Mathematics and Visualization Series Editors Gerald Farin Hans-Christian Hege David Hoffman Christopher R. Johnson Konrad Polthier Martin Rumpf Editors ABC Effective Computational Jean-Daniel Boissonnat Monique Teillaud With 120 Figures and 1 Table Geometry for Curves and Surfaces ISBN-10 ISBN-13 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media c  Springer-Verlag Berlin Heidelberg 2006 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. A E Cover design: design & production GmbH, Heidelberg Printed on acid-free paper 543210 springer.com Typesetting by the authors and SPi using a Springer L T X macro package 46/SPi/3100 SPIN: 11732891 978-3-540-332589 Springer Berlin Heidelberg New York Library of Congress Control Number: 2006931844 Cover Illustration: Cover Image by Steve Oudot (INRIA, Sophia Antipolis) [1] E. Brieskorn and H. Knörrer. Plane Algebraic Curves. Birkhäuser, Basel Boston Stuttgart, 1986. 68N30; 65D17; 57Q15; 57R05; 57Q55; 65D05; 57N05; 57N65; 58A05; 68W05; 68W20; Mathematics Subject Classification: 68U05; 65D18; 14Q05; 14Q10; 14Q20; 68N19; 68W25; 68W40; 68W30; 33F05; 57N25; 58A10; 58A20; 58A25. 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. 3-540-33258-8 Springer Berlin Heidelberg New York Monique Teillaud Jean-Daniel Boissonnat INRIA Sophia-Antipolis 2004 route des Lucioles B.P. 93 06902 Sophia-Antipolis, France E-mail: Jean-Daniel.Boissonnat@sophia.inria.fr Monique.Teillaud@sophia.inria.fr Preface Computational geometry emerged as a discipline in the seventies and has had considerable success in improving the asymptotic complexity of the solutions to basic geometric problems including constructions of data structures, convex hulls, triangulations, Voronoi diagrams and geometric arrangements as well as 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 foundations for effective computational geometry for curves and surfaces. This book covers two main approaches. In a first part, we discuss exact geometric algorithms for curves and sur- faces. 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, behave when defined on curved objects. The mathematical properties of these structures are presented together with algorithms for their construction. To ensure the effectiveness of our algorithms, the basic numerical computations that need to be performed are precisely specified, and tradeoffs are considered 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 attention in the past, the solutions previously proposed were mostly heuristics and limited in scope. We establish theoretical foundations to the problem and introduce two emerging new topics: discrete differential geometry (Chap. 4) and computational topology (Chap. 7). In addition, we present certified algo- rithms 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, graphics etc. Each chapter describes the state of the art algorithms as well as provides a tutorial introduction to important concepts and methods that are both well founded mathematically and efficient in practice. This book presents recent results of the Ecg project, a Shared-Cost RTD (FET Open) Project of the European Union 1 devoted to effective computa- tional 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, Fr´ed´eric Chazal, ´ Eric Colin de Verdi`ere, 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. 1 Number IST-2000-26473 List of Contributors Jean-Daniel Boissonnat INRIA BP 93 06902 Sophia Antipolis cedex France Jean-Daniel.Boissonnat @sophia.inria.fr Fr´ed´eric Cazals INRIA BP 93 06902 Sophia Antipolis cedex France Frederic.Cazals@sophia.inria.fr David Cohen-Steiner INRIA BP 93 06902 Sophia Antipolis cedex France David.Cohen-Steiner @sophia.inria.fr Efraim Fogel School of Computer Science Tel Aviv University Tel Aviv 69978 Israel efif@post.tau.ac.il Joachim Giesen ETH Z¨urich CAB G33.2, ETH Zentrum CH-8092 Z¨urich Switzerland giesen@inf.ethz.ch Dan Halperin School of Computer Science Tel Aviv University Tel Aviv 69978 Israel danha@tau.ac.il Lutz Kettner Max-Planck-Institut f¨ur Informatik Stuhlsatzenhausweg 85 66123 Saarbr¨ucken Germany kettner@mpi-inf.mpg.de Jean-Marie Morvan Institut Camille Jordan Universit´e Claude Bernard Lyon 1 43 boulevard du 11 novembre 1918 69622 Villeurbanne cedex France morvanjeanmarie@yahoo.fr Bernard Mourrain INRIA VIII List of Contributors BP 93 06902 Sophia Antipolis cedex France Bernard.Mourrain@sophia.inria.fr