Table Of ContentLecture Notes in Artificial Intelligence 1669
SubseriesofLectureNotesinComputerScience
EditedbyJ.G.CarbonellandJ.Siekmann
Lecture Notes in Computer Science
EditedbyG.Goos,J.HartmanisandJ.vanLeeuwen
3
Berlin
Heidelberg
NewYork
Barcelona
HongKong
London
Milan
Paris
Singapore
Tokyo
Xiao-Shan Gao Dongming Wang
LuYang (Eds.)
Automated Deduction
in Geometry
Second International Workshop, ADG’98
Beijing, China, August 1-3, 1998
Proceedings
1 3
SeriesEditors
JaimeG.Carbonell,CarnegieMellonUniversity,Pittsburgh,PA,USA
Jo¨rgSiekmann,UniversityofSaarland,Saarbru¨cken,Germany
VolumeEditors
Xiao-ShanGao
InstituteofSystemsScience,AcademiaSinica
Beijing100080,China
E-mail:[email protected]
DongmingWang
LaboratoireLEIBNIZ,InstitutIMAG
46,avenueFe´lixViallet,F-38031GrenobleCedex,France
E-mail:[email protected]
LuYang
LaboratoryforAutomatedReasoningandProgramming
ChengduInstituteofComputerApplications,AcademiaSinica
Chengdu610041,China
E-mail:[email protected]
Cataloging-in-Publicationdataappliedfor
DieDeutscheBibliothek-CIP-Einheitsaufnahme
Automateddeductioningeometry:proceedings/ADG’98,Beijing,China,August1
-3,1998.Xiao-ShanGao...(ed.).-Berlin;Heidelberg;NewYork;
Barcelona;HongKong;London;Milan;Paris;Singapore;Tokyo:Springer,
1999
(Lecturenotesincomputerscience;Vol.1669:Lecturenotesinartificial
intelligence)
ISBN3-540-66672-9
CRSubjectClassification(1998):I.2.3,F.4.1,I.3.5,I.5,G.2
ISBN3-540-66672-9Springer-VerlagBerlinHeidelbergNewYork
Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis
concerned,specificallytherightsoftranslation,reprinting,re-useofillustrations,recitation,broadcasting,
reproductiononmicrofilmsorinanyotherway,andstorageindatabanks.Duplicationofthispublication
orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965,
initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer-Verlag.Violationsare
liableforprosecutionundertheGermanCopyrightLaw.
©Springer-VerlagBerlinHeidelberg1999
PrintedinGermany
Typesetting:Camera-readybyauthor
SPIN:10705416 06/3142–543210 Printedonacid-freepaper
Preface
The Second International Workshop on Automated Deduction in Geometry
(ADG ’98)washeldinBeijing,China,August1–3,1998.Anincreaseofinterest
in ADG ’98 over the previous workshop ADG ’96 is represented by the notable
number of more than 40 participants from ten countries and the strong techni-
cal programof 25 presentations, of which two one-hour invited talks were given
by ProfessorsWen-tsu¨nWu and Jing-ZhongZhang.The workshopprovidedthe
participants with a well-focused forum for effective exchange of new ideas and
timely report of research progress. Insight surveys, algorithmic developments,
and applications in CAGD/CAD and computer vision presented by active re-
searchers, together with geometry software demos, shed light on the features of
this second workshop. ADG ’98 was hosted by the Mathematics Mechanization
ResearchCenter (MMRC) with financial support from the Chinese Academy of
Sciences and the French National Center for Scientific Research (CNRS), and
was organizedby the three co-editors of this proceedings volume.
The papers contained in the volume were selected, under a strict refereeing
procedure,fromthose presentedatADG ’98 andsubmitted afterwards.Mostof
the 14 accepted papers were carefully revised and some of the revised versions
werecheckedagainbyexternalreviewers.Wehopethatthesepaperscoversome
of the most recent and significant researchresults and developments and reflect
the current state-of-the-art of ADG.
We want to thank all those members of MMRC who contributed to the
local organizationof ADG ’98 and the referees for their time and help. We also
want to acknowledge some of the previous workshops and conferences and the
corresponding publications listed below which have stimulated the continuing
activities on ADG.
