Cz = Hc, C3 = Hd, C4=yl + 2 {2ylycydXc + 2yby2cydxb - ybydxb - ycydXc)xd - 4y2by2y2d - 2ybycxbxc + 4ybycy2dxbxc + 2y2y2 + 2y2by2d + 2y2y2d
-yb-yl-yd,
C5 = hxa+yaycydXb yaxbxcxd+yaybydxc+yaybycxd, 2 C 6 = hk + (yby ydxc - ybycxd+ybfed - y2cxbxcxd+y\ydXb - ycydxb) I - ybydxc+ycydxb+ybydxc - ycydxb - ybycydxd + ydxbxcxd, C1=I1m+{2y2l-l-2y2a + C% = S - ybxblk - ydxdm,
\)k-l2+\,
where
^5 = ybycyd - ybxcxd - ydXbxc - ycXbxd, h = ^y\y cydXb - ylxbxcxd+ylydXc + 3y^ycxd - ybydxc - 2ybycxd - ycydXb + 4y2by2xbxcxd - 4y2ylydxb - 4y3by2cydxc - 4y3by3cxd + ybdXb - y2cxbxcxd + 3yby2ydxc + 3yby3cxd, I1 = 2ly2-l-2y2 + l.
134
Chapter 8. Automated Theorem Proving and Discovering
During the computation, a polynomial factor I2 — 1 is removed. Thus, under the subsidiary condition (/2-l)/5/6/7^0, (8.3.2) proving the theorem is reduced to verifying whether prem(G*, C) = 0 . It is indeed so: the verification is not easy and takes about 320 seconds of CPU time in Maple V.3 on an Alpha station. Some of the occurring polynomials are very large. With Wprover the pseudo-reductions of G* to 0 using C&,... ,Ci may be sketched as follows: G* = [93 5 1]-)- [106 m 2 ] 4 [680 k 2} [4529 xa 2] - • [6541 ya 6] -> [19013 xd 9], [3432 xa ->• < [2450 xa [1015 xa [4034 xa [687 xc [327 xc [524 xc [697 xc [432 xc
2} -> [5221 ya 6] -> [17586 xd 9] 2} ->• [3690 ya 4] -+ [11066 xd 8] 2] -> [1543 ya 2} -> [6276 xd 7], 2] -> [6067 ya 6] -» [17813 xd 9]
9], 9], 9], 9], 9],
[210 xc [803 xc [688 xc [688 xc [622 xc
8], 9] 9] 9] 8]
[549 * c 9], [549 xc 9] - • < [732 xc 8], [699 xc 8] [602 xc 7], [799 xc 9] [558 xc 9], [556 xc 9] [437 xc 8], [313 xc 8] [641 xc 7], [649 xc 7] [308 xe 9], [665 xc 9] [780 xc 9], [804 xc 9]
[549 xc 9], [697 xc 9] [667 xc 9] [283 xc 9] [523 xc 8] [544 xc 9] [696 xc 8] [810x c 9] [549 xc 9} [165 x c 6] [621 xc 7] [545 xc 9]
[656 xc 8] [647 xc 9] [420 xc 9] [684 xe 9] [554 xc 9} [376 xc 9] [711 xc 8] [790 xe 9] [494 xc 9] [425 xc 7] [157 xc 8] [796 xc 9}
[505 xc 7] [582 xc 9] [693 xc 9} [519 xc 9] [538 xc 9} [730 xc 8] [646 xc 7] [218 xc 8] [170 xe 7] [293 xc 7] [444 xc 9} [718 xc 9]
{0}. Therefore, the theorem is proved to be true under the subsidiary condition (8.3.2). We do not enter into the details of interpreting the geometric meaning of the algebraic condition. For the above proof of the theorem, we had no attempt to use special techniques to simplify the algebraic formulation, so the involved polynomial computation is very heavy. However, this is already within the reach of today's laptop. The technique of splitting in the pseudo-reduction process
135
8.4. Discovering Geometric Theorems
is due to Wu [88]. Direct computation of the pseudo-remainder without this technique was not possible on our machines.
8.4
Discovering Geometric Theorems
In the case of theorem proving, there is a known conclusion whose truth is to be confirmed. Now we explain how to discover new geometric theorems in an automated manner. More precisely, we want to derive automatically one or several unknown algebraic relations among some geometric entities, where an adequate description of the geometric hypotheses among the geometric entities is given. This may be done by expressing the geometric hypotheses as a system of polynomial equations and equations, then computing a triangular set, triangular series or Grobner basis of the corresponding polynomial set or system using an appropriate variable ordering, and finally reading out the desired relations from the triangularized sets. A typical example is the automated derivation of the Qin-Heron formula (representing the area of a triangle in terms of its three sides [90, 78]). The problem of deriving unknown algebraic relations and its solution may be formulated as follows. Given a set HYP of geometric hypotheses expressed as a system of polynomial equations and inequations r = {Pl(u,x),...,Ps(u,x)}=0,
Q={Q1(u,x),...,Q,{u,x)}^0
in two sets of geometric entities u = (ui,...,Ud) and x = (xi,...,x„) with coefficients in %_ and given a fixed integer k, without loss of generality, say k = 1, we want to determine whether HYP is self-contradictory, and if not, whether there exists a polynomial relation R(u,x\) = 0 between u and x\ such that Zero(P/Q) C Zero(/?), and if so, finds such a R(u,x\). The method works according to the following steps. Dl. Compute over %_ a (quasi-, weak-) medial set T of P by algorithm CharSetN or PriTriSys given in [78], or a Grobner basis T of P U \Q\z\ — 1,..., Qtzt 1} with respect to the purely lexicographical ordering under u\ -<
136
Chapter 8. Automated Theorem Proving and Discovering
D3. Compute an irreducible triangular series ¥ = {Ti,... ,T e } of [P,Q] over %,. If *F = 0, then HYP is self-contradictory and the procedure terminates. Set TP:=T,-n(3:[B,Jci]\aC[H]), 1 If for every \
there exists a polynomial Rj(u,x\) 6 T ) , then return e
R(u,xi):=J\Ri(u,xi). Otherwise, there is no polynomial relation/J(M,xi) = 0 between u andx\ in general such that Zero(P/Q) C Zero(/?). In any case, the procedure terminates. D4. If T ^ •£ 0, then return the polynomial R(u,xi) G T ^ that has minimal degree in x\. Otherwise, there is no polynomial relation between u and x\ in general. A slightly different presentation of this method is given as algorithm Discover in [78]. The following postprocess may be incorporated into the algorithm. D°°. In case no algebraic relation between u and x\ is found, analyze the computed irreducible triangular series or Grobner basis, and try to get possible relations by providing appropriate subsidiary conditions of the form Z), ^ 0 and adding the Z); to Q to exclude some components. The triangular sets or series and the Grobner bases may also be computed over Q,(H) when the variables u are correctly specified as independent parameters. In this case, any polynomial inequation in u may be considered as a nondegeneracy condition. The corresponding method either detects the algebraic dependency of u or derives a relation that is satisfied generically; it does not necessarily hold true in the degenerate cases. Example 8.4.1 (Poncelet theorem [74, 84]). Let R be the radius of the circumscribed circle and r the radius of the inscribed circle of an arbitrary triangle, and let d be the distance between the centers of the two circles. Determine the relation among R,r and d. Let ABC be an arbitrary triangle, D and H be the incenter and circumcenter of AABC, and the coordinates be assigned as A(xi,0),
B(x2,0), D(0,x3),
C(x4,x5),
H(x6,x7).
8.4. Discovering Geometric Theorems
137
Fig. 21. Poncelet theorem Now the geometric hypotheses are: • C lies on the reflection line of AB with respect to AD <£=» Hi = (*2 -*i) [(x\ -x\)x5-
2xi*3 (*4 - * i ) ] = 0;
• C lies on the reflection line of BA with respect to BD « = ^ H2 = (*2 ~ * l ) [(jcf - * 2 ) * 5 - 2*2*3 (*4 ~*2)] = 0;
• H is the circumcenter of AABC j HA = (*2 - Xi) (2x6 -X2-
*l) = 0,
1 H3 — 2*5*7 + 2*4*6 — 2*2*6 —x\ — x\+x\
• r is the radius of the inscribed circle of AABC => H$ • R is the radius of the circumcircle of AABC = • H6 = R2 - x2 - (*6 - *i) 2 = 0; • d = |D//| =^H7
= d2- (*7 - * 3 ) 2 - * 2 = 0.
Assume that AABC does not degenerate into a line, so that ( * 2 - * l ) * 5 7^0.
Computing a plex Grobner basis
= §\
-xi = 0:
138
Chapter 8. Automated Theorem Proving and Discovering
with respect to d -< X2 -< • • • -< xj -< z\ -< Z2, one finds that there is one polynomial G in G which involves d,R,r only: G = d4 - 2d2R2 + R4- 4R2r2 = {d2 -R2 + 2Rr) (d2 -R2- 2Rr). Hence, the geometric hypotheses imply that G = 0. In the above derivation, we have not used the implicit assumption that R > 0 and r > 0. Moreover, it is obvious that R > d because the inscribed circle is contained in the circumcircle of AABC. Therefore, we have R2-2Rr
= d2.
This is the great Poncelet theorem; it has been rediscovered automatically by using the above method. Example 8.4.2 (Brahmagupta formula [14, 66, 78]). Let a,b,c,d be the four sides and © the signed area of an oriented cyclic quadrilateral ABCD. Then @2 = (p-a)(p-b)(p-c)(p-d),
(8.4.1)
where p = (a + b + c + d)/2.
Fig. 22. Brahmagupta formula This well-known formula due to Brahmagupta has been derived in an automated manner in [14, 66, 78]. Now we explain how to "discover" the theorem in a different way. Let the coordinates of the points be chosen as A(0,0),
B(a,0),
C(xux2),
£>(x3,x4),
and b = \BC\,
c = \CD\,
d=\DA\.
139
8.4. Discovering Geometric Theorems
Denote the sum of the signed areas of AABC and AACD by 0 . The following algebraic relations corresponding to the geometric conditions may be easily established: 'Hi=xj+x21-2axi-b2
+ a2 = 0,
1
2
^
2
H2=xl-2x2x4+x -2xiX3+xl+x -c 2
H3=xl+xl~d
= 0,
= 0,
b=\BC\
<-< c=\CD\ <-< d=\DA\
, #4 = X\XA -x2X3 + ax2-2Q
= 0.
<-c 0 = area(AABC) + area(AACD)
Motivated by the Qin-Heron formula [90, 78], we may conjecture that (8.4.1) holds for an arbitrary oriented quadrilateral ABCD. In other words, we wish to show that {Va,b,c,d,xu...,X4,Q)[Hi
=0AH2 = OA//3 = 0AH4 = 0 => G = 0],
where G =
16[e2-(p-a)(p-b)(p-c)(p-d)}
= 160 2 + d4 -2(c2 + b2 + a2)d2 + c4 -2{b2 + a2)c2 + {b2 - a2)2 + Sabcd. The conjecture is clearly true when two of the points A,B,C,D coincide. If it is true not for arbitrary A,B,C,D, there should exist some relation which keeps the four points constrained. So we order one of the variables a,xi,... ,X4 at the beginning of the increasing queue, e.g., x4^.& -
140
Chapter 8. Automated Theorem Proving and Discovering
Example 8.4.3. Determine the volume T of an arbitrary tetrahedron ABCD in terms of its six edges U,V,W,u,v,w.
Fig. 23. Volume of tetrahedron Let the vertices of the tetrahedron be located as A(0,0,0),
B(JCI,0,0),
C(jt2,x3,0),
D(xA,x5,x6).
Then the geometric hypotheses may be expressed as the following polynomial equations: 'Hi=U2-x2l=0, 2
U = \AB\ 2
H2=V -{x2-xi) -x
2
2
H3=W -xj-xl HYP:
2
= 0,
3
V = \BC\
= 0, 2
W = \CA\ 2
Hi, = U — (X4 — X2) - (Xs — X3) - x\ = 0,
u = \DC\
H5=v2-x2-x2-4
v = \DA\
= 0,
Hs = W2 — (X4 — X\)2 — X5 — x\ = 0, 2
,//7 = 3 6 r - x ? ^ = 0.
v = \DB\ r=i|AB|-|x3|-|x6|
Computing a principal triangular system of {Hi,...,Hj} with respect to the variable ordering T -< x\ -< • • • -< x^, one can easily obtain a polynomial R in U,V,W,u,v,w and T as follows: R = 144 T2 + U V + V V + W V + U2 (u4 + v2w2 - w2^ - u2v2) + V2 (v4 + w2u2 - u2v2 - v2w2) + W2 (w4 + «2v2 - v V - w2u2) - U2V2 (u2 + v2) - V2W2 (v2 + w2) - W2U2 (w2 + u2) + U2V2W2. Therefore, R = 0 gives an algebraic representation of the volume r in terms of the six edges U,V,W,u, v,w of the tetrahedron ABCD.
141
8.4. Discovering Geometric Theorems
The relation R = 0 may be turned into a more symmetric form: let X = (w-U + v){U + v + w), x={U-v + w)(v-w + U), Y = (u-V + w)(V+w + u), y = (V - w + u) (w - u + V), Z = (v - W + u) (W + u + v), z = (W - u + v) (u - v + W); a = y/xYZ,
b = y/yZX,
c = y/zXY,
d = ^/xyz.
It is easy to verify that R = 0 is equivalent to 2
_(~a ~
+ b + c + d)(a-b
+ c + d)(a + b-c + d)(a + b + c-d) (I92uvw)2 '
This symmetric formula was communicated to the author by S. Markelov (from the Moscow Center for Continuous Mathematics Education, Russia) in December 2001. Markelov discovered the above formula, but he could not find a traditional geometric proof. The author was able to derive the algebraic relation R = 0 (and verified the equivalence) without making any special effort. Example 8.4.4. Represent the centroid of an arbitrary quadrilateral in terms of its four sides and four angles. After having seen the formula in our article [32] (see Example 8.3.4), in March 2001 J. Brillhart from the University of Arizona, USA asked the author whether there is a nice formula for the centroid of a general quadrilateral in terms of its four sides and four angles, and whether such a formula can be proved by computer programs. The author did not know such a formula, nor any geometric result about the problem. Our first reaction was trying to discover a possible formula using GEOTHER. With a few trials, the author was able to find the following formula in an almost automatic manner. Let ABCD be a quadrilateral with \AB\ = k, \BC\ = I, \CD\ = m, and \DC\ = n, let d{P) denote the square of distance from point P to the centroid of ABCD, and w(A) = nkcosA, 2
h=l(k
w(B) = klcosB, 2
2
2
+ l + m + n )-± 2
w(C) = ImcosC, w(D) = mncosD; [w(A) + w(B) + w(C) + w{D)}
2
= \{\AC\ + \BD\ ) - ^ [w(A) + w(B) + w(C) + w(D)] = ±{\AC\2+\BD\2)-\{k2
+ l2 + m2 + n2).
Then d{P) =h + w(P)/2, for every P 6 {A,B,C,D}.
142
Chapter 8. Automated Theorem Proving and Discovering
Example 8.4.5 (Pappus and Leisenring lines [49, 58]). Let two distinct lines / and /' meet at point O; A, B, C be three distinct points on /; and A', B', C be three distinct points on /', all distinct from 0. We may construct three intersection points Pi = AB' ABA',
P2 = AC'ACA',
P 3 = BC' ACB'.
Then, using our provers it is easy to confirm the following theorems (see [49, 58]): 1 (Pappus theorem). If the three points Pi, P2, and P3 exist (i.e., the corresponding lines are not parallel), then they are collinear. 2 (Leisenring theorem). Let Lx = OP\ ACC',
L2 = OP2ABB',
L3 = OP3 AAA'.
Then the Leisenring points L\, L2, and L3 are collinear. Now denote the line determined by Pi, P2, P3, called a Pappus line, and the line determined by L\, L2, L3, called the Leisenring line corresponding to the Pappus line, by ABC A' B' C
and p
ABC A' B' C
respectively. Making different permutations of A, B, C and A', B', C, we can get the other five Pappus lines and their corresponding Leisenring lines. 3 (Steiner theorem). The three Pappus lines ABC A' B' C
ABC B' C A'
p
ABC C A' B' .
p
are concurrent, and so are the other three Pappus lines, say at S and 5", respectively. 4. The three Leisenring lines ABC A' B' C
A B C B' C A'
ABC' C A' B'
are concurrent, and so are the other three Leisenring lines, say at T and T', respectively. 5. The three points 0, S, and T are collinear, and so are 0, S', and T'.
143
8.4. Discovering Geometric Theorems
6. The lines / and /' harmonically separate S and S', and thus also separate T and T'. The third theorem above was discovered independently by W.-t. Wu in 1980, and the sixth by the author in the middle 1980s. In what follows, we recall a theorem concerning Pappus and Leisenring lines, which is one of the most difficult discovered by our provers. The hypothesis (of the algebraic form) of this theorem contains 30 polynomials in 36 variables. The maximum number of terms of polynomials appearing in the successive pseudo-reductions is 10980, and the CPU time spent for the proof is about 803 minutes on an IBM4341 in late 1985. Let pi,...,/?6 be the six Pappus lines and l\,...,l(, the six Leisenring lines. Consider the point sets *F,- = {piAlj | j = 1,... ,6}, / = 1,... ,6. A natural question is: are there any geometric relations among the points of ^ i , . . . , ^ ? In particular, using the previous notations and writing A B C~ A'B'C A B C C'A'B' 'A B C A'C B'
A p
A p
A B C A'B'C A B C C A' B'
'A B C' A A'C B' P
5
w2 =
L
A B C B'C'A'
w4 =
L
A B C B' A'C
w6 =
L
A B C' C B' A' J
5
5
A
A B C B'C'A'
A
A B C B' A' C
A
A B C C B'A'
p
P
P
we may ask whether or not the six points Wi,...,W$ lie on the same conic. The author had conjectured and then proved by our early prover the following theorem [58]. Theorem. The six points Wi,W2,...,W(, are collinear. By symmetry, we only need prove that the three points W\, W2, and W3 are collinear, and so are the three points W\, W2, and W4. The following is a rough scheme about the successive pseudo-reductions, where the occurring polynomials are indicated by their index triples: [6 x36 1] -»• [48 x34 1] -> [60 X32 1] -»• [64 x30 2} -> [188 x28 2] - • [264 x26 1] -^ [2112 x24 1] ->• [2442 JC22 1] -»• [2152 x20 2] ->• [2420 x i 8 2] -)• [1272 x16 1] -> [10176 xu 1] -> [10980 xu 1] -> [8292 xw 2] ->• [4954 xs 2] -» 0; [6 A:36 1] -)• [48 x34 1] -> [60 x32 1] -> [64 x30 2] -> [188 x28 2] -^ [264 x26 1] ->• [2112 JC24 1] ->• [2442 x22 1] -)• [2158 x20 2] ->• [2116 *ig 2] -> [960 x16 1]
->• [7680 X14 1] ->• [8364 x i 2 1] ->• [6708 x10 2] -> [4597 x8 2] -> 0. The proof now takes only 228 CPU seconds on a laptop Pentium III 700 Mhz.
Chapter 9 Symbolic Geometric Computation
Elimination techniques and tools have remarkable applications in modern engineering geometry. This chapter presents some of the applications to selected problems, such as automatic derivation of locus equations and inequations, implicitization of parametric objects, computation of quasi-offsets to curves and surfaces, blending of algebraic surfaces, and decomposition of algebraic varieties, from automated geometric reasoning and computer-aided geometric design and modeling.