Sylvain Pion INRIA BP 93 06902 Sophia Antipolis cedex France Sylvain.Pion@sophia.inria.fr G¨unter Rote Freie Universit¨at Berlin Institut f¨ur Informatik Takustraße 9 14195 Berlin Germany rote@inf.fu-berlin.de Susanne Schmitt Max-Planck-Institut f¨ur Informatik Stuhlsatzenhausweg 85 66123 Saarbr¨ucken sschmitt@mpi-inf.mpg.de Jean-Pierre T´ecourt INRIA BP 93 06902 Sophia Antipolis cedex France Jean-Pierre.Tecourt @sophia.inria.fr Monique Teillaud INRIA BP 93 06902 Sophia Antipolis cedex France Monique.Teillaud@sophia.inria.fr Elias Tsigaridas Department of Informatics and Telecommunications National Kapodistrian University of Athens Panepistimiopolis 15784 Greece et@di.uoa.gr Gert Vegter Institute for Mathematics and Computer Science University of Groningen P.O. Box 800 9700 AV Groningen The Netherlands gert@cs.rug.nl Ron Wein School of Computer Science Tel Aviv University Tel Aviv 69978 Israel wein@post.tau.ac.il Nicola Wolpert Max-Planck-Institut f¨ur Informatik Stuhlsatzenhausweg 85 66123 Saarbr¨ucken nicola.wolpert@hft-stuttgart.de Camille Wormser INRIA BP 93 06902 Sophia Antipolis cedex France Camille.Wormser@sophia.inria.fr Mariette Yvinec INRIA BP 93 06902 Sophia Antipolis cedex France Mariette.Yvinec@sophia.inria.fr Contents 1 Arrangements Efi Fogel, Dan Halperin  , 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 1.3.1 Construction by Sweeping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.2 Incremental Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.4 Softwarefor PlanarArrangements 25 1.4.1 The Cgal ArrangementsPackage 26 1.4.2 Arrangements Traits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.4.3 Traits Classes from Exacus 36 1.4.4 An Emerging Cgal CurvedKernel 38 1.4.5 How To Speed Up Your Arrangement Computation in Cgal 40 1.5 Exact Construction in 3-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.5.1 Sweeping Arrangements of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . 41 1.5.2 Arrangements of Quadrics in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 1.6 Controlled Perturbation: Fixed-Precision Approximation of Arrangements 50 1.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 1.7.1 Boolean Operations on Generalized Polygons . . . . . . . . . . . . . . . . 53 1.7.2 Motion Planning for Discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 1.7.3 Lower Envelopes for Path Verification in Multi-Axis NC-Machining 59 1.7.4 Maximal Axis-Symmetric Polygon Contained in a Simple Polygon 62 1.7.5 Molecular Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 1.7.6 Additional Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 1.8 FurtherReadingandOpenproblems 66 X Contents 2 Curved Voronoi Diagrams Jean-Daniel Boissonnat  , Camille Wormser, Mariette Yvinec 67 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.2 Lower Envelopes and Minimization Diagrams . . . . . . . . . . . . . . . . . . . . 70 2.3 Affine Voronoi Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.3.1 Euclidean Voronoi Diagrams of Points . . . . . . . . . . . . . . . . . . . . . . 72 2.3.2 Delaunay Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.3.3 Power Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.4 Voronoi Diagrams with Algebraic Bisectors . . . . . . . . . . . . . . . . . . . . . . 81 2.4.1 M¨obius Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.4.2 Anisotropic Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 2.4.3 Apollonius Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.5 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2.5.1 Abstract Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2.5.2 Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 2.6 Incremental Voronoi Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.6.1 Planar Euclidean diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2.6.2 Incremental Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2.6.3 The Voronoi Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 2.7 MedialAxis 109 2.7.1 Medial Axis and Lower Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . 110 2.7.2 Approximation of the Medial Axis . . . . . . . . . . . . . . . . . . . . . . . . . 110 2.8 Voronoi Diagrams in Cgal 114 2.9 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 3 Algebraic Issues in Computational Geometry Bernard Mourrain  , Sylvain Pion, Susanne Schmitt, Jean-Pierre T´ecourt, Elias Tsigaridas, Nicola Wolpert 117 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.2 Computersand Numbers 118 3.2.1 Machine Floating Point Numbers: the IEEE 754 norm . . . . . . . . 