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Mathematics and Visualization SeriesEditors GeraldFarin Hans-ChristianHege DavidHoffman ChristopherR.Johnson KonradPolthier MartinRumpf Jean-Daniel Boissonnat Monique Teillaud Editors Effective Computational Geometry for Curves and Surfaces With 120 Figures and 1 Table ABC Jean-Daniel Boissonnat Monique Teillaud INRIA Sophia-Antipolis 2004 route des Lucioles B.P. 93 06902 Sophia-Antipolis, France E-mail: [email protected] [email protected] Cover Illustration: Cover Image by Steve Oudot (INRIA, Sophia Antipolis) The standard left trefoil knot, represented as the intersection between two algebraic surfaces that are the images through a stereographic projection of two submanifolds of the unit 3-sphere S3 – further details can be found in [1, Chap. III, Section 8.5]. This picture was obtained from a 3D model generated with the CGAL surface meshing algorithm. [1] E. Brieskorn and H. Knörrer. Plane Algebraic Curves. Birkhäuser, Basel Boston Stuttgart, 1986. LibraryofCongressControlNumber:2006931844 MathematicsSubjectClassification: 68U05; 65D18; 14Q05; 14Q10; 14Q20; 68N19; 68N30; 65D17; 57Q15; 57R05; 57Q55; 65D05; 57N05; 57N65; 58A05; 68W05; 68W20; 68W25; 68W40; 68W30; 33F05; 57N25; 58A10; 58A20; 58A25. ISBN-10 3-540-33258-8 SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-332589SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:1)c Springer-VerlagBerlinHeidelberg2006 Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. TypesettingbytheauthorsandSPiusingaSpringerLATEXmacropackage Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper SPIN:11732891 46/SPi/3100 543210 Preface Computationalgeometryemergedasadisciplineintheseventiesandhashad considerable success in improving the asymptotic complexity of the solutions tobasicgeometricproblemsincludingconstructionsofdatastructures,convex hulls,triangulations,Voronoidiagramsandgeometricarrangementsaswellas geometric optimisation. However, in the mid-nineties, it was recognized that the computational geometry techniques were far from satisfactory in practice and a vigorous effort has been undertaken to make computational geometry more practical. This effort led to major advances in robustness, geometric software engineering and experimental studies, and to the development of a large library of computational geometry algorithms, Cgal. The goal of this book is to take into consideration the multidisciplinary nature of the problem and to provide solid mathematical and algorithmic foundationsforeffectivecomputationalgeometryforcurvesandsurfaces.This book covers two main approaches. In a first part, we discuss exact geometric algorithms for curves and surfaces. We revisit two prominent data structures of computational geometry, namely arrangements (Chap. 1) and Voronoi diagrams (Chap. 2) in order to understand how these structures, which are well-known for linear objects, behavewhendefinedoncurvedobjects.Themathematicalpropertiesofthese structures are presented together with algorithms for their construction. To ensure the effectiveness of our algorithms, the basic numerical computations thatneedtobeperformedarepreciselyspecified,andtradeoffsareconsidered between the complexity of the algorithms (i.e. the number of primitive calls), and the complexity of the primitives and their numerical stability. Chap. 3 presents recent advances on algebraic and arithmetic tools that are keys to solve the robustness issues of geometric computations. In a second part, we discuss mathematical and algorithmic methods for approximating curves and surfaces. The search for approximate representa- tions of curved objects is motivated by the fact that algorithms for curves and surfaces are more involved, harder to ensure robustness of, and typically VI Preface several orders of magnitude slower than their linear counterparts. This book provides widely applicable, fast, safe and quality-guaranteed approximations of curves and surfaces. Although these problems have received considerable attentioninthepast,thesolutionspreviouslyproposedweremostlyheuristics andlimitedinscope.Weestablishtheoreticalfoundationstotheproblemand introduce two emerging new topics: discrete differential geometry (Chap. 4) and computational topology (Chap. 7). In addition, we present certified algorithms for mesh generation (Chap. 5) and surface reconstruction (Chap. 6), two problems of great practical significance. Each chapter refers to open source software, in particular Cgal, and discusses potential applications of the presented techniques. In 1995, Cgal, the Computational Geometry Algorithms Library, was founded as a research project with the goal of making correct and efficient implementations for the large body of geometric algorithms developed in the field of computational geometry available for industrial applications. It has since then evolved to an open source project [2] and now is the state-of-art implementation in many areas. A short appendix (Chap. 8) on generic programming and the Cgal library is included. This book can serve as a textbook on non-linear computational geometry. It will also be useful to engineers and researchers working in computational geometry or other fields such as structural biology, 3-dimensional medical imaging, CAD/CAM,robotics, graphicsetc.Eachchapter describesthestate of the