9.1
Deriving Locus Equations
The method of formula derivation explained in Sect. 8.4 may be generalized to determine the locus equations of a motion whose geometric description is given. To express the locus, we need one or several sets of algebraic relations among n variables x = (xi,...,x„) and parameters u, and to derive them projection is required. Both regular and simple systems enjoy the strong projection property. Since regular systems are relatively easy to compute, we shall use regular systems when the projection property is required. By locus equations we mean a system or the disjunction of several systems of polynomial equations and inequations in x with u as parameters such that not only the system is a formal consequence of the geometric hypotheses, but also for any point on the locus there exists at least one configuration which satisfies the geometric hypotheses. The problem of locus derivation may be formulated and solved as follows. Given a set of geometric constraints expressed as a system of polynomial equa-
145
9.1. Deriving Locus Equations
tions and inequations r= {Pi(u,x,y),...,Ps(u,x,y)} Q=
=0,
{Qi{u,x,y),...,Q,(u,x,y)}?0
in u, x and y for a point x = (x\,...,x„) to move in an n-dimensional affine space A^-, where u = (u\,..., Ud) are parameters and y = (yi,... ,ym) are other geometric entities, the following procedure determines a finite set *P of polynomial systems [Pi,Qi],...,pP e ,Q e ] in ^C(H) [X] such that (a) for any (x,y) G Zero(P/Q) there exists an i (1 < / < e) such that x G Zero(P,-/QI-); (b) for any 1 < / < e and x e Zero(P,/Q,) there exists a j e %!" such that (*j)eZero(P/Q). The finite disjunction e
V(P.- = 0AQ,-#0) is called the /ocMi1 equations (and inequations) of point AC (in terms of «). Dl. Compute a regular or simple series *P of [P, Q], or compute a characteristic, triangular, or Grobner series *P of [P, Q] with projection for y with respect to the variable ordering xi<
ixn-
If *F = 0 (in this case Zero(P/Q) = 0), then either the geometric conditions are self-contradictory, or the motion is free (in this case, for any x there is a y such that (x,y) G Zero(P/Q), so the locus fills up the whole space); thus the procedure terminates. D2. Remove redundant sets from (J
Zero(Tn^(H)[x]/Un3C(H)[jt]),
[T,U]6"i'
simplify it, and let the obtained zero set be (JLi Zero(P,/Q,). Return V:={\¥uQi],...,\Pe,Qe]}.
146
Chapter 9. Symbolic Geometric Computation
In Dl the series *P may also be computed in %[u,x,y\. Actually, one needs to perform the elimination only for y because it is sufficient when one has already obtained the equations and inequations in u and x — they do not have to be in triangular form. Example 9.1.1 [100, 64]. Let ABC be a triangle in the plane, the bisector of Z.CAB intersect the side BC at point A\, and the bisector of Z.CBA intersect the side AC at point B\ such that |AAi| = \BB\\. Determine the locus equations of point C. C
A
B
Fig. 24. Triangle with equal bisectors Without loss of generality, let the coordinates of the two vertices A and B be fixed as A(-1,0) and 5(1,0), and the other points be located as C(X,Y),
Ai(xi,x2),
BI(X3,M).
Then we have the following relations: Hi = 1x2 (xi + 1) (X + 1) + [x 2 - (xi + 1)2]F = 0, H2 tf3 H4 #5
2
= 2x4 (*3 - 1) (X - 1) + \x\ - (x3 - \) }Y = 0 , = x2 (X - 1) - (xi - 1 ) 7 = 0, = x4 (X + 1) - (x3 + 1)F = 0, = (xi + l ) 2 + x 2 - ( x 3 - l ) 2 - x 2 = 0.
*-c ICAAx = LAXAB «-( <-< ^ ~
AABBx = ZBtBC Ai lies on BC Bi lies on AC |AAi| = |BBi|
Note that the bisectors may be internal or external, and both of the cases are covered by the polynomial equations H\ = 0 and Hi = 0. Let P = {Hi,...,Hs} and the variables be ordered as X -
<x4.
An irreducible triangular series *P of [P,{K}] computed by t r i s y s [ i t s ] consists of the following 15 irreducible triangular sets: Tj = [X,x2(Y2 - 3 ) - (2xi - 1) (Y2 + l),x2 + xiY -F,x3 +xhx4+xiY T2 = [E,F,H3,T,HA\,
-Y],
147
9.1. Deriving Locus Equations T3 = [ X , 7 2 - 3 , 2 x i - l , 2 * 2 - 7 , 2 * 3 + l , 2 * 4 - 7 ] , T4 = [ X , 3 7 2 - l , 2 * , - 3 7 + l , * 2 + 7 * i - 7 , 2 x 3 + 3 7 - l , * 4 - * 3 7 - 7 ] , T 5 = [X,3Y2- l,2*i - 3 7 + l,* 2 + 7 * i - 7 , 2 * 3 - 37 - l , * 4 - * 3 7 - 7 ] , T 6 = [X,37 2 - l , 2 * i + 37 +l,X2 + Yxi-Y,2x3-3Y
-l,X4-x3Y
-Y],
T 7 = [ X , 3 7 2 - l,2*i + 3 7 + 1 , * 2 + 7*1-7,2*3 + 3 7 - l , * 4 - * 3 7 - 7 ] , T 8 = [ X - l,ATi,*i - l , F i , F 2 , 2 * 4 - * 3 7 - 7 ] , T9 = [ Z - l , / : 2 , x i - l , F i , F 2 , 2 * 4 - * 3 7 - 7 ] , Tw = \X+l,Ki,Gi,2x2
+
Yxi-Y,X3+l,G2],
Tn = [X + l,«2,Gi,2*2 + 7*i - 7 , * 3 + l,G 2 ], Ti2 = [ X 2 - 6 Z - l l , 7 2 + 5X + 7,*i+3,//3,4*3-7X + 5 7 - 2 X - 2 , / / 4 ] , Ti 3 = [ X 2 - 6 Z - l l , 7 2 + 5Z + 7 , * i + 3 , / / 3 , 4 * 3 + 7 Z - 5 7 - 2 X - 2 , / / 4 ] , Ti 4 = [X2 + 6 Z - l l , 7 2 - 5 Z + 7 , 4 * i + 7 Z + 5 7 - 2 X + 2,// 3 ,*3-3,// 4 ], T 15 = [X2 + 6 X - l l , 7 2 - 5 X + 7 , 4 * i - 7 Z - 5 7 - 2 X + 2,//3,*3-3,// 4 ]. where £ = (X 2 +7 2 ) 4 (8X 2 + 9 7 2 ) - 4 ( X 2 + 7 2 ) 2 (14X 4 + 1 3 X 2 7 2 - 3 7 4 ) + 144X6 + 86X 4 7 2 - 148X 2 7 4 - 747 6 - 176X4 + 156X 2 7 2 - 1727 4 + 104X2+13772-24, F = 2*i (X2 + 7 2 - 3) [(X - l) 2 + 7 2 ] [(X + l) 2 + 7 2 ] [(X - 1) (X2 - 1) + (X - 3)7 2 ] + X (2X 2 + 7 2 ) (X2 + 7 2 ) 3 - ( X 2 - 7 2 ) ( 2 X 2 + 37 2 )(X 2 + 7 2 ) 2 - X ( X 2 + 7 2 ) ( 1 2 X 4 + 9X 2 7 2 + 7 4 ) + 12X6 + 7X 4 7 2 - 14X 2 7 4 - 7 6 + X (24X4 + 21X 2 7 2 + 177 4 ) - 2 4 X 4 + 3 3 X 2 7 2 - 3 3 7 4 - X ( 2 0 X 2 + 7 1 7 2 ) + 2 0 X 2 + 297 2 + 6 X - 6 , r = 4*3[(X + l ) ( X 2 - l ) + (X + 3 ) 7 2 ] - 8 [ ( X + l ) 2 - 7 2 ] + [(*i + l ) 2 + * 2 - 4 ] ( X + l ) [ ( X - l ) 2 + 7 2 - 4 ] , F\ = 4 * 2 ( 7 4 + 2 7 2 - 8 ) - 3 7 5 - 4 7 3 + 167, F 2 = 8* 3 7 2 + (*2 + 4 ) ( 7 2 - 4 ) , Gi = 4*i7 2 (7 4 + 2 7 2 - 8) - 7 6 + 327 2 - 64, G2 = 8*4 + [(*i + l) 2 + * | - 8 ] 7 , Ki = 3 7 4 + 8 7 3 + 207 2 + 327 + 16, K2 = 3 7 4 - 8 7 3 + 207 2 - 327 + 16. To obtain triangular systems having the projection property, we compute a regular series *F,- of [T,-,iniset(T,-) U {7}] for each 1 < / < 15; the
148
Chapter 9. Symbolic Geometric Computation
computation is trivial for some T,-. Let
* = I [TnQix,Y],vnci[x,Y}} I [T,u] G \JvA. It is found that *? comprises 14 triangular systems [f ,-,©,•] in Q|X,F], where f i = [X], f3 = [ X , 3 F 2 - l ] , f5 = [X,F 4 + 5F 2 + 8],
% = [E], f4 = [ X , F 2 - 3 ] ,
f 6 = [X2 - 3,9F 8 + 144 Y6 + 616F 4 + 320F 2 + 272], f7 = [X-l,A:i] 1 % = [X-l,K2], % = [X + l,K1], fio = [X + l , ^ 2 ] , 2 2 f n = [ X - 6 X - l l , F + 5X + 7], f 1 2 = [ X 2 + 6 X - l l , F 2 - 5 X + 7], f i 3 = [X2 + 6 X - l l , L i - L 2 ] , fi4=[X2-6X-ll,Li+L2]; CTi = [ F , F 2 - 3 , 3 F 2 - 1 ] , U2 = [X,X - 1,X + 1,X2 - 3,X2 + 6X - 11,X2 - 6X - 11], U3 = • • • = Ui4 = 0, and
U = 2X (9F 6 + 1675F4 + 100874F2 + 1947648), L2 = 27Y6 + 4938F 4 + 297020F2 + 5734400.
Therefore, [E = 0 AX (X2 - 1) (X2 - 3) (X2 + 6X - 11) (X2 - 6X - 11) ^ 0] 14
(9.1.1)
V[X = 0 AF (F 2 - 3) (3F 2 - 1) ^ 0] V \f i=3
Peii
gives the locus equations and inequations we wanted to derive. However, this disjunction of polynomial equations and inequations looks quite complicated. Using the method described in the following subsection, we can simplify (9.1.1) to [£ = 0 A ( X 2 - l ) ( X 2 - 3 ) ^ 0 ] 10
V [X = OAF (F 2 - 3) (3F 2 - 1) ^ 0] V \f i=6
Peti
149
9.1. Deriving Locus Equations
Moreover, both Zero([X,Y2 - 3]) and Zero([X, 3 7 2 - 1]) are contained in Zero([£]/{X 2 — l,X2 — 3}), so the above formula may be further simplified to 10
[X = 0 AY ^ 0] V [E = 0 A (X2 - 1) (X2 - 3) ^ 0] V V i=6
J.1.2)
per.
This is the minimal disjunction of equations and inequations for the locus of C. The first equation X = 0 is trivial: it is the v-axis. Figure 25 plots the real part of the irreducible algebraic curve E = 0 of degree 10.
Fig. 25. Algebraic curve denned by E = 0 The four points (—y/3,0), (—1,0), (1,0), (\/3,0) also lie on the curve £ = 0. However, when C is located at one of these points or (0,0), the triangle ABC degenerates into a line and the two bisectors cannot be of equal length. We have to exclude these four points by the inequation (X2 — 3)(X2 — 1 ) ^ 0 . In the cases (X2 — 3)(X2 — 1) = 0, the locus points are determined by 10
'
V f\P = 0 i=6 _PGT t
The two bisectors AA\ and BB\ of AABC are of equal length if and only if point C lies on the locus defined by (9.1.2). Figure 26 depicts the real part of the complete locus. The triangle ABC is an isosceles if and only if C lies on the v-axis. We see that there are infinitely many triangles with equal bisectors which are not isosceles. When both of the bisectors are internal, the locus of C is the y-axis and thus AABC is always isosceles. This is the well-known Steiner-Lehmus theorem. When both of the bisectors are external, the locus of C contains the y-axis and infinitely many points not on the y-axis, so AABC is not always isosceles. Figure 27 plots the (real) locus of C for two bisectors both external.
150
Chapter 9. Symbolic Geometric Computation
When one of the bisectors is internal and the other is external, the locus of C contains only a few points on the ;y-axis, so in general AABC is not isosceles. Figure 28 plots the (real) locus of C for AABC with one internal and one external bisector.
-2 -1.5 -1
0
1
1.5
0
-1.5 -1
1
1.5
Fig. 27. Locus of C for AABC with two external bisectors
Fig. 26. Complete locus of C
-'/Si
-1.8
-1
0
1
l.i
Fig. 28. Locus of C for AABC having one internal and one external bisector Using the same method, one can determine the locus equations for the intersection point (X,Y) of AA\ and BB\ for nondegenerate AABC as follows [X = OAF (Y2 - 1) (Y2 - 3) ^ 0] V [G = OAY (Y2 - 3) ^ 0] V[2X4 + 5Z 2 + 20 = 0 A r 2 - 3 = 0], where G = (2X2+Y2)
(X2 +Y2)2 - 10X4 - 10X2Y2 -4Y4 + 14X2 + 5F 2 - 6.
151
9.1. Deriving Locus Equations
This locus was derived by Wu and Lii in [100]. Its real part is shown in Fig. 29 (see also [100]). Figure 30 pictures the complete loci of C and the intersection point / of AA\ and BB\. In fact, it is Wu who first studied triangles with two equal bisectors using his method and showed that any triangle with two equal internal bisectors is isosceles, and a triangle with two equal bisectors, not both external, is not necessarily isosceles.
-2
-1.5 -1
0
I 15
Fig. 29. Locus of / derived by Wu and Lii
-1.5 -1
0
1 1.5
Fig. 30. Complete loci of C and /
Example 9.1.2 (Biarcs [46, 66, 78]). Given two points A and B of two different circular arcs which have given tangent directions at A and B, determine the locus of an intermediate point M at which the two circular arcs join together with a common tangent.
Fig. 31. Smoothly connected arcs This problem is taken from the book [46] by Nutbourne and Martin, who
Chapter 9. Symbolic Geometric Computation
152
proved that one branch of the locus is a circle. In [66] and [78, pp. 209-211], we have shown how to derive the locus automatically by using elimination methods. Here we recall the example and make some clarifications. Let us choose the point coordinates as before: A(0,0), £>(«i,0), B ( H 2 , « 3 ) , M(X,Y),
CI(0,XI),
C 2 (x 2 ,x 3 ).
From the geometric conditions we get the following relations: Hi = («2 — U\) (X2 — U2) + U3 (x3 — U3) = 0, 2
H2=X
2
2
+ (xl-Y) -x 1=0, 2
<—: BC2 -L BD
<-c \CiA\ = \CiM\ 2
2
Hi = (X2 - U2) + (X3 - U3) -
(X2-X)
-(Xi-Y)2=0,
"
H4 =X(xi-x{)+X2{xi
-Y) = 0 .
\C2B\ = \C2M\
<-( M lies on C1C2
Let P = {H\,... ,7/4} and X -< Y -< x\ -< x2 < x3. A weak-characteristic series of P computed by charsets [qics] consists of three weak-ascending sets Ci = [R,2Yxi - Y 2 - X 2 , 2 I x 2 + u3Y2 - 2u22Y + 2Ulu2Y - 2u\Y + u3X2 + W3 + U2U3 — 2u\U2U3,U3X3
+ U2X2 — U\X2 — U2 — l^ +U1U2],
€-2 = [X — U2,Y — U3,2u3X\ — U2 — U2,X2 — U2,X$ — W3], C3 = \X ,Y ,X\,2ll\X2
+ U2 + u\-
2u\U2,2u\U3X3
— (U2 + Ul)u2 - u\ + U\U^\,
where R = u3 {X2 + Y2)2 -2ulU3X(X2
+Y2) -2{u\ + u\-uxu2)
(X2 + Y2)Y
— (u2 + U2 — 2uiU2)U3 (X2 — Y2) +2{U2U2 + U\U2 + U^ — U\U2)XY, I = (U2 — U\)Y — U3X + M1M3.
Projection of Zero(Ci/{F,/}) and Zero(C2), Zero(C3) onto X,Y results in [[R],{Y,I}],
p-M2,y-M3],0],
[[X,Y],V>].
Therefore, the locus equations and inequations of point M that we wanted to derive are [R = 0AYI ^ 0] V [X =0AY = 0] V [X = u2 AY = u3]. The equations X = 0,Y = 0 and X = U2,Y = u3 correspond respectively to the points A and B which are on the curve (T defined by R = 0. Note that 7 = 0 corresponds to the line AD (which is the x-axis) and / = 0 represents the line BD. In general, AD intersects £ at two other points determined by
F =X2-2uiX-ul-ul
+ 2uiu2 = 0, 7 = 0 ,
153
9.1. Deriving Locus Equations
which are distinct from point A. Similarly, BD intersects € at two other points determined by G=[uj + {u2 - ui)2]X2 -2ux[u\ + (u2 - ui)2]X + u\u\ = 0,
7 = 0,
which are distinct from point B. Nevertheless, the point M never reaches the points determined by [F = OAY = 0] V [G = 0A/ = 0], so these four points are not on the locus of M. This can be easily verified because Zero(PU {F,Y}) = Zero(PU {G,/}) = 0. The indispensability of the inequation YI^O illustrates the need of projection for the determination of exact loci. The inequation YI ^ 0 is missing in both [66, pp. 166-168] and [78, pp. 209-211] due to incorrect simplification and projection, so the four extraneous points were not excluded from the locus given therein. Using the method of undetermined coefficients, we can factorize R into the following two polynomials (see Example 10.2.3): /?, = [X
/ 2
V
MI-OC\2
2
/
Hi+a\2 2
J
/ \
P + M20t\2 2«3 J
(M3 + M2)a 2 {u\ — U2 + a ) '
/
[3-M2a\2
{4 + 4)a
\
2 M3 J
2 ( n 2 - K i + oO'
where a = Ju\ + {u2-ui)2
= \BD\,
P = u\ + u\ — U\U2-
Fig. 32. Locus arcs of M for u\ = — 1, u2 = 2, M3 = 4
154
Chapter 9. Symbolic Geometric Computation
Ri = 0 and R2 = 0 represent two circles 0/i and Qh passing through A and B, whose centers I1J2 and radii are readily determined from the above expressions. Figure 32 shows the two circles for u\ = — 1, ui = 2, and W3 = 4. Based on our experiments and as pointed out in [78], we have the following conjecture for arbitrary ui,U2,uj, (and thus for any given points A,B,D). Conjecture. Let C0i,...,0)4 be the four arcs of the circle 0/i divided by the two lines AD and BD. Then the two circles ®C\ and 0C2 centered at C\ and C2 are tangent at M externally when M moves along two opposite arcs among ooi,..., 004, and internally when M moves along the other two opposite arcs. This is also true for 0/2. In the case of internal contact, ©Ci is inside ©C2 when M moves along one of the two arcs, and ©C2 is inside ©Ci when M moves along the other arc. An example of the internally tangent circles at two points M and M' on the opposite arcs of 0/i is given in Fig. 33. The interested reader may try Load (Biarc) and then Geometric (Biarc) (or BiarcA instead of Biarc) in the GEOTHER environment for animation.
Fig. 33. Circles contacted internally at M and M' 9.1.1
Simplification of equations and inequations
The methods presented in this and later sections return as output a disjunction of systems of polynomial equations and inequations. The disjunction is complicated in most cases, and how to simplify it is a question that is difficult both to formulate and to answer.