– International Workshop on Automated Deduction in Geometry (ADG ’96)
(Toulouse, France, September 27–29, 1996) — LNAI, vol. 1360 (edited by
D. Wang), Springer-Verlag,Berlin Heidelberg New York, 1997
– WorkshoponAlgebraicApproachestoGeometricReasoning(CastleofWein-
berg,Austria,August17–19,1992)—SpecialissueofAnnalsofMathematics
and Artificial Intelligence, vol. 13, nos. 1–2 (edited by H. Hong, D. Wang
and F. Winkler), Baltzer Sci. Publ., Amsterdam, 1995
– International Workshop on Mathematics Mechanization (Beijing, China,
July 16–18, 1992) — Proceedings (edited by W.-T. Wu and M.-D. Cheng),
Int. Academic Publ., Beijing, 1992
– Workshop on Computer-Aided Geometric Reasoning (INRIA Sophia-Anti-
polis, June 22–26, 1987) — Proceedings (edited by H. Crapo), INRIA Roc-
quencourt, France, 1987
VI Preface
– International Workshop on Geometric Reasoning (Oxford, UK, June 30 –
July 3, 1986) — Special issue of Artificial Intelligence, vol. 37, nos. 1–3
(edited by D. Kapur and J. L. Mundy), MIT Press, Cambridge, 1988
– Asian Symposia on Computer Mathematics: ASCM ’98 (Lanzhou, China,
August 6–8, 1998) — Proceedings (edited by Z. Li), Lanzhou Univ. Press,
Lanzhou, 1998; ASCM ’96 (Kobe, Japan, August 20–22, 1996) — Proceed-
ings (edited byH. Kobayashi),Scientists Inc.,Tokyo,1996;ASCM ’95(Bei-
jing, China, August 18–20, 1995) — Proceedings (edited by H. Shi and
H. Kobayashi), Scientists Inc., Tokyo, 1995
July 1999 Xiao-Shan Gao
Dongming Wang
Lu Yang
Table of Contents
Automatic Geometry Theorem-Proving and Automatic Geometry
Problem-Solving 1
Wen-tsu¨n Wu
Solving Geometric Problems with Real Quantifier Elimination 14
Andreas Dolzmann
Automated Discovering and Proving for Geometric Inequalities 30
Lu Yang, Xiaorong Hou, and Bican Xia
Proving Newton’s Propositio Kepleriana Using Geometry and
Nonstandard Analysis in Isabelle 47
Jacques D. Fleuriot and Lawrence C. Paulson
Readable Machine Solving in Geometry and ICAI Software MSG 67
Chuan-Zhong Li and Jing-Zhong Zhang
Plane Euclidean Reasoning 86
Desmond Fearnley-Sander
A Clifford Algebraic Method for Geometric Reasoning 111
Haiquan Yang, Shugong Zhang, and Guochen Feng
Clifford Term Rewriting for Geometric Reasoning in 3D 130
Thierry Boy de la Tour, St´ephane F`evre, and Dongming Wang
Some Applications of Clifford Algebra to Geometries 156
Hongbo Li
Decomposing Algebraic Varieties 180
Dongming Wang
An Application of Automatic Theorem Proving in Computer
Vision 207
Didier Bondyfalat, Bernard Mourrain, and Th´eodore Papadopoulo
Automated Geometry Diagram Construction and Engineering
Geometry 232
Xiao-Shan Gao
A 2D Geometric Constraint Solver for Parametric Design
Using Graph Analysis and Reduction 258
Jae Yeol Lee
Variant Geometry Analysis and Synthesis in Mechanical CAD 275
Zongying Ou and Jun Liu
Author Index 287
Automatic Geometry Theorem-Proving and
Automatic Geometry Problem-Solving
Wen-tsu¨n Wu
Instituteof SystemsScience, Academia Sinica
Beijing 100080, P. R.China
[email protected]
1 Principle and Method
Studies in geometry involve both theorem-proving and problem-solving. So far
thelatterisconcerned,wemayciteforexamplegeometricallocus-determination
and geometrical constructions by ruler-compass which were much studied in
ancient Greece. In particular we may mention the regular polygon construction
andthethreefamousdifficultproblemsofangle-trisection,cube-duplicationand
circle-squaring.
A turning point of geometrical studies occurred in 1637 when R. Descartes
publishedhisclassicongeometry.Theintroductionofcoordinatesystemturned
then geometricalstudies to algebraic ones in replacing pure geometricalreason-
ings by manipulation of polynomials and solving of polynomial equations. This
paved the way to the final solutions of the geometrical-construction problems
mentioned above.