119 3.2.2 Interval Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.2.3 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.3 EffectiveRealNumbers 123 3.3.1 Algebraic Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 3.3.2 Isolating Interval Representation of Real Algebraic Numbers . . 125 3.3.3 Symbolic Representation of Real Algebraic Numbers . . . . . . . . . 125 3.4 Computing with Algebraic Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.4.1 Resultant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.4.2 Isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 3.4.3 Algebraic Numbers of Small Degree . . . . . . . . . . . . . . . . . . . . . . . . 136 3.4.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 3.5 Multivariate Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 3.6 Topology of Planar Implicit Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 3.6.1 The Algorithm from a Geometric Point of View . . . . . . . . . . . . . 143 Contents XI 3.6.2 Algebraic Ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 3.6.3 How to Avoid Genericity Conditions . . . . . . . . . . . . . . . . . . . . . . . 145 3.7 Topology of 3d Implicit Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 3.7.1 Critical Points and Generic Position . . . . . . . . . . . . . . . . . . . . . . . . 147 3.7.2 The Projected Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 3.7.3 Lifting a Point of the Projected Curve . . . . . . . . . . . . . . . . . . . . . . 149 3.7.4 Computing Points of the Curve above Critical Values. . . . . . . . . 151 3.7.5 Connecting the Branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 3.7.6 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 3.8 Software 154 4 Differential Geometry on Discrete Surfaces David Cohen-Steiner, Jean-Marie Morvan  157 4.1 Geometric Properties of Subsets of Points . . . . . . . . . . . . . . . . . . . . . . . 157 4.2 Lengthand CurvatureofaCurve 158 4.2.1 The Length of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 4.2.2 The Curvature of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 4.3 TheAreaofaSurface 161 4.3.1 Definition of the Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.3.2 An Approximation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.4 Curvaturesof Surfaces 164 4.4.1 The Smooth Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 4.4.2 Pointwise Approximation of the Gaussian Curvature . . . . . . . . . 165 4.4.3 From Pointwise to Local . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 4.4.4 Anisotropic Curvature Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 4.4.5 -samples onaSurface 178 4.4.6 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5 Meshing of Surfaces Jean-Daniel Boissonnat, David Cohen-Steiner, Bernard Mourrain, G¨unter Rote  ,GertVegter 181 5.1 Introduction: What is Meshing? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 5.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 5.2 Marching Cubes and Cube-Based Algorithms . . . . . . . . . . . . . . . . . . . . 188 5.2.1 Criteria for a Correct Mesh Inside a Cube . . . . . . . . . . . . . . . . . . 190 5.2.2 Interval Arithmetic for Estimating the Range of a Function . . . 190 5.2.3 Global Parameterizability: Snyder’s Algorithm . . . . . . . . . . . . . . . 191 5.2.4 Small Normal Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 5.3 Delaunay Refinement Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 5.3.1 Using the Local Feature Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 5.3.2 Using Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 5.4 A Sweep Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 5.4.1 Meshing a Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 5.4.2 Meshing a Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 5.5 Obtaining a Correct Mesh by Morse Theory . . . . . . . . . . . . . . . . . . . . . 223 5.5.1 Sweeping through Parameter Space . . . . . . . . . . . . . . . . . . . . . . . . 223 [...]