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Geometric Modeling and Algebraic Geometry Bert Ju¨ttler Ragni Piene • Editors Geometric Modeling and Algebraic Geometry 123 BertJu¨ttler RagniPiene InstituteofAppliedGeometry CMAandDepartmentofMathematics JohannesKeplerUniversity UniversityofOslo AltenbergerStr.69 P.O.Box1053Blindern 4040Linz,Austria 0136Oslo,Norway [email protected] [email protected] ISBN:978-3-540-72184-0 e-ISBN:978-3-540-72185-7 LibraryofCongressControlNumber:2007935446 MathematicsSubjectClassificationNumbers(2000):65D17,68U06,53A05,14P05,14J26 (cid:1)c Springer-VerlagBerlinHeidelberg2008 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Coverdesign:WMXDesignGmbH,Heidelberg Printedonacid-freepaper 9 8 7 6 5 4 3 2 1 springer.com Preface ThetwofieldsofGeometricModelingandAlgebraicGeometry,thoughcloselyre- lated, are traditionally represented by two almost disjoint scientific communities. Both fields deal with objects defined by algebraic equations, but the objects are studied in different ways. While algebraic geometry has developed impressive re- sults for understanding the theoretical nature of these objects, geometric modeling focuses on practical applications of virtual shapes defined by algebraic equations. Recently, however, interaction between the two fields has stimulated new research. Forinstance,algorithmsforsolvingintersectionproblemshavebenefitedfromcon- tributionsfromthealgebraicside. TheworkshopseriesonAlgebraicGeometryandGeometricModeling(Vilnius 20021, Nice 20042) and on Computational Methods for Algebraic Spline Surfaces (Kefermarkt 20033, Oslo 2005) have provided a forum for the interaction between thetwofields.Thepresentvolumepresentsrevisedpaperswhichhavegrownoutof the 2005 Oslo workshop, which was aligned with the final review of the European projectGAIAII,entitledIntersectionalgorithmsforgeometrybasedIT-applications usingapproximatealgebraicmethods(IST2001-35512)4. Itconsistsof12chapters,whichareorganizedin3parts.Thefirstpartdescribes theaimsandtheresultsoftheGAIAIIproject.Part2consistsof5chapterscovering resultsaboutspecialalgebraicsurfaces,suchasSteinersurfaces,surfaceswithmany realsingularities,monoidhypersurfaces,canalsurfaces,andtensor-productsurfaces ofbidegree(1,2).Thethirdpartdescribesvariousalgorithmsforgeometriccomput- ing.Thisincludeschaptersonparameterization,computationandanalysisofridges and umbilical points, surface-surface intersections, topology analysis and approxi- mateimplicitization. 1R.GoldmanandR.Krasauskas,TopicsinAlgebraicGeometryandGeometricModeling, ContemporaryMathematics,AmericanMathematicalSociety2003. 2M. Elkadi, B. Mourrain and R. Piene, Algebraic Geometry and Geometric Modeling, Springer2006. 3T.DokkenandB.Ju¨ttler,ComputationalMethodsforAlgebraicSplineSurfaces,Springer 2005. 4http://www.sintef.no/IST GAIA VI Preface Theeditorsareindebtedtothereviewers,whosecommentshavehelpedgreatly toidentifythemanuscriptssuitableforpublication,andforimprovingmanyofthem substantially.ThankstoSpringerfortheconstructivecooperationduringtheproduc- tion of this book. Special thanks go to Ms. Bayer for compiling the LATEX sources intoasinglecoherentmanuscript. OsloandLinz, BertJu¨ttler August2007 RagniPiene Contents PartI SurveyoftheEuropeanprojectGAIAII 1 TheGAIAProjectonIntersectionandImplicitization T.Dokken ....................................................... 5 PartII Somespecialalgebraicsurfaces 2 SomeCovariantsRelatedtoSteinerSurfaces F.Aries,E.Briand,C.Bruchou ...................................... 31 3 RealLineArrangements andSurfaceswithManyRealNodes S.Breske,O.Labs,D.vanStraten .................................... 47 4 MonoidHypersurfaces P.H.Johansen,M.Løberg,R.Piene .................................. 55 5 CanalSurfacesDefinedbyQuadraticFamiliesofSpheres R.Krasauskas,S.Zube ............................................. 79 6 GeneralClassificationof(1,2)ParametricSurfacesinP3 T.-H.Leˆ,A.Galligo................................................ 93 PartIII Algorithmsforgeometriccomputing 7 CurveParametrizationoverOptimalFieldExtensions ExploitingtheNewtonPolygon T.Beck,J.Schicho.................................................119 8 RidgesandUmbilicsofPolynomialParametricSurfaces F.Cazals,J.-C.Fauge`re,M.Pouget,F.Rouillier .........................141 VIII Contents 9 IntersectingBiquadraticBe´zierSurfacePatches S.Chau,M.Oberneder,A.Galligo,B.Ju¨ttler ...........................161 10 CubeDecompositionsbyEigenvectorsofQuadraticMultivariate Splines I.Ivrissimtzis,H.-P.Seidel ..........................................181 11 SubdivisionMethodsfortheTopologyof2dand3dImplicitCurves C.Liang,B.Mourrain,J.-P.Pavone...................................199 12 Approximate Implicitization of Space Curves and of Surfaces ofRevolution M.Shalaby,B.Ju¨ttler ..............................................215 Index ............................................................. 229 Part I Survey of the European project GAIA II 3 The European project GAIA II entitled Intersection algorithms for geometry based IT-applications using approximate algebraic methods (IST 2001-35512) in- volvedsixacademicandindustrialpartnersfromfivecountries.Theprojectaimedat combiningknowledgefromComputerAidedGeometricDesign,classicalalgebraic geometryandrealsymboliccomputationinordertoimproveintersectionalgorithms for Computer Aided Design systems. The project has has produced more than 50 scientificpublicationsandseveralsoftwaretoolkits,whicharenowpartlyavailable undertheGNUGPLlicense. We invited the coordinator of the project, Tor Dokken, to present a survey de- scribingthebackground,themethods,theresultsandtheachievementsoftheGAIA project.Hissummaryisthefirstpartofthisvolume.