art algorithms as well as provides a tutorial introduction to important conceptsandmethodsthatarebothwellfoundedmathematicallyandefficient in practice. This book presents recent results of the Ecg project, a Shared-Cost RTD (FET Open) Project of the European Union1 devoted to effective computational geometry for curves and surfaces. More information on Ecg, includ- ing the results obtained during this project, can be found on the web site http://www-sop.inria.fr/prisme/ECG/. We wish to thank Franz Aurenhammer, Frédéric Chazal, E´ric Colin de Verdière, Tamal Dey, Ioannis Emiris, Andreas Fabri, Menelaos Karavelas, John Keyser, Edgar Ramos, Fabrice Rouillier, and many other colleagues, for their cooperation and feedback which greatly helped us to improve the quality of this book. 1Number IST-2000-26473 List of Contributors Jean-Daniel Boissonnat Joachim Giesen INRIA ETH Zu¨rich BP 93 CAB G33.2, ETH Zentrum 06902 Sophia Antipolis cedex CH-8092 Zu¨rich France Switzerland Jean-Daniel.Boissonnat [email protected] @sophia.inria.fr Dan Halperin School of Computer Science Frédéric Cazals Tel Aviv University INRIA Tel Aviv 69978 BP 93 Israel 06902 Sophia Antipolis cedex [email protected] France [email protected] Lutz Kettner Max-Planck-Institut fu¨r Informatik David Cohen-Steiner Stuhlsatzenhausweg 85 66123 Saarbru¨cken INRIA Germany BP 93 [email protected] 06902 Sophia Antipolis cedex France Jean-Marie Morvan David.Cohen-Steiner Institut Camille Jordan @sophia.inria.fr Université Claude Bernard Lyon 1 43 boulevard du 11 novembre 1918 Efraim Fogel 69622 Villeurbanne cedex School of Computer Science France Tel Aviv University [email protected] Tel Aviv 69978 Israel Bernard Mourrain [email protected] INRIA VIII List of Contributors BP 93 Elias Tsigaridas 06902 Sophia Antipolis cedex Department of Informatics and France Telecommunications [email protected] National Kapodistrian University of Athens Panepistimiopolis 15784 Sylvain Pion Greece INRIA [email protected] BP 93 06902 Sophia Antipolis cedex Gert Vegter France Institute for Mathematics and [email protected] Computer Science University of Groningen P.O. Box 800 Gu¨nter Rote 9700 AV Groningen Freie Universita¨t Berlin The Netherlands Institut fu¨r Informatik [email protected] Takustraße 9 14195 Berlin Ron Wein Germany School of Computer Science [email protected] Tel Aviv University Tel Aviv 69978 Israel Susanne Schmitt [email protected] Max-Planck-Institut fu¨r Informatik Stuhlsatzenhausweg 85 Nicola Wolpert 66123 Saarbru¨cken Max-Planck-Institut fu¨r Informatik [email protected] Stuhlsatzenhausweg 85 66123 Saarbru¨cken Jean-Pierre Técourt [email protected] INRIA BP 93 Camille Wormser 06902 Sophia Antipolis cedex INRIA France BP 93 Jean-Pierre.Tecourt 06902 Sophia Antipolis cedex @sophia.inria.fr France [email protected] Monique Teillaud Mariette Yvinec INRIA INRIA BP 93 BP 93 06902 Sophia Antipolis cedex 06902 Sophia Antipolis cedex France France [email protected] [email protected] Contents 1 Arrangements Efi Fogel, Dan Halperin(cid:1), Lutz Kettner, Monique Teillaud, Ron Wein, Nicola Wolpert .................................................. 1 1.1 Introduction ................................................. 1 1.2 Chronicles................................................... 3 1.3 Exact Construction of Planar Arrangements ..................... 5 1.3.1Construction by Sweeping ................................ 7 1.3.2Incremental Construction ................................. 20 1.4 Software for Planar Arrangements .............................. 25 1.4.1The Cgal Arrangements Package.......................... 26 1.4.2Arrangements Traits ..................................... 33 1.4.3Traits Classes from Exacus............................... 36 1.4.4An Emerging Cgal Curved Kernel ........................ 38 1.4.5How To Speed Up Your Arrangement Computation in Cgal .. 40 1.5 Exact Construction in 3-Space ................................. 41 1.5.1Sweeping Arrangements of Surfaces ........................ 41 1.5.2Arrangements of Quadrics in 3D........................... 45 1.6 Controlled Perturbation: Fixed-Precision Approximation of Arrangements................................................ 50 1.7 Applications ................................................. 53 1.7.1Boolean Operations on Generalized Polygons................ 53 1.7.2Motion Planning for Discs ................................ 57 1.7.3Lower Envelopes for Path Verification in Multi-Axis NC-Machining........................................... 59 1.7.4Maximal Axis-Symmetric Polygon Contained in a Simple Polygon ................................................ 62 1.7.5Molecular Surfaces....................................... 63 1.7.6Additional Applications .................................. 64 1.8 Further Reading and Open problems............................ 66 X Contents 2 Curved Voronoi Diagrams Jean-Daniel Boissonnat(cid:1), Camille Wormser, Mariette Yvinec.......... 67 2.1 Introduction ................................................. 68 2.2 Lower Envelopes and Minimization Diagrams .................... 70 2.3 Affine Voronoi Diagrams ...................................... 72 2.3.1Euclidean Voronoi Diagrams of Points...................... 72 2.3.2Delaunay Triangulation................................... 74 2.3.3Power Diagrams ......................................... 