... implementing computational geometry software in many areas Cgal contains an elaborate and efficient implementation of arrangements that supports general types of curves The recent Ecg project, which stands for Effective Computational Geometry for Curves and Surfaces, running from 2001 to 2004, extended the scope of implementation research towards curved objects.1 Arrangements of curves and surfaces were an important... nature The exact evaluation of predicates has become a research topic on its own in recent years, in particular for the case of arrangements of curves Algorithms for computing arrangements of curves (or segments of curves) require several operations to be performed exactly One essential predicate, for example, is the xy-comparison that takes two endpoints or intersection points of two segments and compares... setting for line segments We then show how it can be adapted to arbitrary x-monotone curves and derive the necessary predicates We explain in detail how these predicates can be implemented for conics, i.e., implicit algebraic curves of degree 2.2 We then briefly examine the case of algebraic curves of degree 3, also known as cubic curves, and describe what difficulties emerge as the degree of the curves. .. Cgal, the Computational Geometry Algorithms Library, was founded as a research project with the goal of making the large body of geometric algorithms developed in the field of computational geometry available for industrial applications with correct and efficient implementations [2, 222, 156] It has since then evolved to an Open Source Project and is now representing the state-of-art in implementing computational. .. sweep-line approach to cubic curves A generalization of Jacobi curves for locating tangential intersections is described by Wolpert [341] Berberich et al [50] recently extended these techniques to special quartic curves that are projections of spatial silhouette and intersection curves of quadrics, and lifted the result back into space The recent extension to algebraic curves of general degree by Wolpert... locally around p: 10 E Fogel, D Halperin, L Kettner, M Teillaud, R Wein, N Wolpert Algorithm 1 OrderToRight (C1 , , Ck ; p) 1: For each 1 ≤ i < k do: 1.1: Compute mi , the multiplicity of intersection of the curves Ci and Ci+1 at p 2: Let M ←− max {m1 , , mk−1 } 3: Let m ←− M 4: While m ≥ 1 do: 4.1: Form maximal subsequences of curves, where two curves belong to the same subsequence, if they are... http://www-sop.inria.fr/prisme/ECG/ 4 E Fogel, D Halperin, L Kettner, M Teillaud, R Wein, N Wolpert arcs of conics, where the conics can be of any type Following the newly emerging requirements from curves on the software, Fogel et al [167, 166] improved and refined the software design of the Cgal arrangement package This new design formed a common platform for a preliminary comparison documented in [165] of the different... Germany, produces a set of C++ liu braries in the project Exacus (Efficient and Exact Algorithms for Curves and Surfaces) [4] as contribution in the Ecg project with support for the Cgal arrangement class The design of the Exacus libraries is described in Berberich et al [48] The theory and the implementations for the different applications behind Exacus are described in the following series of papers:... position assumption and allow several curves to intersect at a common point p, we have to be a bit more careful when the sweep line passes over p For straight line segments the y-order of the intersecting segments just has to be reversed, but this, of course, is not necessarily true for arbitrary curves The following algorithm, taken from [49], determines the y-order of k curves immediately to the right... arrangements From the inception of computational geometry in the seventies till the mid eighties the focus was almost exclusively on the theoretical study of arrangements of unbounded linear objects, of hyperplanes Many of the results obtained during this period are summarized in the book by Edelsbrunner [130] The central role of arrangements in computational geometry has fortified in the following period, . Johnson Konrad Polthier Martin Rumpf Editors ABC Effective Computational Jean-Daniel Boissonnat Monique Teillaud With 120 Figures and 1 Table Geometry for Curves and Surfaces ISBN-10 ISBN-13 This work is. algorithmic foundations for effective computational geometry for curves and surfaces. This book covers two main approaches. In a first part, we discuss exact geometric algorithms for curves and sur- faces 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