155
9.1. Deriving Locus Equations
Roughly speaking, we are given a finite sequence of polynomial systems [Pi, Qi ], ..., \PS, Qs], and we need to determine finitely many other polynomial systems [Fi, Gi ] , . . . , [F,, G, ] such that t
s
| J Zero(F(- /Gj) = \J Zero(P J/QJ ), 1=1
(9.1.3)
7=1
and in terms of polynomial representation of the zero set the left-hand side is simpler than the right-hand side of (9.1.3). The difficulty of the problem lies on how to measure the simplicity. One may consider that the left-hand side of (9.1.3) is simpler if t < s, or if F,- and G,- contain fewer polynomials, or these polynomials have fewer terms or take less space of computer memory, or they look simpler from a mathematical point of view. Which criterion to use depends on the form of the results desired, and it is thus difficult to give a general and satisfactory formulation for this simplification problem. Solving the problem with respect to a formulation may be even more difficult; it is much easier when all the Q ; are empty sets. In our situation the polynomial systems [Pi,
qje«
and, in case the polynomial systems in *P are triangular and irreducible, O is likely simpler than *P in terms of their size and the number of polynomials contained therein. 51. Compute an irreducible regular series Qtp of ^3 for each ^3 G *F, and let Cl:= UspeY^lqj. For each [T,U] e £2, let U be replaced by the set of all distinct irreducible factors of the polynomials in U. Set O :=0. 52. While |Q| > 1 do: S2.1. Let[T,U] 6Qwith|T|thesmallestpossible,andsetri:=£2\{[T,l[J]}, V:=U.
156
Chapter 9. Symbolic Geometric Computation
52.2. For each U G V do: 52.2.1. Compute an irreducible regular series Cl of [TU {£/},U\ {£/}] andsetQ:=0. 52.2.2. While Q f 0 and 3 [f, U] e Q, [f, 0 ] 6 f i such that Zero(f/U) = Zero(f/U) do: fi:=Q\{[f,0]} a n d Q : = Q U { [ f ,U]}. 52.2.3. IfQ = 0 , t h e n s e t U : = U \ { t / } , Q : = Q \ Q . 52.3. Set
( J Zero(f/U) c \J Zero(f/U). [f,u]en [i,u]en
In the former case, Zero(T/U) = Zero(T/U\ {£/}) and thus U can be simply removed from U. For the latter, one can remove U from U and the subset Cl from £2 simultaneously. With these technical notes, the correctness of the simplification procedure becomes evident. In step S2.2.2, since both [T, U] and [T, U] are irreducible regular systems, whether Zero(T/U) = Zero(T/U) can be easily decided.
9.2
Implicitizing Parametric Objects
Geometric objects like curves and surfaces may be represented algebraically by implicit equations or parametric equations. The advantage of each representation depends upon the type of problems to be solved. In geometric modeling, one often needs to convert one representation into the other. The rational parametrization of a geometric object in an ^-dimensional affine space may be represented as Xl
_
l
frOO tt
x
_
p
n{y)
-Qi(yV"' ~Qn(yy
where y = (_yi,... ,ym) are parametric variables. The problem of implicitization amounts to finding the implicit equations and inequations in x which define the same geometric object as the parametrized representation does. This can be done by using the following algorithm. The incorporation of projection into implicitization algorithms was suggested first by Li in [40] (see also [96]). Given two sets of polynomials Pi,...,Pn and Q i,..., Qn in %\y\, where Q\-Qn ^ 0 and m
9.2. Implicitizing Parametric Objects
157
mial systems [Pi,Qi],..., [P e ,Q e ] in 3C[x] such that for any x = (xh... ,xn) G §?, ( 3 y € 9C" such that e XG UZero(P,/Q,) < = » < _ _ P i ( ^ . _ f„(y)
r1_C2i(y)""",JC"~e-W 11. Let P:={/5i-x1ei,---,i5«-^G4, Q:={Gi,-,en}, P*:=PU{z1Ci-l,...,z„Q„-l} and xi -< • • • -< x„ -< j i -<•••-< ym -< z\ -< ••• < z„, where the zj are new indeterminates. Compute a regular series *F of [P,Q], or a triangular series ¥ of [P,Q] or a Grobner series *P of P* (under the purely lexicographical term order) with projection for z\,..., z„ and >>. 12. Remove redundant sets from U[T,u]e4/Zero(Tn 3C[*]/Un ^C[x]), simplify it, and let the obtained zero set be (JLi Zero(P,-/Q,-). Then return ¥:={[Pi,Qi],-,[Pe,Qe]}. It is easier to compute the Zariski closure of the quasi-varieties Zero(P,/Q;) or the implicit ideal using other techniques without projection. For example, one can do so by means of Grobner bases [24], (multivariate) resultants (see the functions miscel [bezres] and miscel [macres]), and the techniques of moving curves and surfaces [54] and undetermined coefficients [82]. However, without projection (and using inequations) the implicitly defined geometric object is not necessarily the same as the object defined by the parametric equations. Example 9.2.1. Consider the parametric surface defined by the equations _P_ _{s2-l)t2 2 sl + 1 in three-dimensional affine space. Let P = {t3 -2x,(s2
1st2 sl + 1
- \)t2 -y{s2 + l),2st2 -z(s2 + 1)},
Q = { s 2 + 1}.
A regular series of [P,Q] computed by sisys [regser] with respect to x -< y -< z -< s -< t consists of five regular systems [T i, Ui ] , . . . , [T5, U5 ]. Let X; = [T,-n Q[x,y,z\,Ui n Q]x,y,z)} for 1 < i < 5; then Ti = [[E], {x,yi+4x2y-4x2}], X2=[[v6-16x4,z4 + 3vV + 3 / ] , W ] , X 3 =[[y 3 +4x 2 ,z],{x}], X4 = X5 = [[x,y,z],0],
158
Chapter 9. Symbolic Geometric Computation
where E = z6 + 3y2z4 + 3y4z2 + y6 - 16x4. It follows that 5
(JZero(X,-) = Zero(Ti) UZero(T2) UZero([y3 + 4.x2,z]) = Zero([£]/{x,v3 - 4x2}) UZero([y3 - 4x 2 ,z 4 + 3y2z2 + I2x2y}). This simplification has been done automatically by using the method described in Sect. 9.1.1. Therefore, the desired implicit equations and inequations are [E = 0 A x(y3 - 4X2) ^ 0] V \y3 - 4x2 = 0 A z4 + 3 y2z2 + 12j?y = 0]. For instance, (x,y,z) = (4,4,0) satisfies the equation E = 0, but it is not on the parametric surface. Using our method presented in [63] with projection, we obtain [[E]^,
[\y3+4x2,z],{x}],
[[x,y,z],0],
where U contains a large polynomial of degree 4 in z which has 75 terms and is irreducible. Using Wu's method with projection, one may get an even larger polynomial with more computing time. Example 9.2.2. The following parametric equations originate from [10]:
x = Px/Q, y = Py/Q,
z = Pz/Q,
where px = st3 -s4-
2s2t2 + s2t + 4s3t - It3,
Py = s2t - s3t - 2s3 +
3st2-t3,
Pz = s3~ st3 - 4s2t + 6f3 - st2, Q = s3-3st32st2 + 6s2t2 + t4 - ts3. These equations define a rational surface in three-dimensional affine space, whose implicit equation may be easily determined by using miscel [implic i t ] (see Example 6.4.1). In order to see which points on the implicit surface are not defined by the parametric equations, we want to compute a regular series of [P,Q], where P = {Px,Py,Pz} and Q = {£>}. We have tried to compute such a series using sisys [regser] in Maple without success. By first computing a Grobner basis using J.-C. Faugere's Gb package in C++ (see Sect. 6.2), we are able to obtain a regular series *P of [P, Q] under z -< y -<
9.2. Implicitizing Parametric Objects
159
x -< t -< s. *P consists of four regular systems [T,-,U,] with T,- = T,T1 Qiz,;y,Jt] and U,- =Uir)QjLz,y,x]:
f2 = [ 1 7 z - 6 , 1 7 y + l ] , f3 = [ z - l , y + 2], f4 = [7z5 + 52z4 + 313z 3 +100z 2 + 2 z - l , 692z 3 v-173z 2 v+187zy + 37y+121z 4 +1009z 3 - 163z2-52z-l,17zx-6;t-12/-23v-l-6z2-3z-l]; UI = {y + 3z - 1, - / + 30 zy2 + 2y2 + 3z2y + 6zy-y
+ z3,G,
2 9 7 1 / - 9 6 4 8 z / + 4 2 7 9 / - 1975z2?3 - 876z)>3 + 658j 3 + 3442 z V - 3993 z V + 1332zj2 - 1 1 8 / + 284 z4y + 4z3y -237z2j+108z>'-13}'+133z5-218z4+154z3-59z2 + 12z-l}, U2 = {1734x2 + 408x+145,17x 2 + 2x-ll,1419857x 5 -2839714x 4 -938383x 3 + 1851045x 2 -292043x+100121},
th = {*}, U4=0, where F =yx+3zx
— x — 2y2 — 4y + z2 — z,
G = - 1 2 / -2zy2 - 2 3 / + 6z2y - 4zy -2y + z3 + 8z2 - 6z + 1. The first polynomial in Ui may be removed by using the simplification procedure presented in Sect. 9.1.1. Prom the regular systems [T,-,U,-], one can easily establish the exact implicit equations and inequations for the parametric surface. The zero set of the single polynomial F considered as an implicit surface contains several curves which do not lie on the parametrically defined surface. For example, the cubic curve defined by P3 in Example 9.5.2 and shown in Fig. 38 is an irreducible component of the reducible curve denned by F = 0 and G = 0. However, on this cubic there are only finitely many points (i.e., the zeros of T4) which are on the parametric surface. The surface defined by F = 0 is the one in between shown in Fig. 35. Example 9.2.3. Now we recall the surface of revolution studied in [54]. This surface may be obtained by rotating around the z-axis the cubic polynomial
160
Chapter 9. Symbolic Geometric Computation
Bezier curve with control points (2,0,0.9),
(2,0,0.45),
(1.5,0,0.225),
(1.5,0,0.15)
and was used to model the lower body of a teapot. It is denned parametrically by r
_
(2s-l)P 2Q
'
„_ ~
y
s(s-l)P Q
3(f 3 -9f 2 + 18f-12) 40 '
'
[
'
where P = 2t3-3t2
+ 4,
Q=
2s2-2s+l.
We wish to determine the implicit equations and inequations that define exactly the same surface as these parametric equations. Let P denote the set of the three polynomials corresponding to (9.2.1) and Q = {Q}. It is easy to compute an irreducible characteristic series *F of [P,Q] withx-< y
Zero(P/Q) = yZero(T,-/iniset(T,-)UQ), 1=1
where Ti = [£,(1280000z3 + 10126800z 2 -7200y 2 z-7200x 2 z + 3407400z + 5 9 3 1 9 / + 59319x2 + 292626)? + 3 (1984000 z 3 - 766800 z2 + 24840v2z + 24840x 2 z-402840z- 128979y 2 - 128979x 2 -36180),G], T 2 = [ 4 0 9 6 / + 12288x2/+ 127117107/+ 12288x 4 / + 254234214x 2 / + 98973163953/ + 4096x6 +127117107 x4 +98973163953 x2 + 116700507,F, 54587755224899112- 6 (5746688/ + 11493376x 2 / + 148682503809/+5746688x4+148682503809x2+217869762017484)f - 2(69890048/+ 139780096x2/ +2423086870725/+69890048x 4 + 2423086870725 x 2 - 595618612975413), G], T 3 = [x,y,32000z 3 -76500z 2 -30240z-2781,200z?+180z + 33f + 36], T 4 = [15625/ +15625x 2 -49,F,25f 2 -60f + 54, G]\ £ = 3 2 4 ( / + x 2 )(1600z 2 + 6 9 0 0 z + 3 / + 3x 2 +1197) 2 - (128000z 3 -306000z 2 + 2160/z + 2160x 2 z- 120960z- 1 1 7 4 5 / - 11745x 2 -11124) 2 , F = 20(64000000/+ 128000000X2/ -43555072953/ + 64000000x4 - 43555072953x2 + 265874801598) z - 9 (872784000/ + 1745568000X2/ + 1643501398790/+ 872784000x4
161
9.3. Computing Offsets + 1643501398790.x2- 105071597253), G = 2(2t3-3t2 + 2y + 4)s-2t3 + 3t2-2y +
2x-4.
From these triangular sets we see that the surface of revolution may be defined by the implicit equation E = 0. In fact, the implicit polynomial E may be easily computed by function miscel [ i m p l i c i t ] . However, the surface defined by E = 0 is not the same as the surface defined by the parametric equations (9.2.1); some points on the implicit surface do not lie on the parametric surface. In order to locate such points, we need to project the triangular sets onto x,y,z. For this purpose, we have tried to compute the regular series of [T,-,iniset(T,-)UQ]. Unfortunately, the computation is very heavy and our program ran several days without completion (e.g., for i— 1). Therefore, we still do not know the exact implicit equations and inequations of the parametric surface. Elimination methods may also be used to deal with other problems such as the independency of parameters, the propriety of parametrization, and the inversion problem which are related to the implicitization of parametric objects (see, e.g., [24]).
9.3
Computing Offsets
Let F (x, y) = 0 be the implicit equation of an algebraic curve £ in Euclidean plane. Roughly speaking, the r-offset to £ is the set of all the points that have the same perpendicular distance r to £. It is easy to prove that at every singular point (xo,yo) of € all the points on the circle (x — xo)2 + (y — yo)2 = r2 (called extraneous circle) are contained in the r-offset. We shall eliminate such known extraneous circles in our algebraic formulation. Assume that the algebraic curve £ defined by F (x,y) = 0 is irreducible and r is a positive number. An algebraic formulation of the r-offset to the curve £ is given by the following equations: 'P1=F(M,V)=0, < jP2
= ( x - M ) 2 + ( y - v ) 2 - r 2 = 0,
P3=Fv(x-u)-Fu(y-v)=0, >P* = (Fuw-l)(Fvw-l)
' "' = 0,
where F„ = dF(u,v)/du, Fv = dF(u,v)/dv, and u,v,w are new indeterminates. The meanings of the first two equations are obvious: the point Q(u, v) is on the generating curve £ and the distance between Q and the point P(x,y) on the offset
162
Chapter 9. Symbolic Geometric Computation
is r. The third equation means that the line PQ is perpendicular to the tangent line of € at Q. The last equation is added to rule out the case in which Fu and Fv vanish simultaneously (i.e., Q is a singular point). Let P = {Pi,... ,P4}. We call the set of points Proj^ZeroCP) the r-quasi-offset to €. What is usually called the r-offset to £ is the Zariski closure of the r-quasi-offset to <£.. In this section we are concerned mainly with the computation of algebraic equations and inequations for quasi-offsets. This can be done by computing a regular or simple series of P, or a triangular series of P with projection. Like the case of implicitization, a quasi-offset is expressed as a disjunction of systems of polynomial equations and inequations, and it is often necessary to simplify the disjunction. The equations for offsets may be obtained from the equations and inequations of quasi-offsets quite easily. Similarly, for any algebraic surface & defined by an implicit equation F(x,y,z) = 0 in three-dimensional Euclidean space, the r-offset to 6 is the set of points that have the same perpendicular distance r to 6 . An algebraic formulation for it may be given by the following equations (cf. [29]): ' F(u,v,w) = 0, {x - u)2 +(y- v)2 + (z - w)2 - r 2 = 0,
r/'1 = (x- M ) 2 + ( v - v ) 2 - i = o, P2 = v 2 - w3 = 0,
' P3 = 2v(x-u) + 3u2(y~v)=0, . P 4 = (3 M2W - 1) (2 vw - 1) = 0.
(
' '^
We want to determine the implicit equations and inequations (in x and y) of the quasi-offset. For this purpose, let P = { F i , . . . , ^ } (as in Example
9.3. Computing Offsets
163
A47 of Appendix A). Under x -
Zero(P) = Zero(Ti/Ui) U [ J Zero(T,-), i=2
where Ty = [E,Ti2,P3,P<],
Ui =
{X,T2I,TAI},
T2 = [T21,T22,Tl2,P3,P4], T3 = [T2l,T32,T33,P3,P4}, T4 = [74i,742,743,F3,F4], T 5 = [T4i, y, 12xu + 2 u - 9x 2 - 2x + 9, v2 + u2 - 2xu + x2 - 1, P4], T6 = [x,729y4-956y2-529,85u-Sly2
+ 72,6y2v +
23v+12y3-39y,P4};
E = 729x8 + 216x7 + 729 A 2 - 2900x6 - 1458^y 2 - 2376X5 - 2619x4y2 + 3870x4 - 1458x3y4 - 4892x3y2 + 4072x3 + 729x2y4 - 297 x2y2 - 1188x 2 -4158xy 4 + 5814xy 2 -1656x + 427y 2 -1685y 4 + 729y6 + 529, T12 = [2187y 4 -6(729x 3 + 162x 2 +2079x + 478)y 2 +2187x 6 - 1944*5 - 10125x 4 -4800x 3 + 2501x2 + 4968x-1587]w + 4x 2 [27(18x-l)y 2 + 243x4 + 756x 3 -270x 2 +124x + 279], 72i = (81x2 + 18x + 28)(729x 4 + 972x 3 -1026x 2 +1684x + 765), 722 = 729 (30618x 5 +38151x 4 +8316x 3 +2286x 2 +59092x + 20664) y2 + 279686682x 5 -194912487x 4 +343568520x 3 +126051867 x2 + 74246894x +30796164, r 32 = 6 ( 1 8 x - l ) ( 8 1 x 2 + 81x + 83)y 2 -2187x 6 + 7776x5 + 18252x4 - 4812x3 - 4787x2 + 540x + 2766, 733 = (243x2 + 36x+ 8 5 ) u 2 - (81y2 + 162x3 - 36x2 - 1 5 4 x - 72)u -72x3+4x2, T4i = 2 7 x 4 + 4 x 3 - 5 4 x 2 - 3 6 x + 23, T42 = 19683y4 - 27 (1458X3 - 729 x2 + 4158x+ 1685)y2 - 64(2917x3 + 2052X2 - 2493x - 514), 743 = 19683 (13x2 - 9)y2w - 864 (1418x3 + 129x2 - 1692x- 59) u - 8748 ( 1 8 x - l)x 2 y 2 - 32(18952x3 + 12663X2 - 4 7 3 4 x - 943).
164
Chapter 9. Symbolic Geometric Computation
Thus the implicit equations and inequations of the quasi-offset are given as 6
(T(,2) = 0 A Ui ^ 0) V V T} 2) = 0.
(9.3.4)
(=2
Using the method presented in Sect. 9.1.1, the above disjunction of polynomial equations and inequations is simplified to E = 0,
x£ 0
(9.3.5)
or x = 0,
7 2 9 / -956y2-
529 = 0.
(9.3.6)
These equations and inequations may also be derived by computing a characteristic or triangular series with projection. A characteristic set of P is easy to compute, but the computation of a characteristic series may need much time. Using our method [63] of 1993, it takes only a few seconds to compute a triangular series of P with projection, but the output is complicated.