78 2.4 Voronoi Diagrams with Algebraic Bisectors ...................... 81 2.4.1Mo¨bius Diagrams ........................................ 81 2.4.2Anisotropic Diagrams .................................... 86 2.4.3Apollonius Diagrams ..................................... 88 2.5 Linearization ................................................ 92 2.5.1Abstract Diagrams....................................... 92 2.5.2Inverse Problem ......................................... 97 2.6 Incremental Voronoi Algorithms................................ 99 2.6.1Planar Euclidean diagrams................................101 2.6.2Incremental Construction .................................101 2.6.3The Voronoi Hierarchy ...................................106 2.7 Medial Axis .................................................109 2.7.1Medial Axis and Lower Envelope ..........................110 2.7.2Approximation of the Medial Axis .........................110 2.8 Voronoi Diagrams in Cgal ....................................114 2.9 Applications .................................................115 3 Algebraic Issues in Computational Geometry Bernard Mourrain(cid:1), Sylvain Pion, Susanne Schmitt, Jean-Pierre Técourt, Elias Tsigaridas, Nicola Wolpert...........................117 3.1 Introduction .................................................117 3.2 Computers and Numbers ......................................118 3.2.1Machine Floating Point Numbers: the IEEE 754 norm........119 3.2.2Interval Arithmetic ......................................120 3.2.3Filters..................................................121 3.3 Effective Real Numbers .......................................123 3.3.1Algebraic Numbers ......................................124 3.3.2Isolating Interval Representation of Real Algebraic Numbers ..125 3.3.3Symbolic Representation of Real Algebraic Numbers .........125 3.4 Computing with Algebraic Numbers ............................126 3.4.1Resultant...............................................126 3.4.2Isolation................................................131 3.4.3Algebraic Numbers of Small Degree ........................136 3.4.4Comparison.............................................138 3.5 Multivariate Problems ........................................140 3.6 Topology of Planar Implicit Curves .............................142 3.6.1The Algorithm from a Geometric Point of View .............143 Contents XI 3.6.2Algebraic Ingredients.....................................144 3.6.3How to Avoid Genericity Conditions .......................145 3.7 Topology of 3d Implicit Curves.................................146 3.7.1Critical Points and Generic Position........................147 3.7.2The Projected Curves ....................................148 3.7.3Lifting a Point of the Projected Curve......................149 3.7.4Computing Points of the Curve above Critical Values.........151 3.7.5Connecting the Branches .................................152 3.7.6The Algorithm ..........................................153 3.8 Software ....................................................154 4 Differential Geometry on Discrete Surfaces David Cohen-Steiner, Jean-Marie Morvan(cid:1)..........................157 4.1 Geometric Properties of Subsets of Points .......................157 4.2 Length and Curvature of a Curve...............................158 4.2.1The Length of Curves ....................................158 4.2.2The Curvature of Curves .................................159 4.3 The Area of a Surface.........................................161 4.3.1Definition of the Area ....................................161 4.3.2An Approximation Theorem ..............................162 4.4 Curvatures of Surfaces ........................................164 4.4.1The Smooth Case........................................164 4.4.2Pointwise Approximation of the Gaussian Curvature .........165 4.4.3From Pointwise to Local..................................167 4.4.4Anisotropic Curvature Measures ...........................174 4.4.5(cid:1)-samples on a Surface....................................178 4.4.6Application .............................................179 5 Meshing of Surfaces Jean-Daniel Boissonnat, David Cohen-Steiner, Bernard Mourrain, Gu¨nter Rote(cid:1), Gert Vegter ........................................181 5.1 Introduction: What is Meshing?................................181 5.1.1Overview ...............................................187 5.2 Marching Cubes and Cube-Based Algorithms ....................188 5.2.1Criteria for a Correct Mesh Inside a Cube ..................190 5.2.2Interval Arithmetic for Estimating the Range of a Function ...190 5.2.3Global Parameterizability: Snyder’s Algorithm...............191 5.2.4Small Normal Variation ..................................196 5.3 Delaunay Refinement Algorithms...............................201 5.3.1Using the Local Feature Size ..............................202 5.3.2Using Critical Points .....................................209 5.4 A Sweep Algorithm...........................................213 5.4.1Meshing a Curve ........................................215 5.4.2Meshing a Surface .......................................216 5.5 Obtaining a Correct Mesh by Morse Theory .....................223 5.5.1Sweeping through Parameter Space ........................223