Our method of manipulating polynomials and solving arbitrary systems of
polynomialequationsintheformofWell-OrderingPrincipleandZero-Decompo-
sition Theoremspermits to treat geometricalstudies evenin an automated way.
Thegermofsuchstudies aswellastheirwide-rangeapplicationsliesinourgen-
eralmethod ofpolynomialequations-solving.Tobe precise,letthe unknownsof
thepolynomialsinquestionbedefiniteonesformingafinitesetX andtheircoef-
ficientsbeinafixedfieldKofcharacteristic0.Byapolynomialset(abbr.polset)
PS ⊂ K[X] we shall mean one consisting of a finite number of non-zero poly-
nomials in K[X] unless otherwise stated. For such a polset PS the solving of
equations PS = 0 is then equivalent to the determination of the set Zero(PS)
consisting of the totality of solutions of PS = 0 or zeros of PS in an arbitrary
extension field of K. For any polynomial G ∈ K[X], we shall denote the set-
difference Zero(PS)\Zero(G) by Zero(PS/G). We remark in passing that the
set Zero(PS) is nothing else but an affine algebraic variety, irreducible or not,
in the usual sense. Now the difficulty of solving polynomial equations arises in
essence from the fact that the polynomials in question are in general given in
confused order. The key point to overcome such difficulties is thus: The given
polset should be so manipulated in a way that the resulting re-arranged poly-
nomials will be in some well-orderedformfor whichthe correspondingsolutions
are almost self-evident and are closely related to those of the original ones to
X.-S.Gao,D.Wang,andL.Yang(Eds.):ADG’98,LNAI1669,pp.1–13,1999.
(cid:1)c Springer-VerlagBerlinHeidelberg1999
2 Wen-tsu¨n Wu
be sought for. In fact, in the case where all polynomials are linear in the un-
knowns X, the usual elimination method (say the Gaussian one) is just such a
method which, after due manipulations, will lead to a set of well-ordered linear
forms from which the solutions of the corresponding equations are immediately
given.
With this understood we shall introduce a well-ordering procedure of poly-
nomials and polsets in the following way.
1. Ordering of Unknowns. First of all we shall arrange the unknowns X in
anarbitrarybutfixedorder,denotedhenceforthbyx1,x2,...,xn inthe natural
order
x ≺x ≺···≺x .
1 2 n
2. Partial Ordering of Polynomials. For a non-constant polynomial P ∈
K[X] let xc be the unknown actually occurring in P with greatest subscript c.
Then P may be written in the canonical form below:
P =Ixdc +J1xdc−1+···+Jd−1xc+Jd,
inwhichthecoefficientsI,Ji arethemselvespolynomialsinK[x1,...,xc−1]with
I (cid:6)=0.We callc (>0)the class ofP,d(≥1)the degree ofP,I ((cid:6)=0)the initial
of P, and the formal partial derivative ∂P/∂xc the separant of P. In case P is
a non-zero constant in K, we set the class of P to be 0, and leave the degree
etc. undefined. We now introduce a partial ordering for polynomials in K[X]
first according to the class, and then to the degree.
3. Partial Ordering of Polsets of Particular Form. Consider now a polset
AS ofparticularforminwhichthepolynomialsmaybearrangedinanascending
order:
A1 ≺A2 ≺···≺Ar. (1)
We suppose that either r=1 and A is a non-zeroconstantin K, or r≥1 with
1
corresponding classes ci of Ai steadily increasing:
0<c1 <···<cr.
For technical reasons we shall require that for each pair (i,j) with 0 < j < i,
the degree of xcj in the initial Ii of Ai should be less than the degree of Aj.
Under such conditions the polset AS = {A1,...,Ar} is called an ascending
set (abbr. asc-set). The product of all the initials Ii (resp. all the initials and
separants) will be called the initial-product (resp. the initial-separant-product)
of AS. For any polynomial P ∈ K[X], we shall get by successive divisions with
respect to Ar,...,A2,A1 a unique remainder R given in the form
(cid:1)
R=I1s1···IrsrP − QiAi,
i
with Qi ∈K[X], si (≥0) as smallas possible, so that R is reduced with respect
to AS in the following sense: the degree of xci in R is less than that of Ai for
each i with 0<i≤r. This remainder R will be denoted as Remdr(P/AS).