Fig. 34. Curve offsetting
The generating and offset curves are shown in Fig. 34. When x — 0, the first equation E = 0 in (9.3.5) becomes (y2 - 1) ( 7 2 9 / - 956y2 - 529) = 0. However, (0,1) and (0,-1) satisfying E = 0 do not lie on the curve defined by (9.3.3) (i.e., there are no corresponding u, v and w such that the equations (9.3.3) are satisfied). This is why one needs (9.3.6) instead of (9.3.5) in the case x = 0. In summary, we have:
165
9.3. Computing Offsets
• any point (x,y) on the curve defined by (9.3.3) is a point on the curve defined by the equation E = 0; • any point (x,y) other than (0,1) and (0,-1) on the curve defined by E = 0 is a point on the curve defined by (9.3.3). Computing implicit equations and inequations for quasi-offsets is very expensive in general. If there is no need to exclude the extraneous components of lower dimension, one may consider offsets instead of quasi-offsets. More precisely, let P be the set of the four polynomials Pi,... ,P$ in (9.3.1) or the set of the six polynomials in (9.3.2). Then Zero(Ideal(P) n %L[x,y,z}) is the Zariski closure of the quasi-offset, that is the offset to the generating curve or surface. Implicit equations of offsets may be more easily computed by using different elimination methods (e.g., Grobner bases) without projection. It is clear that the Zariski closure of the point set determined by (9.3.4) is the algebraic curve defined by E = 0. Example 9.3.2. Consider the implicit surface defined by xy + 3xz - x - 2y 2 - Ay + z2 - z = 0, which has been derived in Example 9.2.2. To this quadratic surface the 1-ofFset may be formulated by the following equations: Pi = F — uv + 3 uw — u — 2v2 — 4v + w2 — w = 0, P2 = (x-u)2
+ (y-v)2+{z-w)2-i
P3=Fv(x-u)-Fu(y-v) P4=Fw(y-v)-Fv{z-w)=0, P5=Fu(z-w)-Fw(x-u)
= 0,
= 0, = 0,
P6 = {Fut- 1) (Fvt - 1) (Fwt - 1) = 0, where F„ = v + 3 w - l ,
Fv = u-4v-4,
Fw = 3u +
2w-l.
We have tried to compute the polynomial equations and inequations for the quasi-offset without success. Computing the implicit equation of the offset is also not easy in Maple. Let P = {Pi,...,Ps}. Using the Gb package of J.-C. Faugere, we are able to compute the Grobner bases Gu,
VU{Fut-l},
PU{F v f-l},
V\J{Fwt-l}
with respect to an elimination term order determined by z ^ v ^ x X w ^ v -< u -< t. It is found that Gu n Q[x,y,z] = Gv n Qlx,y,z] =GWD Q[x,y,z] = [G],
166
Chapter 9. Symbolic Geometric Computation
where G is polynomial of total degree 12 in x,y, z, consisting of 451 terms. It follows that the 1-offset to the quadratic surface is given by G = 0. This offset surface and its generating surface are plotted in Fig. 35.
Fig. 35. Surface offsetting In fact, the generating surface does not have any singular point, so the polynomial P^ is not needed. Computing the Grobner basis of P with Gb under the above-mentioned term order, one may get the same polynomial G for the 1-offset (see Example 6.2.1). However, we have not succeeded in computing a triangular or regular series of IP in Maple 8. The reader is urged to offset algebraic surfaces of moderate degree as to appreciate the difficulty of computation.
9.4
Blending Algebraic Surfaces
Blending several pieces of algebraic surfaces smoothly is one of the central problems in computer-aided geometric design. A few methods are available for solving the problem of surface blending, see [29]. Here we explain how to deal with the problem using triangular-set-based methods. This approach was initiated by W.-t. Wu and has been furthered by other Chinese researchers [ 101,22, 11]. What is presented below is based mainly on Wu's work. Recall that ^ denotes the field of real numbers. For any nonconstant polynomials F,G 6 %\x,y,z], let "U{F) and 1^(F,G) denote the algebraic surface denned by F = 0 and the algebraic curve defined by F — 0 and G — 0, respectively, in three-dimensional Euclidean space %} with base (x, v,z).
9.4. Blending Algebraic Surfaces
167
We say that an algebraic surface 6 is in smooth contact with another surface 6 ' along a curve € at regular points, if for every point P on <£ which is regular for €, 6 , and 6 ' , the surfaces 6 and 6 ' have the same tangent plane at point P. Similarly, a surface 6 is said to have the same curvature as another surface 6 ' along a curve £ at regular points, if for every point P on € which is regular for C, 6 and &', the surfaces 6 and &' have the same Gaussian curvature at point/5. The problem of blending implicit algebraic surfaces may formulated as follows. We are given two families of polynomials G;,//, G ^.[JC,;y,z] such that the algebraic curves <£, = ^(G,,//,) are all irreducible for i e /, where / is a finite set of indices. Let / be a subset of /, K a subset of/, and &j = V(Gj) for j e / . Determine an irreducible polynomial F 6 ^,[x,v,z] of given degree m such that the following conditions are satisfied, where & = 1S(F): (i) 6 contains all the curves £, for i e /. (j) & is in smooth contact with each &j along the curve €j at regular points for j e J. (k) 6 has the same curvature as S* along the curve <£* at regular points for each k£K. The subsets K and L may be empty. It is clear that the three conditions correspond to three different levels of smoothness for the surface 6 to contact the surface 6/ along the curve <£/. One may also use other (equivalent) geometric conditions (e.g., the so-called reseating continuity [11, 22]) to characterize the smoothness of surface contact. For any irreducible polynomialF 6 2{[x,y,z], F = 0 defines an irreducible algebraic surface in ^ 3 . An irreducible algebraic curve € in %} may be determined by an irreducible ascending set A = [Ai,A2] C ^_[x,v,z]. When the defining polynomial equations F = 0 and G = 0 for € are given, it is easy to obtain A from F and G by computing a characteristic set of {F,G}. Let & = <^(F) and & = l^(F') be irreducible, where F,F' £ %\x, y, z]. The following results have been proved by Wu[99, §8.4]. (1) The surface © contains the whole of £ if and only if prem(F, A) = 0. Suppose that £ is contained wholly both in & and in 6 ' , but not wholly in the singular locus of 6 or 6 ' . (2) Form the polynomials 8l(F,F')=FxF;-FyFl, S2(F,F>)=FXF:-FZF;, 53(F,Fl)=FyFl-FzF;,
168
Chapter 9. Symbolic Geometric Computation
where Fv — dF/dv andFv' = dF'/dv for v e {x,j,z}. Then 6 and S ' are in smooth contact along £ at regular points if and only if prem(8/(F,F'),A) = 0 ,
/ = 1,2,3.
(3) Define fi{F) = F? + F2 + F2 and v(F) = FxxFyyF2 + FxxFzzFy +FyyFZ2Fx — 2FxFyFxyFzz — 2FxFyFxzFyz ZryrzrxztiyZ + ZrxryrxzryZ p2p2 p2p2 cT2r*2 r r r x ryz y rxx y rxy •
~\- ZrxrzrXyryZ
+
ZryrzrXyrxz
Then © and &' have the same curvature at regular points if and only if prem(p(F)2v(F') -jt(F')2v(F),A)
= 0.
Let Ht,Gi,I,J,K and m be given as in the formulation of the problem of surface blending. Using the results (l)-(3) above, we can reduce the surface-blending problem to the problem of solving a set of polynomial equations according to the following steps. Bl. Let F £ ^,[x,y,z] be a polynomial of degree m, written in the form i>0,j>0,k>0 i + j + k<m
where the coefficients itjjk are to be determined. Set E:= 0. B2. For each;' e /, compute an (irreducible) characteristic set Q of {G,,//,} and the pseudo-remainderR :=prem(F,Q), and set E:= EU {R}. B3. If / 7^ 0, then compute the polynomials 8i (F, Gj), 82 (F, Gj), and 83 (F, Gj) for j G / . For / = 1,2,3 and j 6 / , compute R:=prem(8/(/r,G7),C7) and setE:=EU{#}. B4. If K ^ 0 , then compute H:=ji{F) andV:=v(F). For each k e K, compute R :=prem(// 2 v(G /t ) -n(Gk)2V,Ck) and set E:= EU {R}. B5. For each R € E, let F^ be the set of all the nonzero coefficients of R considered as a polynomial in x,y,z. Set
F:=
\j¥R. REM
Then all possible polynomials F satisfying the conditions (i), (j) and (k) are covered by the set of real zeros of F for uijk. Any real zero Hijk such that p =F\Ui k=Ui -k is irreducible leads to a desired F and thus a real irreducible algebraic surface 1S(F).
169
9.4. Blending Algebraic Surfaces
Therefore, the problem is reduced to finding the real solutions of the polynomial equations F = 0 for the unknowns M,-^, which can be done by using Wu's method of characteristic sets or other elimination methods. Example 9.4.1. form
A general cubic polynomial in x,y,z may be written in the
F(x,y,z) = UQ + u\z + u2y + u^x + u^z2 + u^yz + u$y2 + uqxz + u&xy + U9X2 + MIOZ3 + uuyz1 + u\2y2z + uny3
(9.4.1)
+ U14XZ2 + uisxyz + u\kxy2 + mjx2z + u\%x2y + U19X3. For any M, G ^,, F = 0 defines a cubic surface in ^, 3 . Consider two elliptic cylinders £1 and €2 with the x-axis and y-axis as their axes and two ellipses €1 and £2 of section by the planes orthogonal to these axes respectively. The ellipses €\ and £2 are defined by two irreducible ascending sets Ai = [Hi,b\c\Gi] and A2 = \H2,a\c\G2\ with y
Hi=x-dh
z
Gi = - 2 + ^ - 1 ;
xr
H2=y-d2,
7T
G2 = ^ + - ^ - l .
We want to determine such cubic surfaces that contain the ellipses €1,^2 and touch the cylinders £ i , £ 2 smoothly along £\,£2 respectively. For this purpose, we follow the above algorithmic steps. Now the sets / and J of indices are the same, i.e., I = J = {1,2}, and the index set K is empty. In step B2, we get two pseudo-remainders, viz. 1:= {prem(F,Ai),prem(F,A 2 )}. Six pseudo-remainders are formed in step B3; adding them to E, we obtain a set of 8 polynomials in b\,c\,d\,a2,c2^d2, x,y,z and the M,-. Let F be the set of all the nonzero coefficients of these polynomials with respect to x,y,z; F has 48 elements. Computing an irreducible triangular series *P of [F,{b\,c\,a2,C2}] by t r i s y s [its] with respect to the variable ordering b\ -< c\ -< d\ -< a2 < c2 ~< d2 -< MO -< • • • -< t«i9, we find that W consists of 13 irreducible triangular sets, of which one is trivial: [uo,ui,...,u\g]. Among the remaining 12 triangular set there is one whose first polynomial is C = {b]C2dl)2 + (a2blCl)2
- (a2cxd2)2 -
[a2bxc2)2.
The pseudo-remainders of C with respect to all the other 11 triangular sets are 0. It follows that there exists a cubic blending surface for the two elliptic
170
Chapter 9. Symbolic Geometric Computation
cylinders only if C = 0. In this case there is one and only one possible cubic surface, given by b\c\d\£ + a\b\c\c\d2£y - b\c\ (a\c\d\ + b\c\d\)J-\-a^)\c\d\xz2 + 3)b\c\c\d\xy2 — 2a%b\c\c\d\d2xy - b2c\ (b\c\d\ - 2a\c\d\ + a\b2c\) d\x +d\b\c\d2yz2 + ajcfay3 - a\c] (b\c\d\ + a\c\d\)y2 - a\b2c2 (a\c\ - eld2) d2y - a\b\ {b2c\d\ + a\c\d\) z2 + b\ {a\c\d\ + b\c\d\ + a\b\c\c\d\ - alc2cld2d2) = 0. The condition C = 0 generalizes the formula of Wu and Wang (see [101] or [99, p. 368]) for circular cylinders. Taking C2 = 1, b\ =a% = 2, a = di = 3 and d\ — 7, we have 7 A:3 + 21x2y + 63xy2 + 2%xz2 + 243 y3 + 108yz2 - 130.x2 - 3 7 8 x y - H 7 0 y 2 - 5 2 0 z 2 + 539.x:+351v + 3112=:0. This blending surface is plotted together with the two elliptic cylinders and the two planes of section in Fig. 36 (left). For the same cylinders, there are infinitely many quartic blending surfaces, which may be determined by using the same method. The blending surface shown on the right-hand side of Fig. 36 is defined by the following quartic equation: - 1037340A: 4 + 326592x 3 y- 13481271^y 2 - 10258092.x2z2 + 2939328x>'3 + 1306368xyz2 - 6108732/ - 57693580y2z2 - 24434928 z4 + 14135688A 3 - 1580256;t2y-7170912;ty2 + 56542752A:z2 + 36652392y3 + 16289952yz2 + 65 1 9 7 4 4 A-2 - 37642752xy + 37642752A: = 0.
Fig. 36. Blending two elliptic cylinders
171
9.4. Blending Algebraic Surfaces
Example 9.4.2. Let two elliptic cylinders be given by Gl =
v2 z2 T + ?"1
= 0
'
x2 G2 = - + Z 2 - I = O
as in Example 9.4.1. We search for a third elliptic cylinder of the form 2
2
G3 = — + — - 1 = 0 a c and quartic surfaces in ^ defined respectively by Ai=[*-7,36Gi],
(a > 0, c> 0)
to blend these three cylinders along the ellipses A2 = ty-3,4G 2 ],
A3 = \y + 3,acG3].
Now the quartic polynomial F contains terms M2OZ4 H 1- M34X4 of degree 4 to be determined, in addition to the terms of lower degree listed in (9.4.1) and the set F obtained as in step B5 consists of 101 polynomials in a,c,uo,...,U34. With respect to the variable ordering a^c
^
U34,
an irreducible triangular series of [F, {a,c}] computed by t r i s y s [its] consists of three triangular sets, including the trivial one [UQ, ... ,1/34]. Another triangular set contains 2 c + 47, so it does not have any zero with c > 0. The remaining triangular set contains a —A and c— 1. Therefore, there exists a quartic blending surface for the three elliptic cylinders only if a = 4 and c = 1, i.e., G2 = G3 (or in other words the third cylinder must be identical to the second cylinder). In this case, the family of quartic surfaces which may blend the elliptic cylinders is given by (2240x4 - 45360x2y2 - 20160xV + 181440/ - 181440y2z2 - 116480z4 + +
38080x 2 -3084480y 2 + 14813120)w0+(265304x4 + 2900457xy 2347172xV + 6 3 5 0 4 / + 11601828/z 2 + 5143824 z4 - 3703280x3 14813120xz2 + 224740x 2 -12744900y 2 +14813120x)w 3 +(2016x 4 874062x2y2 - 388472x2z2 + 163296/ - 3496248y2z2 - 1586144z4 3298680x2 + 5 5 6 9 2 0 / + 14813120z2)M4 = 0 ,
in which 110,113,114 are parameters. Two of the blending surfaces for UQ = 0,^3 = —1,^4 = 7 and UQ = 0,u$ = \,u\ = 0 are plotted in Fig.37. One sees that the blending surface on the right-hand side is not connected. How to choose suitable values for the free parameters to get a reasonable blending surface is a difficult question that cannot be discussed here.
Chapter 9. Symbolic Geometric Computation
172
Fig. 37. Blending three elliptic cylinders
9.5
Decomposing Algebraic Varieties
Algebraic varieties are geometric objects defined by sets of polynomial equations. Decomposing algebraic varieties into irreducible components is a fundamental construction in computational algebraic geometry which has applications in modern geometric engineering. For example, in computer-aided geometric design, it is desirable to decompose the considered geometric objects such as algebraic curves and surfaces into simpler subobjects. In automated geometric theorem proving (see Chap. 8), the configuration of the geometric hypotheses of a theorem may have to be decomposed in order to decide on which components the theorem holds true. Definition 9.5.1. For any triangular set T C K[x], the saturation of T is the ideal sat(T) := {P € %[x]: JqP € Ideal(T) for some integer q > 0}, where/ = nreT'iK^)Let/i,...,/, be i factors of J such that/1 •••/, ^ 0 if and only if J / 0, let F = TU{z,/,-l| l
173
9.5. Decomposing Algebraic Varieties
Therefore, one can compute a finite basis, called a saturation basis, for the ideal sat(T) by means of Grobner bases. Now let T i , . . . , Te be a (weak-) characteristic series or a regular series of P in $C[x}- It is not difficult to prove (see Theorem 6.2.8 in [78]) that e
Zero(P) = ( J Zero(sat(T,-)).
(9.5.1)
i=i
For each i let P,- be a finite basis for sat(T,), which can be determined by computing a Grobner basis. If sat(T,) = 7([x\, then the constant 1 is contained in (the Grobner basis of) P,-. Let us assume that such P,- has been removed. Then, a decomposition of the following form is obtained: e
Zero(P) = [ J Zero(P,).
(9.5.2)
i=i
An algebraic variety is the set of zeros of a polynomial set in !K\x], considered as points in ^-dimensional affine space. A variety is said to be unmixed or equidimensional if all its irredundant irreducible subvarieties have the same dimension. It has been proved in [25] (see also [78, p. 161]) that each Zero(P,-) in (9.5.2), if nonempty, is an unmixed variety of dimension n — |T,-|. The decomposition (9.5.2) may be contractible; that is, some variety may be a subvariety of another. Redundant subvarieties may be removed by using various techniques (see [15] and Sect. 6.2 of [78] for details). An algorithm for decomposing any algebraic variety into unmixed components as explained above has been described as UnmVarDec in [78]. Here we are more interested in decomposing an arbitrary algebraic variety defined by a polynomial set into a family of irreducible subvarieties. This is done with an analogy to the unmixed decomposition of P, requiring additionally that the characteristic series is irreducible; then any finite basis for sat(T), T € O, will define an irreducible variety with any generic zero of T as its generic point. For any irreducible triangular set T c !K[x], sat(T) is a prime ideal (see Theorem 6.2.14 in [78]). In this case, any finite basis for sat(T) is also called a prime basis of T and denoted by PB (T). Let the T, in (9.5.1) be all irreducible. Then we have the following variety decomposition: e
Zero(P) = UZero(PB(T ; )). i=i
Now, each PB(T,) which can be exactly determined by using Grobner bases defines an irreducible algebraic variety and we have thus accomplished an irreducible decomposition of the variety defined by P.
Chapter 9. Symbolic Geometric Computation
174
This decomposition is not necessarily minimal. All the redundant subvarieties can be removed (see [98, p. 181] and Lemma 6.2.13 in [78]), so one shall finally arrive at a minimal irreducible decomposition. The above discussions may be summarized as follows. Given a nonempty polynomial set P C %\x], one can compute a finite set *F of polynomial sets P i , . . . ,P e according to the following steps such that the decomposition (9.5.2) holds, it is minimal, and each P,- defines an irreducible algebraic variety. 11. Compute an irreducible characteristic series 4> of P (e.g., by charsets [ics] ortrisys[its])andset:={Te
I P - I F — — —X \
' dx' dy' dz j
is the set of all singular points of the surface. With x -< y -< z, P may be decomposed by using charsets [ics] into three irreducible ascending sets Ci, C2 and C3 such that Zero(P) = Zero(Ci //) U Zero(C2) U Zero(C 3 ), where Ci = [C,D], C2 = [15625 y2 + 15625 x2 - 49,2500z + 423], C3 = [x,y, 32000z3 - 76500z2 - 30240z - 2781];
9.5. Decomposing Algebraic Varieties
175
C = 4096/ + 12288X2/ + 127117107/+ 12288// + 254234214// + 98973163953/ + 4096*6 + 127117107 x4 +98973163953x2+116700507, D = 2560/z - 23386807616465120809010141553/ -46773615232930241618020283106/y2 -18720334565178332908780355773275/ - 23386807616465120809010141553*4 -18720334565178332908780355773275x2 - 22073421407845867557135008553; / =1623097324004360776837843/ + 3246194648008721553675686;// + 1294518690743377529625696657/+1623097324004360776837843x4 +1294518690743377529625696657/+ 1526302072638384385676283. The ascending sets C2 and C3 already define two irreducible subvarieties 1^ = Zero(C2) and lA, = Zero(C3) of 'U. To obtain an irreducible decomposition of 'U, we determine a prime basis of Ci by computing the Grobner basis G of Q Li{tl — 1} with respect to the purely lexicographical term order induced by x -
(0,0,-0.1778141491 ±0.005245561338V'=:T)-
Example 9.5.2. The reader may find an irreducible decomposition of an algebraic curve in Example 1.4.1. Here we consider another curve defined
176
Chapter 9. Symbolic Geometric Computation
by
{
z2 + 3xz — z — 2y2 + xy — Ay — x, z3 + 6yz2 + 8z2 - 2y2z - 4 vz - 6z - 12y3 - 23 y2 - 2y + 1
which is the intersection of two irreducible algebraic surfaces in threedimensional affine space. The two polynomials have appeared in Examples 9.2.2 and 9.3.2. With the variable ordering x -< y -< z, this curve may be decomposed into three simpler irreducible components defined by P 1 = {v + 2,z-l}, P2 = {17y+l,17z-6}, r3 = {2x2y-y-xi + 3x2+x-2,z-2xy + x2-l}; the first two are lines parallel to the x-axis and the third is a cubic curve. These curves around the origin are plotted in Fig. 38.
Fig. 38. An algebraic curve having three components
Chapter 10 Selected Problems in Computer Mathematics
Symbolic elimination methods provide constructive tools not only for algebraic geometry, but also for polynomial-ideal theory and many related problems in different areas of computer mathematics. Our elimination tools have natural applications in all such areas. In this chapter we consider some of the applications, more concretely to the problems of computing with polynomial ideals, factorizing multivariate polynomials, analyzing differential equations qualitatively, and reasoning about curves and surfaces automatically in differential geometry.
10.1
Computation with Polynomial Ideals
Polynomial ideals are fundamental mathematical objects studied in commutative algebra. The method of Grobner bases provides an essential tool for symbolic computation with polynomial ideals. This method has been implemented in all major general-purpose computer algebra systems including Maple, Mathematica, MuPAD, and Magma for diverse applications and in several specialized systems like Macaulay 2, Singular, and CoCoA for computations in commutative algebra and algebraic geometry (see Sect. 6.2). These implementations allow us to perform various operations and computations with polynomial ideals such as: • • • • • •
sum, product, and intersection of ideals, power, dimension, and radical of an ideal, ideal quotient, saturation, and membership test, minimal (associated) primes of an ideal, top dimensional components or primary decomposition of an ideal, Hilbert functions and polynomials, syzygies, and free resolutions.
178
Chapter 10. Selected Problems in Computer Mathematics
An introduction to the involved concepts and algorithms for these operations and computations is beyond the scope of this section. The reader is directed to standard textbooks such as [4, 18] and documents for the above-mentioned specialpurpose computer algebra systems. We advise the reader to use one of these systems if he or she needs to do extensive computations with polynomial ideals. The functions implemented in the CPSILON modules mainly deal with sets of zeros (varieties or quasi-varieties) instead of polynomial ideals, so most of the applications considered in this book are concerned only with zeros of polynomial systems. However, there are three kinds of computations which can be carried out with functions from the CPSILON library: (i) radical-ideal membership test, (ii) prime decomposition of the radical of a polynomial ideal, and (iii) primary decomposition of a polynomial ideal. Function t r i s y s [rim] is implemented for (i), but the test can alternatively be done by several other functions (see Sect. 6.3 in [78]). As the reader may be aware, the algebraic decision problem for geometric theorem proving addressed in Chap. 8 is essentially the problem of testing radicalideal membership. An irreducible decomposition of an algebraic variety (by ivd) leads to a prime decomposition of the radical of the corresponding polynomial ideal. From a prime decomposition of the radical, one can construct a primary decomposition of the ideal by means of localization and extraction according to [55]. Function charsets [pid] is an implementation of this construction. In what follows, we present a few examples to illustrate the application of these functions. Example 10.1.1. The following four polynomials Px = 9v3 - 2 7 x y 2 - 6v2 - 36x2y-6xy 2
+ y-9x3
2
P2 = 9xz + 6z+lSy -63xy-l2y-27x
+ x,
+ 3x + 2,
/>3 = 9 v z - 3 z - 9 ; y 2 + 45xy + 6 v + 1 8 . x 2 - 3 x - l , P4 = 9z2-l2z-45y2+l62xy
63x2-6x-5,
+ 30y +
have appeared in Example 1.4.1, where P = {P\,... ,P4} is the set of defining polynomials for an irreducible algebraic curve. With respect to x -< y -< z, the polynomial set {P2,P3} may be decomposed by t r i s y s [ t r i s e r ] into three triangular systems [T,-,U,-] with Ti = [Fi,P2],
U 1 = { 3 x + 2}, 2
T 2 = [3x+2,3y + 5 y - 2 , z - v - 3 ] , T3 = [ 3 x + 2 , 3 v - l ] ,
V2 = {3y-l}, U3=0,
such that 3
Zero({P 2 ,^}) =
\JZero(Ti/Vi).
10.1.
179
Computation with Polynomial Ideals
It is trivial to verify that prem(Pi,T,) = 0 for / = 1,2,3. It follows that Pi vanishes on Zero({P2,P3}), so Pi belongs to y/ldeal({P2,P3}), the radical of the ideal generated by P2 and P3. As a consequence, we have Zero({PuP2,P3})
= Zero({P2,P3}),
Zero(P) = ZeTo({P2,P3,P4}).
This shows that the first polynomial Pi is redundant and may be removed from the defining set P in Example 1.4.1. Using function t r i s y s [rim], one can directly verify that Pi e ^ldcal({P2,P3}),
Pi £
^/Idcal({P2,P3,P4}),
P2 $ ^ldeal({PuP3,P4}),
P3 $
y/ld^l({PhP2,P4}),
P4?^ldeal({Pi,P2,P3}). Therefore, none of the polynomials P2,P3,P4 can be removed from the generating set P. Example 10.1.2. Let 3 be the ideal generated by the following set of polynomials: ' Xl +X2 +
X3+X4,
p _ J XlX2 + XiXA + X2X3 + X3X4, J X\X2X3 + XlX2X4 + XiX3X4 + X2X3X4,
( '
_ XiX2X3X4 - 1
System P = 0 is the well-known "cyclic 4 roots." Under the variable ordering xi -< • • • -< X4, it is trivial to decompose P (e.g., using t r i s y s [ i t s ] ) into two polynomial sets P i — [xiX2- l,X3+Xi,X4+X2],
P 2 = [xiX2+
l,X3+Xl,X4+X2],
such that V3 = Ideal(Pi) n Ideal(P2), where both Ideal(Pi) and Ideal(P2) of dimension 1 are prime. It is interesting to see that the ideal 3 may be decomposed by charsets [pid] into eight primary ideals 3,- = Ideal(F,) such that 8
a=n^=n 1=1
8 ideai
(^)-
1=1
The generating sets F,- and the corresponding sets P,- of generators for the primes associated to 3, are listed below: Fi=Pi,
F2=P2;
180
Chapter 10. Selected Problems in Computer Mathematics F3 = [xi - l,x2-l,xl
+ 2x3 + \,x4+x3
+ 2],
P 3 = [ * i - l , X 2 - l , * 3 + l,X4+l]; F4 = [x\ - 1,^2 + 2x2+ 1,X3+X2 + 2,X4- 1], P 4 = [x\ - 1,X2+1,X3 + 1,X4- 1]; F 5 = [xi + l,x 2 + \,x\ - 2x3 + 1,*4 + x 3 - 2], F 5 = [xi + l , X 2 + l , x 3 - l , x 4 - l ] ; F6 = [x\ + 1,X2-2X2+1,X3+X2-2,X4+ 1], F6 = [xi + l , x 2 - l , x 3 - l , x 4 + l ] ; F7 = [Xj + l,X2 + 2xiX2~ l,X3+X2 + 2 x i , X 4 - X i ] , F7 = [Xj + l,X2+Xi,X3+Xi,X 4 -Xl]; Fg = [xj + l,X2-Xi,X3 + 2xiX3 - l , 2 x i +X3+X4], Pg = [ x ^ + l , X 2 - X l , X 3 + X i , X 4 + X l ] .
The primary ideals 33,...,3s are all embedded components. Example 10.1.3 (Example 7.4.4 in [78] or A49 in Appendix A). For the univariate quartic polynomial F = x4 + xix3 + X2X2 + x3x + X4 with indeterminate coefficients xi,X2,x3 and X4, the discriminant A of F, A = resultant(F,3F/3x,x), is an irreducible polynomial of total degree 6, consisting of 16 terms. A = 0 defines the discriminant surface of F in four-dimensional affine space. Consider the singularity ideal generated by f 3A 3A 3A 3A1 \ ' 3xi' 9x2' 9x3' 3*4 J With respect to xi -<•••-< X4, IP may be decomposed by charsets [pid] into two polynomial sets Pi = [G?,G2,G3,G4], P 2 = [8x3 -4xiX2+Xj,64x4- I6X2 + 8x^x2 -Xj], such that Ideal(P) = Ideal(Pi) nldeal(P 2 ),
10.2. Factorization of Polynomials
181
where both Ideal(Pi) and Ideal(P2) of dimension 2 are primary and Gi = 108x^ - 108xix2X3 + 27x^X3 + 32x| - 9x\x\, G2 = 72X2X4 — 27XjX4 — 27Xj + 9xiX2X3 — 2X2,
G3 = 62208x^X4 - 7776x^x3x4 + 243xfx4 - 38880xix^ + \2Q96x\x\ + 23916x\x2x\ - 3645x\x\ - 13824xix2,x3 + 2 3 7 6 x ^ x 3 - 81x^x2x3 + 2048X25 - 384xfx^ + 18x|x^, G4 = 3456x^ - 1728xix3x4 + 81x^X4 + 216x2x^ + 297 x\x\ — 216x1x^x3 — 27x^X2X3 + 40x2 + 6x\x\The prime ideal associated to Ideal(Pi) is generated by Gi and 12X4 — 3xiX3 +X2,
while Ideal(P2) itself is prime. One may verify that Pi is a plex Grobner basis for the ideal generated by G2 and G4, so Ideal(Pi) = Ideal({G2,G4}). Therefore, the big polynomials G\ and G3 may be deleted from Pi if the generating set is not required to be a Grobner basis.
10.2
Factorization of Polynomials
Factorization of polynomials is an elementary problem in mathematics that is easy to formulate and that may be solved theoretically without difficulty. The problem is, however, computationally hard when one is concerned with how to factorize concrete polynomials of high degree in many variables over the field of rational numbers or its extensions. The development of effective modern algorithms for polynomial factorization is one of the most remarkable achievements of computer algebra. The reader may be impressed by the high efficiency of these algorithms implemented in popular computer algebra systems like Maple, which can be used to factorize various polynomials of moderate size in a matter of seconds. Despite this great success, factorization of polynomials over algebraic extension fields remains a time-consuming task. Elimination methods based characteristic or triangular sets may be used to devise algorithms for factorizing polynomials over successive algebraic extension fields. Two such algorithms, named FactorA and FactorB, have been described in [78] and implemented as cf actor in the CharSets module of 6PSILON. A successive algebraic extension field of Q. m a v De determined by an irreducible ascending set A = [A\,..., Ar] as follows. Let y,- be the leading variable of A, for 1 < i
182
Chapter 10. Selected Problems in Computer Mathematics
in A. Let ^Co = QXU\i---iud) denote the extension field obtained from Q,by adjoining the transcendental elements ui,...,Ud and assume that one knows how to factorize polynomials over ^CoLet (r|i,... ,ri r ) be a zero of A for (yi,... ,yr) in some extension field of JCoThen the algebraic extension field obtained from 3Co by successively adjoining the algebraic elements r\i,... ,~r\r is uniquely determined by A We denote this field by 5C o (y 1, • • •, yr)» when the adj oining ascending set A for y i,..., yr is explicitly given. The algorithm FactorB uses linear transformation and characteristic sets computation to reduce polynomial factorization over algebraic extension fields to that over the field of rational numbers. The following two examples serve to illustrate the basic idea underlying this algorithm. Example 10.2.1 ([64] or (B.5) in Appendix B). Factorize F = \6w2y —U2 — 2iqu2 — 2u2 — u\-\-2u~{ — 1 over Q(u\,U2,yi) with y\ having adjoining (minimal) polynomial A = y\ - u\y\ - u\y\ -y\ + u\. First, we make a linear transformation y 4— y + cy\ (so that the polynomial to be factorized involves the variable yi), where c is a randomly chosen integer. Taking c = 1 for instance and substituting y in F by y + y\, we have F
=F\y=y+yi = \6u\y2 + 32u^y\y + \6u2y\ — U2 — 2u\u\ — 2u\ — u\ + 2u\ — 1.
Next compute a characteristic set C of {F,A} with respect to the variable ordering y -< y\. It is found that C = [Ci,C2], where C\ factors over Q, into {Dl+D0)(Dl-D0) with £>i = 256i4y4 - 32u\ (9u\ + \0u\u\ + \Qu\ + u\- 2u\ + \)y2 -\5ul-AA(u\+\)u\-6{lu\-l>%u\ - \2{u\DQ = -\2%U\{U\
+ l)u\
u\- u\ + \)u\ + u\- Au\ + 6u\-4u\ + u\ + 2M 1 + 1) {u\ + u\ - 2wi +
+ \, 1)y.
Let us take the first factor of Ci and substitute y back by y — y\. The result is a polynomial D consisting of 50 terms in m,U2,yi, and y. Now we need to find a GCD of D and F over Ql.u\,uz,y\). This may be done by computing a characteristic set C* of {D,F,A} with respect to the ordering y\
183
10.2. Factorization of Polynomials
polynomial CJ, consisting of 32 terms in m,u2,yi,y and linear in y, is a true factor of F over Q(ui,U2,y\). This factor is somewhat complicated. Normalizing C* using miscel [normat], one gets a much simpler factor F\ of F over QSui,u2,y\): F\ = Au2y -2y\ + u\ +
u\+l.
Finally, removing this factor from F, one obtains the other true factor
F2 = pquo(F,Fi,y)/(l6i4) = 4u2y +
2y\-u\-u\-\.
Therefore, F is factorized as the product F\F2 over
Q(ui,u2,y\).
In the following example, the field JC is defined by successive algebraic extensions, where the adjoining ascending set contains more than one polynomial. Example 10.2.2 (Problem 7 in [65] or B23 in Appendix B). Factorize F = — lOy3 — 370y2y2 + 60yiy2 + 4y2y + 2uyiy + 74uyiy2 — 2Ay\y2 - 37y2 + 37uyi +
l2u3-24u
over 9C= Q,(u,yi,y2) with yi,y2 having adjoining ascending set
A=
\y2l+u2-2,4yj+y2-uyl].
Computing a quasi-characteristic set C of AU{F} with respect to the variable ordering y ~
which is of degree 8 in y, and has fewer terms and smaller coefficients than the other factor of degree 4 in v. Now compute a characteristic set C* of AU {F,C} with respect to y\ -< y2 -
184
Chapter 10. Selected Problems in Computer Mathematics
Therefore, F is factorized over %^ as 2F\F2Our algorithm FactorA works according to the principle of undetermined coefficients: writing the polynomial to be factorized as the product of two polynomials of lower degree with indeterminate coefficients «, and comparing the corresponding coefficients in the two sides of the equality (modulo the irreducible ascending set that defines the algebraic extension field), one shall obtain a set of polynomial equations in the M,. Then, the given polynomial can be factorized as the product of these two polynomials if and only if the set of polynomial equations has a solution in the field of rational functions of the transcendental elements. The latter may be determined by solving the polynomial equations, and one solution leads to one factorization. The two algorithms have been applied to a number of example polynomials collected from the literature or arising from geometric applications. Fifty five of them are listed in Appendix B. The reader may try to factorize these polynomials using charsets [cfactor]. For this function, an irreducible ascending set that determines the successive algebraic extension field has to be given (of course). However, the method of undetermined coefficients may also be used to search for an algebraic extension field over which a given polynomial can be factorized. We illustrate this point with the following example. Example 10.2.3. Let us recall the polynomial R in Example 9.1.2 and explain how to obtain the factorization of R given therein. Suppose that R may be factorized as the product of two polynomials of degree 2 in X and Y, viz., R = M3 (X2 + Y2 + xiXY + x2X + x3Y + xA) (X2 + Y2 + x5XY + x^i. + x7Y + x 8 ). (10.2.1) Expanding the products and comparing the coefficients with respect to X and Y of the two sides of (10.2.1), one obtains a set of 12 polynomial equations in UI,U2,UT, and x\,...,x%. We want to see when this set of equations has a solution for x\,... ,x%. For this purpose, compute a characteristic series *F of the corresponding set of 12 polynomials with respect to the variable ordering x\ -< • • • -< x%, for example, using charsets [qics]. It is found that *F consists of a single ascending set C = [x\,x\-\-2u\X2
- $ + U\U2,U?,{x2 + U\) XT, + $X2 + U2$ + U\u\,X4,
X5,X6 +X2 + 2 M I , « 3 (X2 + U\)xj + (3x2 + {u\ - U2) (P - «lH2),*8],
where P = u2 + w2 — u\U2- All the polynomials in C, except the second of degree 2 in X2, are linear with respect to their leading variables. Therefore,
185
10.3. Qualitative Study of Differential Equations
in the extension field Q(u\,U2,Ui,a), where a = Ju\ + (M2 — Mi)2) the ascending set C has two zeros for 0,-ui + a,--
(xi,...,x%):
B + U70,
B — U20. n ,
B - mv.
B + «20c „ 1
,0,0,-HI-a,--
0,-m-a,-H
,0 ,
,0,0,-m + a,-^——,0 ). U-i
W3
Both of them lead to the same factorization R = Ui Xz +
Yz-(ui-a)X-
B + a2a
Xz + Yl-(ui
+ a)X-
M3
B - «20c M3
The two factors on the right-hand side have been written as /?i and R2 in Example 9.1.2.
10.3
Qualitative Study of Differential Equations
For plane differential systems of center and focus type dx . . — =y + P(x,y),
dy -£ =
„. . -x+Q(x,y),
(10.3.1)
where P(x,y) and Q(x,y) are polynomials beginning with terms of total degree > 1 in x and y with coefficients involving parameters u = (u\,...,ue), function miscel [licon] can compute polynomials V3, V5,...,V2j+\, • • • 6 Q[u] such that the differential of a locally positive polynomial L(x,y) € Qju,x,y] along the integral curve of (10.3.1) is of the form d
J^A=V3/
+ V5f
+
...
+ V2j+iy2J+2
+
...,
where V2j+\ is called the y'th Liapunov constant or focal value of (10.3.1). The origin (0,0) is a critical point of (10.3.1). It is said to be a center for (10.3.1) if V 3 = V5
:
V2j+1
0.
(10.3.2)
Since condition (10.3.2) is given by infinitely many equalities (in a finite number of variables), by means of computing the Liapunov constants V2j+i — 0,
186
Chapter 10. Selected Problems in Computer Mathematics
j = 1,2,..., one can only establish the necessary conditions for the origin to be a center. The sufficiency of these conditions has to be proved separately by using sophisticated mathematical techniques. Liapunov constants and center conditions are required in the study of several problems such as distinguishing between a center and a focus, searching for fine foci of high order, and constructing limit cycles (which is closely related to Hilbert's 16th problem) in the qualitative theory of differential equations. To derive center conditions and to study the problems just mentioned, one needs frequently to solve a system of polynomial equations, to determine whether a polynomial relation follows from a system of polynomial equations and inequations, and to simplify a polynomial using a set of polynomial relations. Elimination techniques and tools based on triangular sets, Grobner bases, and resultants may be used to handle these computational tasks effectively (see [60, 76]). The problem of deriving necessary center conditions may be reduced partially to simplifying, decomposing, or solving large polynomial systems. The derivation has proved to be thorny and intractable because the occurring polynomials are often very large in terms of degree and number of terms. Computationally, one takes a suitable N, computes the Liapunov constants V2j+i, forms the polynomial set P={v3,V5,...,V2JV+l},
and simplifies or solves P = 0 to obtain the necessary conditions for the origin to be a center. In what follows, we present two examples to show how necessary center conditions may be derived in this way. Example 10.3.1 (Saharnikov system [52, 56, 48, 60, 47]). Consider the class of cubic differential systems without quadratic terms, i.e., P{x,y) and Q(x,y) only involve terms of total degree 3 in x and y. Following [52] and to simplify computations, we suppose that the system under consideration has been transformed into the form dx — =y+Bx3 + (C-G)x2y
+ (3D-H)xy2
+ Ey3,
dt
$ = -x-Ax3 dt
- (3B + F)x2y - (C + G)xy2 - D y 3 .
(10.3.3)
Saharnikov [52] proved that the origin is a center for (10.3.3) "if and only if" A,...,H satisfy one of the following four sets of conditions: F = G = H = 0;
(SI)
F=H=B=D = 0;
(S2)
187
10.3. Qualitative Study of Differential Equations
ri=F+H = 0, r2=F(A-E)+2G(F
(S3) + 2B + 2D) = 0,
r1=o, r 2 = o, T3=3(A+E) 2
+ 2C = 0, 2
r4=C(F -4G )-6FG(B-D) T5=F[2F 2
+ 5(B+D)]-G[2G 2 3
[
= 0,
'
+ 5(A- E)] = 0, 2
r6 = (F + 4G ) -25[(B-D)(F -4G2)-4FG(A+E)]2
= 0.
We explain how to derive (necessary) center conditions of this type from Liapunov constants, which may be computed by miscel [licon]. The first Liapunov constant for (10.3.3) is V3 = (F +H)/3. It is necessary to compute the other Liapunov constants of higher order only if H = —F. Let this condition be assumed. Then the next six Liapunov constants V5,...,vi5 computed by miscel [licon] consist of 5, 20, 65, 162, 347, and 680 terms respectively. With respect to the variable ordering F~
P1 = [F,G],
v2 = [ni,-,ck],
P3 = ["5,r2,o6]
such that Zero(P) = Zero(Pi) UZero(P2) UZero(P 3 ), where each Zero(P,-) as an algebraic variety is irreducible and Qi = 9(10D+F)
(WD + 3F)+4(25C2
-9G2),
Q 2 = 15A + 5 C + 3 G , Q.3=5B + 5D + 2F, Q 4 = 15£ + 5 C - 3 G , Q.5=£l7B + 2AFG + DF2, Q.6 = 8BE2 + 8{AD-AB-BC)E + S[ABG+{A-G)(CB-AD)} Q.7=SG2-F2, A=A-C
+ G.
+ 4(4B + 4D + + 2AFG +
3F)(D2-B2) F2(D-B);
188
Chapter 10. Selected Problems in Computer Mathematics
Therefore, the origin is a center for (10.3.3) only if (SI) or one of the following conditions holds: r i = Q i = --- = Q 4 = 0;
rl=r2
(10.3.4)
= Q.5 = a6 = o.
(10.3.5)
Here we add a remark: instead of P3, included in the output of ivd is the plex Grobner basis of P3, that contains another polynomial of 12 terms. Similarly, the plex Grobner basis of the polynomial set F below (under the same variable ordering) contains two more polynomials. The removal of these additional polynomials makes the polynomial set smaller but does not change its zero set (cf. Example 10.1.1). An irreducible triangular series of P3 (computed by t r i s y s [ i t s ] ) with F
T 3 = [ns,r 2 ],
T^ = [F, G, £2], T 4 = [Q7,ag,Q9],
where Q is a polynomial consisting of 12 terms in A,B,C,D,E Q.S=F2 + S(A-C)G 2
Q9=F
+ 8(C-E)G
and
+ 4FD, + 4FB.
Note that Zero(Ti) C Zero(Pi) and Zero(T'1) C Zero(Pi). Therefore, one of (SI) and (10.3.5) holds if and only if (SI), or (S2), or one of the following conditions holds: Ti = T2 = Q 5 = 0, FQ.7 ^ 0; (10.3.6) T i = 07 = ^8 = ^9 = 0,
G^0.
(10.3.7)
It may be easily verified that • (10.3.6) or (10.3.7) holds if and only if (S3) holds; • Zero(Pi)UZero(P 3 ) =Zero({Q 5 ,r 2 }). Therefore, the following three conditions are equivalent: (a) one of (SI), (S2), and (S3) holds; (b) one of (SI) and (10.3.5) holds; (c) r i = r 2 = Q 5 = 0.
(10.3.8)
189
10.3. Qualitative Study of Differential Equations
In other words, the first three sets of conditions of Saharnikov are equivalent to the two sets of conditions (SI) and (10.3.5), which are equivalent to the single set of conditions (10.3.8). It remains to establish the relationship between (S4) and (10.3.4). The polynomial set {Ti,...,!^} may be decomposed into three polynomial sets defining three irreducible varieties. Each of them is a true subvariety of one (and only one) of the varieties Zero({ri}UP,-), i= 1,2,3. The one contained in Zero({ri} UP2) may be defined by F=[r1,©1,02,Qi,...,n4], where
0 i = 25 (F2 + 4G2)C2 - 36F2G2, 0 2 = 12FG(5Z) + F ) + 5 C ( F 2 - 4 G 2 ) .
Note that Zero({ri} UP2) is of dimension 3, while Zero(F) is of dimension 2. Clearly, the conditions given by Zero({r 1 } UP 2 ) \ (Zero(F) UZero({ri,r 2 ,Q 5 })) are not covered by Saharnikov's conditions. For example, if A and B satisfy A2 + ( B + l ) 2 = i ,
A^O,
B#-l,
and C=-3A,
D = -(B + 2),
E=A,
F = 5,
G = 0,
H = -5,
then (A,...,H) is a zero of -{Ti}UP2 but not a zero of F, nor a solution of (10.3.8). On the other hand, both V13 and V15 vanish on Zero(P,) for / = 1,2,3, so V13 = 0 and V15 = 0 are formal consequences of V3 = • • • = vn = 0 . Thus V3 = • • • = V15 = 0 does not imply that one of (S1)-(S4) must hold. This suggests that Saharnikov's conditions are incomplete. It is indeed so, the incompleteness of Saharnikov's conditions was discovered by Sibirskii [56] in 1965. The author pointed out this incompleteness independently in his Ph.D. thesis in 1987. System (10.3.3) was investigated also by several other authors (see [48, 47] for instance). It is easy to verify that (10.3.8) and (10.3.4) are equivalent to the conditions (2) and (3) in Theorem 4.1 of [47] (while (1) given therein is contained in (2) and thus is redundant). According to this theorem 4.1 of [47] (see also [56, 48]), the conditions (10.3.8) and (10.3.4) are also sufficient for the origin to be a center for (10.3.3), so V3 = • • • = vn = 0 imply V2j+i = 0 for all j > 5. Making use of this result, we have the following equivalent conditions for system (10.3.3):
190
Chapter 10. Selected Problems in Computer Mathematics
(1) the origin is a center; (2) v3 = - " = v n = 0 ; (3) one of (10.3.8) and (10.3.4) holds; (4) one of (SI), (10.3.4), and (10.3.5) holds; (5) one of (SI), (S2), (S3), and (10.3.4) holds; (6) one of (SI), (S2), (10.3.4), (10.3.6), and (10.3.7) holds. Finally, we reproduce the simplest condition (3) below for ease reference. Theorem. For Saharnikov's system (10.3.3), the origin is a center if and only if one of the following two sets of conditions holds:
F{A-E)+2G(F 2
2
(&G -F )B
+ 2B + 2D) = 0,
+ 2(A-C
+ G)FG + DF2=0;
f F+H = 0, 5B + 5D + 2F = 0, 5C+15A + 3G = 0, 5 C + 1 5 £ - 3 G = 0, 9(10Z) + f)(10£) + 3/;') + 4 ( 2 5 C 2 - 9 G 2 ) = 0 . Example 10.3.2 (Kukles system [36, 12, 33, 17, 42, 51, 76]). Another example we want to consider is the classical system of Kukles [36], which has been studied in a number of papers after [33] in the last decade. The so-called Kukles' system is the particular case of (10.3.1) with P(x,y) = 0, Q{x,y)= a2ox2 + a i ixy + a§2y2 + a30x3 + a2 \x2y + a \ 2xy2 + amyi.
(10.3.9)
For this class of systems Kukles [36] proposed necessary and sufficient conditions for the origin to be a center. Kukles' conditions may be found in standard textbooks (see, e.g., [45, p. 124]). Here we reproduce two sets of the conditions: a03 = 0, Ki =aii(a 3 0 aii+a2iao2) = 0, K2 = a n («2i«o2 + a 2i«i2 - anaoiau) = 0, , K3 = a02an +a2oan +«2i = 0;
191
10.3. Qualitative Study of Differential Equations
003 = 002 = «20 = «21 = 0.
(K4)
Research interest in Kukles' system was stimulated in the 1990s by the work of Jin and the author [33] (see also [17]), suggesting that Kukles' conditions are incomplete. This incompleteness was discovered first by Cherkas [12]. Instead of Kukles' first set of conditions, Cherkas derived the following conditions: 002011 + 3 ^ 0 3 + 0 2 0 0 1 1 + 0 2 1 = 0 , 6 0 2 0 0 0 3 + 0 2 0 0 1 1 0 0 2 - 0 2 1 0 0 2 - 0 1 1 0 1 2 - 2 ^ 3 0 0 1 1 ~ 9011 = 0, \
6a 3 O 0O3 - 302Q0O3 + 030011002 + 020002021 + 020012011 2 2
. . (^1/
- 021012 - 030021 - J 0 H 0 2 1 = 0, 030021002-6020030003 + 0 3 0 0 1 1 0 1 2 + 0 2 0 0 2 1 0 1 2 - §011021 = 0, _ 030021012 - 303O0O3 - §021 = ° ,
which contain the example of conditions given in [33]. To see how these conditions for Kukles' system may be derived, let us compute the Liapunov constants V2;+i using miscel [licon]. The first one is V3 = ( 0 0 2 0 1 1 + 3 0 0 3 + 0 2 0 0 1 1 + 0 2 l ) / 3 ,
which should necessarily be 0. To simplify calculations, we make a substitution 021 = -(3003 +011002 + 011020)
(10.3.10)
in (10.3.9). Then V3 becomes 0 and the next eight Liapunov constants computed by miscel [licon] are characterized as follows: Table 13. Liapunov constants Liapunov const.
V5
VJ
No. of terms Total degree Max LIC
13 4 2
49 6 4
v9 131 8 6
Vll
V13
V15
V17
V19
292 10 9
577 12 13
1046 14 17
1775 16 22
2859 18 27
In Table 13, MaxLIC stands for "maximum length of integer coefficients." These polynomials in 020,011,002,030,012,003 are rather large. Now the question is how to simplify the condition given by V5 = • • • = V2W+1 = 0 and to examine their relationship with the known center conditions. Consider the special case 003 = 0; then vs,...,vi9 simplify to vs,...,vi9 in ^20, 011, 002, 030, 012, consisting of 7, 26, 65, 134, 245, 412, 651, and 980 terms, respectively. With respect to the variable ordering «02 -< 012 -<
Chapter 10. Selected Problems in Computer Mathematics
192
011 -< 020 -< 030, the polynomial set P = {vs,..., vn} may be decomposed by charsets [ivd] or t r i s y s [ivd] into three polynomial sets Pi = [an], P 2 = [002,020], P 3 = [(212020 + 2ai2«02 + 02O«O2 + a 0 2 ' a 3 0 ~ 002 ~ a 02«2o],
such that 3
Zero(P) = (JZero(P ; ), 1=1
where each Zero(P,) as an algebraic variety is irreducible. It is easy to verify that vi3,...,vi9 all vanish on Zero(P,) for / = 1,2,3. Therefore, in the case ao3 = 0, V3 = • • • = vn = 0 implies that V13 = • • • = V19 = 0, so Zero({ao3,V3,--.,vi9}) = Zero({a1i,a2i,ao3})UZero({ao3,02i,02O,0O2})UZero(P3f), where P^ = {003,1^3}UP3. Note that K3 = 0 corresponds to (10.3.10) in the case 003 = 0, and K3 becomes a2\ when an = 0 or a2o = 002 = 0. On the other hand, the set of the four polynomials ao3,Ki,K2,K3 in (K2) may be easily decomposed into {011,021,003} and P3" such that Zero({ao3,Ki,K2,K3}) = Zero({an,a2i,0O3}) UZero(P3f). Therefore, in the case ao3 = 0, V3 — • • • = V19 = 0 if and only if (K2) or (K4) holds. This shows how the simple necessary conditions (K2) and (K4) may be derived from the complicated Liapunov constants by means of irreducible variety decomposition. Other mathematical and computational techniques are required to prove that the conditions (K2) and (K4) are also sufficient for the origin to be a center for Kukles' system. This proof has been given in [17]. In order to establish the complete center conditions for Kukles' system, one has to deal with the case ao3 ^ 0, but the computation becomes much heavier in this (general) case. Another set of conditions was derived by Lloyd and Pearson [42], and it was proved in [76] that their conditions and the conditions given in [33] are all covered by Cherkas' conditions (CI). In fact, we have shown that the three sets (CI), (K2) and (K4) of conditions cover all the known center conditions for Kukles' system. The remaining question is whether there are other center conditions for Kukles' system. Sadovskii [51] proved that the three sets (CI), (K2), and (K4) of conditions are all the center conditions for Kukles' system (a conjecture posed by Lloyd and Pearson [42] in 1992). From the computational
10.4. Automated Reasoning in Differential Geometry
193
point of view, it would be interesting to confirm Sadovskii's result by a direct proof of the following alternative claim. Claim. Let Q be the set of the five polynomials in (CI), K2 = {ao3,Ki,K2,K3}, and K 4 = {«03,002,020,021}-
T n e n
Zero({v 3 ,..., vi3}) = Zero(Ci) U Zero(K 2 ) U Zero(K 4 ). In other words, the set of six polynomial equations V3 = 0,...,vi3 = 0 does not have any solutions other than the solutions of (CI), of (K2), and of (K4). The difficulty of proving this claim computationally is caused by the large polynomials involved. The challenging problem is to decompose the polynomial set P = {v3,...,vi3} into simpler or irreducible ones. We have tried to decompose P into irreducible triangular systems or algebraic varieties without success. The reader is strongly encouraged to attack this challenge that has been open for more than a decade. Center conditions have been derived for various classes of differential systems in different ways and by different authors. Some of the conditions are erroneous or incomplete, while some others are equivalent. The above examples (and those in [60, 76] and Sect. 7.6 of [78]) demonstrate that elimination methods and tools are practical devices for examining the correctness and the relationship among different sets of center conditions.
10.4
Automated Reasoning in Differential Geometry
The theorems and problems we have considered in Chap. 8 are all from elementary geometry. The ideas and principles used therein may be naturally extended to differential geometry. However, for this extension one needs a thorough development of elimination theory, algorithms, and software tools for the involved differential algebraic computation and reasoning. It is Wu [86] who initiated automated reasoning in differential geometry by extending his successful method of theorem proving and discovering from the algebraic to the differential case. He was able to prove and "discover" several remarkable theorems automatically in the local theory of curves and in plane mechanics [97]. Wu's work has been followed up by several researchers including the author [68, 3]; we follow the same algebraic approach of Wu, but use different algorithms for the involved zero decomposition. For differential geometry, the formulation of problems using differential polynomial equations and inequations is not always easy and straightforward, and the involved algebraic computations
194
Chapter 10. Selected Problems in Computer Mathematics
are much more complicated and expensive. One often encounters very large differential polynomials when dealing with problems about curves and surfaces. A full account of the concepts and algorithms for differential polynomial elimination is much beyond the scope of this section. Here we only give a couple of illustrative examples without entering into the technical details. In fact, the development of such algorithms with efficient implementation for differential polynomial systems is a very difficult task that has been under serious investigation. In the €PSILON library the reader may find the modules dcharsets and d t r i s y s and the function geother [Dprover], which are preliminary implementations and may deal with ordinary differential polynomial systems and theorems in the local theory of curves and plane mechanics. Maple provides a package dif f alg for manipulating systems of (ordinary and partial) differential polynomials. We use some of the functions from the above-mentioned packages for the computations required in the following examples. The first example was studied in detail by Wu [97]. Example 10.4.1 (Bertrand curves [97, 68]). Let £ and £ be a Bertrand pair of curves in one-to-one correspondence with arc lengths s and s as parameters in the ordinary metric space. Attach the trihedrals (X,e\,e2,e^) and {X,e\,e2,e-i) to £ and £ at the corresponding points X and X, and denote the curvature and torsion of £ and £ by K,x and by K,x respectively. We have the following theorems. 1 (Schell theorem). The product of x and x is a constant. 2 (Bertrand theorem). There exists a linear relation between K and x with constant coefficients. 3 (Mannheim theorem). The cross-ratio of X,X and the centers of K,ic is a constant. To prove these theorems, let X =X + a\ei + 0 2 ^ 2 + 03^3, 3
e;=X M '7 e J>
'=1,2,3.
7=1
From the Frenet formula of £ and £, one can easily deduce a set of 12 differential polynomials (see [97]). In the classical Bertrand case, ei = ±^2Let us take the positive sign, and similarly for the orthogonality relations among the uij, so that CL\ = 0,
a-i = 0,
"22=1;
Mn=M33,
W12 = "21 = "23 = "32 = 0, "13 = — " 3 1 ,
M1i + M13 = l .
195
10.4. Automated Reasoning in Differential Geometry
Combining these relations with the 12 differential polynomials, one obtains the following set of 14 differential polynomials (the primes denoting the derivatives with respect to s): H\ =s'uu + « 2 K - 1,
H2 = -a'2,
Hj = s'«13 — Ci2l,
H4 = — M J I ,
H$ = S'K — KU12 + TU13,
H(, = — H' 1 3 ,
HJ = s'KUu
H$ =S"'iCMi3 — s't M33 + X,
— S'XM31 — K,
Hy = — MJJ,
//10 = — s ' x — KM31 +TM33,
H l i = —M33,
H12 = "11 — "33,
#13 = " 1 3 + "31,
//14 = M J J + M j 3 — 1.
With respect to the variable ordering S~< a\ -< Cl2 -< a-i -< « n -< "12 ~< "13 -< "21 -< "22 -< "23 ~< "31 -< "32 -< "33 ^K-<X-
P = {Hi,... ,Hu} may be decomposed by d t r i s y s [dits] into 10 irreducible differential triangular systems, with the corresponding differential triangular sets T i = [a'2,u\vUn
~ 1*11,1?^ + U2n - 1,1*31+«13,M33 - Mil,<22K + .s'Kll - 1,
a-ii — s'u\-i,s'vi + M13T — M H K , S ' T — " n x — H13K], T2 = [s",a'2,s'uu
— l , ^ ' 2 " i 3 — S,2+
1,M31 + M13,M33 — Uu,K,a2l
—s'lltf,
s'K + M13X, 5'X — Hi iX], T 3 = [a 2 ,Mn - 1 , " 1 2 - 1, " 1 3 , " 3 1 , " 3 3 - l , 0 2 K + s ' -
1,X,J'K-K,X],
T4 = [a'2,Uu + 1 , " 1 2 + 1,"13,"31,"33 + l,a2K. — s' -
1,X,S'K+K,x],
T 5 = [s' + l , a 2 , M i i + l,Hi3,M31,"33 + l ! K,T,K,X], T 6 = [s' ~ l,a'2,Un
- 1,M13,«31,"33 -
T 7 = [s' - 1 , 0 2 , " 1 1 - 1 , " 1 3 , " 3 1 , " 3 3 Tg = [S' + l,a2,Un T 9 = [s' -l,a2,UnTTlO = [s' + l,a2,Uu
+ l,Ui3,U3l,U33
1,K,X,K,X], 1,K,K,X-X],
+ l,K,K,t
- X],
1,«12-1,"13,"31,"33-1,K-K,X-X], + 1 , " 1 2 + 1,"13,"31,"33 +
1,K-K,X-X],
such that 2
4
10
d-Zero(P) = | J d-Zero(T,-/{ j,'a 2 , "13}) U (J d-Zero(Tl-/{s,'a2}) U (J d-Zero(T,-). 1=1
1=3
1=5
196
Chapter 10. Selected Problems in Computer Mathematics
The conclusions of Schell, Bertrand, and Mannheim's theorems to be proved are CS = (xx)' = 0, CB = K'X" - K"X' = 0, CM = [(l+«2K)(l-a 2 K)]' = 0, respectively. It is easy to verify that
d-prem(Q,T,){^;;:;;:::;50; d-prem^Toj^;;:;;-
8
'
d-prem(CM, T,-) = 0 , i = 1,..., 10. Therefore, the algebraic form of Schell's theorem and of Bertrand's are both conditionally true and of Mannheim's is universally true. The subsidiary conditions for the former may be provided as s' ^ 0 and aj^O (i.e., <£^£). Note that Bertrand's theorem is also true when «2 = 0 and K = ic = 0 (i.e., € = € is a straight line). Mannheim's theorem is considered usually under the condition KK ^ 0. However, the algebraic form of the theorem is true as well in the degenerate case K = ic = 0. For any parametric surface 6 r = r(u,v) =
(x(u,v),y(u,v),z(u,v))
in three-dimensional Euclidean space, dr au
dr dv
are the tangent vectors along the lines of u and v in the tangent plane at point P(x,y,z) of 6 . The first fundamental form of 6 is \dr\2 = Edu2 + 2Fdudv + Gdv2, where E=ri,
F=ru-rv,
G = i*.
Except at singular points of 6 , we have
£ > 0 , G>0,
EG-F2>0.
8=
Assume that the two tangent vectors ru and rv are not parallel at a regular point P. Then r„xrv n
=
i
\ruxrv\
T
10.4. Automated Reasoning in Differential Geometry
197
is a unit normal vector perpendicular to the tangent plane of & at P. The quadratic form -dn -dr = Ldu2 + 2Mdudv+Ndv2 is called the second fundamental form of &, where L = -ru-nu,
M=-ru-nv,
N=
-rv-nv.
The three vectors [ru,rv,n] form a moving frame of 6 at P. Taking their partial derivatives with respect to u and v, we have r„u = r1nru + T2lrv + Ln, ruv = r\2ru + F22rv + Mn,
(10.4.1)
rvv = T\2ru + T22rv +Nn; nu = \{MF - LG) ru + (LF - ME) r v ]/5, nv =[(NF-MG)ru + (MF -NE)rv]/8,
(10.4.2)
where (GEu-2FFv+FEv)/(25), T\2 = 1
r 1
2 2
•
(GEv-FGu)/(2d),
r2 r2 1
(2GFv-GGu+FGv)/(28),
{2EFu-EEv-FEu)/(26), (EGu-FEv)/(28),
12
rj2 = (EG
2FFv-FGu)l{2b).
The equations (10.4.1) and (10.4.2) are the well-known Gauss formulas and Weingarten formulas in the local theory of surfaces. Their integrability conditions are given respectively by the following equations:
' KF - (r 1 2 ) M -(r 1 1 ) v +r 1 2 rj 2 -r 1 1 r 2 2 , KE = (r2j)v - (r22)„ + Txnt\2 + r2nrl2 - r ^ r ^ • 0 1 2 ) 2 , < KG = (r22^" (ri2)v + r2 2 r 12 +r 22 r 11 r 2 r 1 • (rj 2 ) 2 , 2 1 1 ,KF = (r12)v — (r22)H + r 12 r 12 — r 22 r 11 ; 12 22 Lv-Mu=LT\2+M{r22-r\1)-Nr2n, Mv-Nu = Lr\2+M(r22-r\2)-Nr22.
(10.4.3)
(10.4.4)
(10.4.3) and (10.4.4) are the Gauss equations and Codazzi-Mainardi equations, which may be found in standard textbooks of differential geometry. Example 10.4.2 [3]. If a surface r = r(u,v) consists entirely of umbilics, then it is planar or spherical.
198
Chapter 10. Selected Problems in Computer Mathematics
A point P on the surface is called an umbilic if the two principal curvatures at P are equal. At every umbilic we have L _ M _ N_ Take the differential polynomial equations (10.4.1)-(10.4.4) together with EM — FL = 0 and EN — GL = 0 as the hypothesis of the theorem, denote the corresponding set of differential polynomials by P, and let Q = {E,G,8}, where 5 = EG — F2 as before. Order the derivatives of E ~
T2 =
UI=QU{F},
U2 = {£,G},
[F,M,T2,T3,T;,T5,T6,TJ,T^,T;0],
and 7i
=EM-FL,
T2=EN-GL, T4 = E2LV - ELEV - EFLU + LFEU, Ti =Env+Lrv, T6=Enu+Lru, T7=E2[2d(Guu-2Fuv+Evv) + (2FFU-GEU+FEV)GU+(2EFU-FEU-EEV)GV - 2 (2FFU - GEU -FEV)Fv -EG2U - GE2 + 4G2L2} - 4(2EG - F 2 ) F 2 L 2 , T&=E[25/Vv + (GGU + FGV -2GFV)ru - (FGU+EGV -2FFV)rv] - 2 L G 8 n , T9=E [2hruv + (FGU - GEV)ru - {EGU -FE V )r v ] - 2LF8n, Tio = 25ruu + (2FFU - GEU - FEV)ru - (2EFU - FEU - EEV) rv - 2Lbn; i\ — ELU — LiEu,
T7' = 2EG(GUU + Evv) - EG\ - GE2 - GEUGU - EEVGV + 4G2L2,
10.4. Automated Reasoning in Differential Geometry
199
T8' = 2EGrvv + GGuru - EGvrv - 2G2Ln, Tg = 2EGruv — GEvru — EGurv, T/0 = 2EGruu -GEuru+EEvrv-2EGLn. In fact, T4 may be replaced by r4'. Let L
„
«
Then one can easily verify that d-prem(A:„,T,) = d-prem(£v,T,) = d-prem(/„,T,) = d-prem(/v,T,) = 0 for i = 1,2. It follows that £ is a nonzero constant and / is equal to a constant vector ro, and thus n r-r0 = - - . Therefore, we have 1 This proves that the surface is spherical In [3] we were able to derive an algebraic relation between the coefficients of the first and the second fundamental forms in a very compact form. It is still not known where in the literature the relation can be found. Another notable example worked out by Wu [97] is the automated derivation of Newton's gravitational laws from Kepler's observational laws in celestial mechanics. Along with the advance of software and hardware technologies, powerful elimination methods and tools will surely be developed to tackle various old and new problems in differential geometry, and other computational problems around systems of differential polynomials as well.
Appendix A Polynomial Systems: 50 Test Examples
The following list indicates the source of 50 test examples (polynomial sets with variable orderings) that have been used for our experiments reported in several papers [63, 69]. These examples in Maple format are included in the CPSILON library. Al. [93] (Example 3); x22 -< x2\ -< x23 -< x\2 <xU<x\\< xl03-<xl0. A2. [93] (Example 4); xlO -< ••• ^ x l 0 5 .
xlOl -< *102 -<
< xl3 -< x21 -< x22 -< x23 < x30 < xlOl <
A3. [44] (no. 4, p. 30, PS"); Xn < X72 -< X73 -< X80. A4. [44] (no. 4, p. 51); 5i -< C\ -< S23 < C23 < SA -< Q < S5 -< C5 -< S6 •< C6 -< S3^C3. A5. [9] (p. 70); c\ < c2 -< si < s2 -< py -< cf -< ct -< cp < sf -< st -< sp. A6. [9] (p. 74, e = 1); x -< y -< z -< w. A7. [9] (p.74,8 = 1 / 2 ) ; x - < y < z < w . A8. [9] (p. 77);
z
A9. [9] (p. 80); y •< x. A10. [7] (p. 248, left-bottom); x -< y. Al 1. [7] (p. 248, right-top);
r<x-
A12. [7] (p. 248, right-middle); a
<x.
A13. [7] (p. 249, left-middle); r < x -< y. A14. [7] (p. 249, left-top); r -< x -< y •< z. A15. [19] (Problem2);
z
201
Appendix A
A16. [19] (Problem 5(a)); All.
[19] (Problem 5(b));
d
A18. [19] (Problem 7); y<x. A19. [19] (Problem 9); x -< y -< z (where a,... ,k are considered as free parameters). A20-A31. [57] (p. 195, A4-C2); y -<x.
a^b^c^d-<w-
A32. [6] (p. 86, Example 1); C3 -< C2 -< A31 -< A32 -< All ^Bl
63.
< B\ < All -< A31 -<
A34. [6] (p. 87, Example 3); C5 -< C3 •< CI -< C4 -< A54 -< A53 •< A52 < A43 -< A42 < A3!
U0.
A37. [6] (p. 91, Example 2); [72 -< U\ -< U0. A38. [6] (p. 91, Example 3); U3 -< • • • -< U0. A39. [6] (pp. 91-92, Example 4); I/O -< • • • < U4. A40. [6] (p. 92, Example 5); i/0 -< • • • ~< U5. A41. [6] (pp. 92-93, Example 1); LI -< • • • -: LI. A42. [6] (pp. 93-94, Example 2); X1 -<•••-< X4. A43. [6] (p. 94, Example 3); A46 -< U4 -< U3. A44. [6] (pp. 94-95, Example 4); C5 -<;
< C O -< B5 -< • • • -< BO -< A 5 -< • • • -<
A2 -< A0. A45. [6](p.95,bottom);5^,S-
Appendix B Algebraic Factorization: 55 Examples
In the CPSILON library the reader may find a file containing 55 examples for polynomial factorization over algebraic extension fields. Let these examples be referred to as B1-B55 correspondingly. The first 23 examples are collected from the literature, followed by three examples coming from some application problems, and Examples B27-B40 are randomly generated. For these 40 examples the polynomials and their factorizations together with the adjoining ascending sets that determine the successive algebraic extensions fields are listed in [83]. Provided in what follows are 15 algebraic factorizations, which are needed in some of the examples, mainly for geometric applications, presented in this book (see also [64, 75, 84]). As we have seen from the examples in Sect. 8.4, algebraic factorization is required to deal with the reducibility problem in geometry theorem proving when "natural" algebraic formulations are used. Note that most of the reducibility cases can be avoided by some tricky formulations which take into account geometric information. One does not need to utilize such tricks when efficient factoring routines are available. Moreover, the proof of a statement may be figured out even if its algebraic formulation does not precisely correspond to the geometric statement and thus is not a theorem in the logical sense. This will help us understand the geometric ambiguity reflected in the algebraic form of the theorem. Here let us recall the theorem about incenter and excenters. Example B.l. Refer to Example 8.2.1 and take coordinates for the three vertices of AABC as A(-«i,0),
B(ui,0),
C(u2,u3).
Let the three bisectors of the angles A,B,C meet the y-axis at A'(0,yi),
B'(0,y2),
C'(0,v3)
203
Appendix B
respectively (see Fig. 39). Then the hypothesis of the theorem consists of ZCAA' = ZA'AB,
ZABB' = ZB'BC,
ZBCC' = ZC'CA
Fig. 39. Incenter and excenters and the conclusion to be proved is: the three lines AA',BB',CC' are concurrent. By taking tangent for the equalities of the angles the hypothesis conditions correspond to three polynomial equations H\ = M3vj + 2«iM2Vi -\-2u\y\ —u\us, #2 = u$y\ - 2uiU2)>2 + 2u\y2 - u\m, Hi = u^yj - ujyi - ujy3 + u\yi - u\ui. With the variable ordering y\ -< j2 -< y$, these polynomials already form a characteristic set C = [Hi,Hi,Hi\. Direct verification shows that the pseudoremainder of the conclusion polynomial C with respect to C is non-zero. In order to prove the theorem, we need to examine the reducibility of C. This involves first checking the reducibility of H% over %^\ = Q,(Mi,M2,W3,yi), where y\ is an algebraic element having adjoining polynomial H\. It is verified that H2 is irreducible over %^\. Next, we check whether #3 is reducible over %^2 = $ii{y2), where V2 is an algebraic element having adjoining polynomial H2. It has been found that 7/3 may be factorized as . {2u\y3-F
H
3 =
-u\ui)-[2u\uiyi
+ u?,F -u\(ul
+
2u\-2u\)}
—^
,
{O.l)
where F = M3yiy2 + wi (K2 + u\)y2 - u\ (u2 - ui)yi (see Example 7.5.1 in [78]). Using the factorization (B.l), C is immediately decomposed over Q,(MI,M2,«3) into two irreducible components. The algebraic form of the theorem is true on one component and false on the other. This corresponds to the geometric fact that among the 8 sets of three (internal or external) bisectors of the three respective angles, the bisectors in 4 sets are concurrent at four points and those in the other sets are not.
Appendix B. Algebraic Factorization: 55 Examples
204
Most of the algebraic factorizations listed below come from examples of geometric theorem proving. • For computing the irreducible triangular systems [Ti,Ui],... ,[¥9,11)9] in Example 7.2.7 of [78], several polynomials have to be factorized over algebraic extension fields. One of the factorizations is 4y2-4(Ul
+ l)y5-3u2
+ u2 + 2Ul + l
= (2y5+2u2y2-ui-
1) ( 2 y 5 - 2 M 2 V 2 - M I - 1)
over Q,(MI,M2,)'2) with adjoining polynomial Ay2, — 3 for V2 (see Example 7.5.2 in [78] for the factoring details). Here is another factorization over the same extension field Qju\, 112,yi)'• 4y\ - 4 Ml v 3 - 3u\ + u\ = (2y3 - 2M2v2 - «i) (2y3 + 2u2y2 - «i)- (B.3) • Let 73 and / be as in Example 8.3.2; then • r3T3" [J/ + 2 M i(i^ + l ) y o ] [ J / - 2 i i i ( ^ + l)yo] I3 = —— =
T
(B.4)
over Qiu\, U2,yo) with VQ — 3 as adjoining polynomial for yo, where H = /y 3 — 2u\{3 u\u\ + Au2 — u\). • Let QJyUi, U2,yi) be an extension field of Q, obtained by adjoining the transcendental elements u\,u2 and algebraic element y\ with minimal polynomial 4
y\-ay\
9
9
+ u\.
where a = u\ + u\-\-\. The following factorizations over Q,(«i,M2,yi) have been given in [64]: \6u]y2 -a2 + 4u2 = (4u2y + 2y\ - a) (4u2y - 2y2 + a), 16H|()>I
+ m)y2 -32ujyl
-(u2-l)2]y1-ul[u2(u1 =
(B.5)
+ 16muly2 + [u\(7u\ + 6u\ + 22) + 2u2+lS)
+
(u2-l)2]
(B.6)
*—(4u\U2y + F) (4u\u2y - F), u\
where
F =4y\-6uiy\-4{u22+\)yl+ui(a
+ 4).
The factorization (B.5) has been detailed in Example 10.2.1.
205
Appendix B
• For the irreducible decomposition in Example 8.3.3, the following algebraic factorizations are required: 4 u\ (2 u\x\ — H)x\ — A a2abcdx2 - a2 [2 u2a2x\ + 2 u2yii3
- 4 ( y + u\) u\m - pa2] =
2V
^—^2r2',
4 «2 (2 w2xi + H) x2 + 4 a2abcdxj, - a 2 [2 w2a2xi - 2 w2YW3 + 4 (y+ U2) ujus + pa 2 ] =
2V
l L a bc d
'-T3T}
over Q(u\,U2,U3,xi) withxi having adjoining polynomial 7\, where / / = 2M2M3 + P;
a =
M2
+ 1,
P = M|-1,
y = " 1 + 1;
anda,fo,c,d,a,Y,72,r 2 ',r3,r3 areas in Example 8.3.3. • The following algebraic factorizations are needed for computing the zero decomposition in Example 9.1.1: 2JC3 + 2x3 - 1 = - (2x3 - 3x2 + 1) (2x3 + 3x2 + 1)
(B.9)
over Q,(x2) with adjoining polynomial ?>x\ — 1 for x%, and Xj — X1X5 — X5 + 4xi + 5 . (4x5 — X1X2 + 5x2 — 2xi — 2) (4x5 +X1X2 — 5x2 — 2xi — 2) ~ 16 (B.10) over Q,(xi,X2) with adjoining ascending set [x2 —6x1 — 11,X]X2 + 3x2 + 52xi +76] forxi andx2. • Over the extension field Q,(xi,...,X4) — where xi,X2,X3 are adjoined to Q, as transcendental elements and X4 as an algebraic element with minimal polynomial 4 x 4 - 8 a x 4 - 4 ( x 3 - x i X 2 - a 2 ) x 4 + 4a(xf —x\xi)x^ -a 2 x?,, where a = X2 + xi, we have ay2 + 2 (x| — X1X2) y — 0x3 ^ (av + 2 x 2 . - 2 a x 4 ) ( a y - 2 x 2 . + 2ax4 + 2x 2 i -2xiX2) a (see [64] and Examples 7.2.5 and 7.2.6 in [78]).
(B-n)
206
Appendix B. Algebraic Factorization: 55 Examples
• Let C\,... ,CA be as in Example 8.3.4 and %. = Q(yb>yc,yd,Xb,Xc,Xd) with transcendental elements yb,yc,yd and algebraic elements Xb,xc,Xd adjoined by the ascending set [Ci,C2,C3]. Then the polynomial CA may factorized over ^C as follows:
CA = (ya + ybxcxd + ycxbxd + ydXbXc - ybycyd) (ya - ybxcXd - ycxbXd - ydXbxc+ybycyd) •
, „ 1 „.
• Let %, = QlXjX1,... ,x"",y, r) with transcendental elements x,x!,... ,x"" and algebraic elements y, r adjoined by the weak-ascending set [A\, A2] with A1=Iy2-x2J2, and
A2=J2r2-(I
+ J2)y2,
/ = -3xx" (xx"" + 2x"2) + 2 (xxf" - x'x") (2xx"' + x'x"), J = xx"' + 2x'x".
During the computation of the irreducible differential triangular systems in Example 4.3.1 the following factorization over H£ is required: Ia2 + x"2K = lr(xa+x"r)(xa-x"r), xl
(B.13)
where K = 3*VY'" - 5 x V ' 2 - 2xx'x"x'" - 2x'2x"2 + 6xx"\ In fact, the factorization (B.13) holds over Qlx,x!',... ,x"",r) with transcendental elements x,x!,... ,x"" and the algebraic element r adjoined by Ir2 +x2K =[IA2-
(I + J2)Al]/J2 =
-pzem(A2,Auy)/J2.
• Our list of geometric examples is completed with the following factorization Ax\R2-x\-
{x\+x\)xl-x\x\
= (2x3R-F)(2x3R
+ F)
(B.14)
with F = 2x\ — 2x2x\ — 2X\XA —x\ + x\x2
over the extension field Q(xi,... ,xj) determined by the irreducible polynomial 4.4 - 8 (x2 +xi)x\- A{x2-x\Zx\x2-x\)x\ + 4 (X2 + Xl) (x2 - X\X2) X4 — (X2 +
X\)2X2,
which is required for the proof of Poncelet's theorem formulated in [84]. The factorization of H2 over ^Ci as in Example B.l corresponds to Example B41, and the factorizations (B.1)-(B.14) correspond to Examples B42-B55 in the above-mentioned file.
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[82] Wang, D.: A simple method for implicitizing rational curves and surfaces. Preprint, LIP6 - Universite Paris VI, November 2002. [83] Wang, D., Lin, D.: A method for multivariate polynomial factorization over successive algebraic extension fields. In: Mathematics and Mathematics-Mechanization (Lin, D., Li, W.,Yu, Y., eds.), pp. 138-172. Shandong Education Publ. House, Jinan (2001). [84] Wang, D., Zhi, L.: Algebraic factorization applied to geometric problems. In: Proc. ASCM '98 (Li, Z., ed.), pp. 23-36. Lanzhou University Press, Lanzhou (1998). [85] Wu, W.-t.: On the decision problem and the mechanization of theorem-proving in elementary geometry. Sci. Sinica 21: 159-172 (1978). [86] Wu, W.-t.: On the mechanization of theorem-proving in elementary differential geometry (in Chinese). Sci. Sinica Special Issue on Math. (I): 94-102 (1979). [87] Wu, W.-t.: Basic principles of mechanical theorem proving in elementary geometries. /. Syst. Sci. Math. Sci. 4: 207-235 (1984). [88] Wu, W.-t.: Some recent advances in mechanical theorem-proving of geometries. In: Automated Theorem Proving: After 25 Years (Bledsoe, W. W., Loveland, D. W, eds.), Contemp. Math. 29, pp. 235-241. American Mathematical Society, Providence (1984). [89] Wu, W.-t.: On zeros of algebraic equations — An application of Ritt principle. Kexue Tongbao 31: 1-5(1986). [90] Wu, W.-t.: A mechanization method of geometry and its applications I: Distances, areas, and volumes. /. Syst. Sci. Math. Sci. 6: 204-216 (1986). [91] Wu, W.-t.: On reducibility problem in mechanical theorem proving of elementary geometries. Chin. Quart. J. Math. 2: 1-20 (1987). [92] Wu, W.-t.: A zero structure theorem for polynomial equations-solving. Math. Mech. Res. Preprints 1: 2-12 (1987). [93] Wu, W.-t.: A mechanization method of geometry and its applications III: Mechanical proving of polynomial inequalities and equations-solving. Syst. Sci. Math. Sci. 1: 117 (1988). [94] Wu, W.-t.: On the foundation of algebraic differential geometry. Syst. Sci. Math. Sci. 2:289-312(1989). [95] Wu, W.-t.: Some remarks on characteristic-set formation. Math. Mech. Res. Preprints 3: 27-29 (1989). [96] Wu, W.-t.: On a projection theorem of quasi-varieties in elimination theory. Chin. Ann. Math. (Ser.B) 11: 220-226 (1990). [97] Wu, W.-t.: Mechanical theorem proving of differential geometries and some of its applications in mechanics. /. Automat. Reason. 7: 171-191 (1991). [98] Wu, W.-t.: Mechanical Theorem Proving in Geometries: Basic Principles. Springer, Wien New York (1994) [transl. from the Chinese edition by X. Jin and D. Wang].
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Index
A algebraic routine 45 algebraic variety 23, 173 irreducible 33, 59 algorithm BasSet 31 CharSer 31 f, 123 CharSetN 31, 42, 117 f, 135 Discover 136 ExtendedCharacteristicSeries 32 F 4 87 FactorA 34, 181, 184 FactorB 34, 181 f FGLM 86 GenCharSet 30 f IrrCharSer 32, 124 IrrCharSerE 32, 124 IrrTriSer 53, 124 IrrVarDec 34,53,60 LexTriangular 86 MacRes 90 ModCharSet 30 f PrildeDec 34 PriTriSys 117,135 ProjA 114 ProverB 125 ProverC 125 ProverD 125 QualrrTriSer 32,53 RegSer 58 SimSer 58
SubresChain 91 TriangularForm 31,42 TriangularFormC 31,42 TriSer 32, 52, 123 TriSerP 52, 113 f TriSerS 57,123 ascending set 29 differential 36 associated prime 24, 33 Aubry, P. 84 B Bezout, E. 89 Bezout matrix 89 Bezout resultant 89 Barth,W. 12 Barth sextic 9 basic set 42 Bertrand curve 194 Bertrand theorem 194 Blaschke,W. 132 Bradford, R. 83 Brahmagupta formula 138 Brillhart,J. 141 Bronstein, M. 110 Buchberger, B. 85 f Buse, L. 90 butterfly theorem 4, 81 C Cayley cubic 13 center 185
Index
216
Ceva theorem 81 characteristic series 31, 57 differential 37 f extended 31 irreducible 22, 32, 59 quasi-irreducible 32 characteristic set 28, 30, 37, 66, 115 differential 36 f modified 30,42 Cherkas, L. A. 191 Cherkas condition 191 Cheung, C. W. S. 91 Chinese matrix method 96 Chou, S.-C. 83 Chtcherba, A. 89 class 19,35 Clifford algebra 115 Codazzi-Mainardi equation 197 conclusion 115 constant 19 contradictory 29 curvature 167, 194, 198 cyclic system 99 Czapor, S. R. 109 D derivative 35 Desargues theorem 81 differential 24, 37 f differential ascending set 36 differential characteristic series 37 f irreducible 39, 57 quasi-irreducible 38 differential characteristic set 36 f modified 37 differential polynomial 35 differential polynomial set 36 differential polynomial system 36 differential pseudo-remainder 36 f differential quasi-ascending set 36 differential quasi-characteristic set 36 f modified 37 differential routine 45 differential triangular series 24, 55 fine 24,55
quasi-irreducible 56 differential triangular set 36 fine 37 irreducible 38 differential triangular system 24, 55 fine 24,55 differential weak-ascending set 36 differential weak-characteristic series 37 f irreducible 57 quasi-irreducible 38 differential weak-characteristic set 36 f modified 37 discriminant surface 180 Dixon, A. L. 89 Dixon matrix 89 Dubuque, B. 84 E Emiris, I. Z. 91 equality type 116 equidimensional 173 Eulerline 81 extended characteristic series 31 extended weak-characteristic series 31 extraneous circle 161 F F-modified 20,30,37 F-modified quasi-characteristic set 20 Faugere, J.-C. 85, 87, 158, 165 Fee.G. 109 Feuerbach theorem 81 fine 20,55 fine differential triangular series 24, 55 fine differential triangular set 37 fine differential triangular system 24, 55 fine triangular series 20 fine triangular set 20 fine triangular system 20 first fundamental form 196 focal value 185 Frenet formula 194 full projection 52
Index
function advance 87 Algebraic 69,77,116 bezout 89 BezoutMatrix 89 bezres 89 f, 157 cfactor 34, 47, 181, 184 char_series 83 charser 3 If, 46 f char set 30, 43, 46 f, 182 CharSet 19, 31 charsets [ivd] 65, 187, 192 Chinese 69 Click 75 convert 87 Coordinate 67, 77, 81, 116 csolve 34 f, 47, 48 dcharset 36 f, 48 dcs 37,48 decompose 83 f degree 29 Demo 75 depend 40 df 40 dies 38f,48 diff 2, lOf discrim 2 diss 37 f, 40 dits 55 f, 195 dmcharset 36f, 48 f dmes 37,48 Dprover 71, 79, 194 dqics 38,48 dqits 53, 55 drim 53,55,57 drs 37 f dtriser 55 dTriSer 24,55 ecs 31f,46f eics 32, 42, 47 f English 69 factor 2 fglm 87 format 40 gbasis 86 f
217
GCprover 71, 80 f, 118 Generic 70, 73 f, 78, 81 Geometric 7, 72, 154 Gprover 71, 80 f Groebner [normalf ] 120,126 ics 25, 32, 34, 42,47 f, 65, 174 ICS 12,22,25,53 implicit 91, 158, 161 indext 92 f iniset 22,29,34,93 irreducibleCharacteristic Series 83 its 25, 51,53, 58 f, 65, 99, 102, 108, 110,112, 125, 146,160, 169, 171, 174, 179, 188 ivd 33 f, 42 f, 47 f, 51, 53, 59 f, 65, 174, 178, 188, 192 IVD 14,23,25,53 Let 67f,77 lextriangular 87 lcoeff 41 licon 93, 185, 187, 191 Load 74 f, 154 Logic 69 f macres 90 f, 157 mcharset 30 f, 34, 42 f, 46 f, 49, 102, 107 mes 31f,42,46f,63,98 mecs 3 If, 42, 46 f minibasis 86 f multires 90 mvresultant 91 normalf 118,120,126 normat 61, 92 f, 108, 112, 175, 183 pid 33 f, 64, 178 f PID 23, 33 pquo 183 prem 6,20,41, 117f, 123 f, 126 Print 72 f Prove 24, 70 qics 32, 42, 47 f, 63, 152, 184 qits 53, 58 f, 63 regser 51, 57, 62, 64, 98, 111, 113, 157 f, 163
218
Index
RegSer 21, 25, 57 remset 20, 29 f resultant 2, 89 rim 53, 178 f RIM 22,53 Rosenfeld.Groebner 198 Search 74 f simser 57,62,64,98, 113 SimSer 21, 25, 57 sisys [its] 25, 60, 64 sisys[ivd] 65 sisys [qits] 53,63 sisys [triser] 59, 63, 98 solve 35 solvet 3 ssolve 57 subres 91 subresultantChain 91 Subresultants 91 Sylvester 89 SylvesterMatrix 89 system 26, 72
Tprover 71, 78 f, 125 t r i a n g 83 TRIANGSYS 84 t r i s e r 52, 57, 62 f, 98, 113, 125, 178 TriSer 20,25 trisys [its] 25,51,53,65,99, 102,108, 110, 112,125, 146,
160, 169, 171, 174, 179, 188 t r i s y s [ivd] 51,65,192 t r i s y s [qits] 53, 63 trisys [triser] 62f, 98, 113, 125, 178 tsolve 53, 84 Tsolve 20, 53 uvd 22, 59 f, 64 Wprover 70, 80 f, 118, 134 G Gao, X.-S. 83 Gauss equation 197 Gauss formula 197 Gauss line 81
Gauss point 81 Gaussian curvature 167 Gaussian elimination 96 Geddes, K. O. 109 Gelernter, H. 115 generic 116, 125 geometric dependent 118 global variable .contracted- 43 _factorized_ 54f, 61, 111 .reduced. 54 f, 62, 98 Gontard, M . C. 82
Grobner basis 3, 28, 66, 82, 85, 96, 115 H Hemmecke, R. 100 Hilbert,D. 186 Hilbert driven algorithm 86 Hillebrand, D. 83 Hu, S. 34 hypothesis 115 I index triple 92 inequation 145 initial 6, 19, 35 irreducible 12, 16, 22 f, 32, 38 f irreducible characteristic series 22, 32, 59 irreducible differential characteristic series 39,57 irreducible differential triangular set 38 irreducible differential weak-characteristic series 57 irreducible triangular series 22, 59 irreducible triangular set 22 irreducible variety 33, 59 irreducible weak-characteristic series 59 J Jin, X. 81, 191 Jouanolou, J.-P. 90 K Kalkbrener, M. 84 f
219
Index
Kapur method 71 Kepler law 199 Kukles, I. S. 190 Kukles system 190 Kutzler-Stifter method 71 L Lii,X.-L. 151 Lazard,D. 84,86 lead 35 leading coefficient 41 leading term 97 leading variable 3, 6, 29 Leisenring line 142 Leisenring theorem 81,142 Li,Z. 156 Liapunov constant 93, 185 Lloyd, N.G. 192 locus equation 145 Lorenz, E. 107
epsilon 18 geother 66 miscel 91 sisys 57 t r i s y s 51 Moreno Maza, M. 84 Morley,F. 127 Morley theorem 81,120,127 Morley-Wu theorem 81 Mourrain, B. 90 N Nemytskii, V. V. 190 Newton law 199 Noda,M.-T. 84 Noonburg, V. W. I l l normal 16,42,92 normal form 118 Nutbourne, A. W. 152 O
M Macaulay resultant 90 Mannheim theorem 194 Markelov, S. 132, 141 Martin, R. R. 152 medial set 31,42 Menelaus theorem 81 Messollen, M. W. 83 Minimair, M. 90 minimal 174 Miquel theorem 81 modified characteristic set 30, 42 modified differential characteristic set 37 modified differential quasi-characteristic set 37 modified differential weak-characteristic set 37 modified quasi-characteristic set 30 modified weak-characteristic set 30 module 17, 50 charsets 29 dcharsets 35 d t r i s y s 55
offset 162 order 35 ordinary 19, 35, 194 orthocenter theorem 81 P Pappus line 142 Pappus theorem 81,142 parameter 110, 118 Pascal conic 81 Pascal theorem 81 Pearson, J. M. 192 plex 118 polynomial set 18 differential 36 polynomial system 18 differential 36 Poncelet theorem 81,136 primary ideal 24, 33 prime basis 173 principal curvature 198 projection 51 full 52 projection property 16, 52, 55
220
Index
strong 97 pseudo-division 40 pseudo-remainder 6, 29 differential 36 f Ptolemy theorem 81
Qin-Heron formula 135, 139 quasi-ascending set 19 differential 36 quasi-basic set 118 quasi-characteristic set 30 differential 36 f F-modified 20 modified 30 quasi-irreducible 32, 38, 53, 56, 58 quasi-irreducible characteristic series 32 quasi-irreducible differential characteristic series 38 quasi-irreducible differential triangular series 56 quasi-irreducible differential weak-characteristic series 38 quasi-irreducible triangular series 53, 58 quasi-irreducible weak-characteristic series 32 quasi-offset 162 quasilinear 183 R rank 35 reduced 29 regular 16, 198 regular series 21 regular system 21 rescaling continuity 167 resultant 1, 82, 89 Ritt,J. F. 82 Russian killer 132
Sadovskii, A. R 192 Saharnikov, N. A. 186
Saharnikov condition 189 Saharnikov system 186 saturation 172 saturation basis 173 Saxena, T. 89 Schell theorem 194 secant theorem 81 second fundamental form 197 separant 35 serveur 87 Shimoyama, T. 83 Sibirskii, K. S. 189 simple 16 simple series 22 simple system 21 simpler 125, 155, 172 simplicity 155 Simsonline 81 Simson theorem 81 singularity ideal 180 smooth contact 167 software Aldor 84 Axiom 84 CASA 85 charsets 29, 63 f CharSets 15, 28 f, 50, 78, 83 CoCoA 85 dcharsets 35, 48 f, 194 d i f f a l g 194, 198 d t r i s y s 55, 194 €PSILON
15 f
FGb 82,86 Fortran 82 Gb 82, 86 f, 158, 165 geother 66 GEOTHER 7,15,66 f, 116 GEX 83 Ghostview 73 Groebner 3, 23, 70, 78, 85, 8i HTML 26,66,72 Java 15 Java applet 72 LinearAlgebra 89 Lisp 82
Index
Macaulay 2 83 Macsyma 82,84 Magma 85 Maple 2, 15, 25 f, 72, 78, 82, 85 Mathematica 85 Matlab 91 Maxima 84 miscel 89,91 MuPAD 85 Netscape 73 PostScript 66,73 Reduce 83 Risa/Asir 83 SACLIB 82 Scratchpad II 82 Singular 83 Singular-libfac 83 sisys 57,63f SiSys 15, 50 f, 84 t r i s y s 51,55, 63 f TriSys 15, 50 f, 78, 84 solvable 95 Stanger, D. 84 Steiner theorem 81,120,142 Steiner-Lehmus theorem 81,131,149 Steiner-Wu theorem 81 Stepanov, V. V. 190 strong projection property 97 subresultant 89, 91 Sylvester, J. J. 89 Sylvester matrix 2 Sylvester resultant 2 symbol .contracted- 43 _factorized_ 54f, 61, 111 .reduced- 54 f, 62, 98 BiarcA 154 Biarc 154 basset 30f, 34, 36f, 46f charsetn 30 f, 34, 36 f, 46 f els 19,35 d-prem 36 f d-Zero 24,36 degree 29 diss 37f,40
221
drs 37 f DRL 87 dTheorem 76 elim 86 false 22,43,54,57 fast 87 filename 86 g_obj 77 grlexA 198 Ideal 87 Ideal 22 f ini 19,35 i n i s e t 22,29,34,93 i s o t r o p i c 76, 78 KS 81 Kapur 81 l a t e x 40 LaTeX 73 lcoeff 41 lead 35 l i n e 76 l i s t 87 lvar 29 online 76, 78 ord 35 PB 173 Proj 51 qbasset 30, 36, 46 f qcharsetn 30 f, 36, 46 f rank 35 remset 20, 29 f r e s u l t a n t 2, 89 sat 172 sep 35 s p l i t 86 tdeg 44 term 44 Theorem 76, 118 t r i s e t c 30 f, 34, 42, 46 f t r i s e t 30, 42, 46 f true 22,53,55,57,61,98, 111 verbose 86 wbasset 30f, 36f, 46f wcharsetn 30 f, 36f,46f Zero 10, 19
222
Index
T Thebault conjecture 120 Thebault-Taylor theorem 81,130 torsion 194 triangular 3, 19, 22 triangular series 20, 52 f, 57 f differential 24,55 fine 20 irreducible 22, 59 quasi-irreducible 53, 58 triangular set 3, 19, 28, 82 differential 36 fine 20 irreducible 22 triangular system 20 differential 24,55 fine 20 true subvariety 23 U umbilic 198 universally 123 unmixed 173 V variety 23 irreducible 33, 59 W Wang,D.-K. 170 weak-ascending set 29 differential 36 weak-characteristic series 31 f differential 37 f extended 31 irreducible 59 quasi-irreducible 32 weak-characteristic set 30 differential 36 f modified 30 Weingarten formula 197 Windsteiger, W. 85 Winkler, R. 81 Wu, W.-t. 28 f, 32, 40, 48, 82, 1 127,143, 151,166 f, 193
Wu method 66,70, 115f Wu-Ritt method 15, 28 f, 84 Y Yang, L. 85, 105
ELIMINATION PRACTICE Software Tools and Applications
With a software library included, this book provides an elementary introduction to polynomial elimination in practice. The library Epsilon, implemented in Maple and Java, contains more than 70 well-documented functions for symbolic elimination and decomposition with polynomial systems and geometric reasoning. The book presents the functionality, implementation, and performance of Epsilon and demonstrates the usefulness of the elimination tool by a number of selected applications, together w i t h many examples and illustrations. The reader will find Epsilon an efficient tool, applicable to a wide range of problems in science, engineering, and industry, and this book an accessible exposition and a valuable reference for elimination theory, methods, and practice.
Imperial College Press www.icpress.co.uk
ISBN 1-86094-438-8
9 "781860"944383'
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