Effective Computational Geometry for Curves and Surfaces

1 Arrangements
Efi Fogel, Dan Halperin, Lutz Kettner, Monique Teillaud, Ron Wein, Nicola Wolpert 1.1 Introduction
1.2 Chronicles
1.3 Exact Construction of Planar Arrangements
1.3.1Construction by Sweeping
1.3.2 Incremental Construction
1.4 Software for Planar Arrangements
1.4.1 The Cgal Arrangements Package
1.4.2 Arrangements Traits
1.4.3 Traits Classes from Exacus
1.4.4An Emerging Cgal Curved Kernel
1.4.5 How To Speed UpYour Arrangement Computation in Cgal
1.5 Exact Construction in 3-Space
1.5.1 Sweeping Arrangements of Surfaces
1.5.2Arrangements of Quadricsin 3D
1.6 Controlled Perturbation: Fixed-Precision Approximation of Arrangements
1.7 Applications
1.7.1 Boolean Operations for Conics
1.7.2 Motion Planning for Discs
1.7.3 Lower Envelopes for Path Verification in Multi-Axis NC-Machining
1.7.4 Maximal Axis-Symmetric Polygon Containedin a Simple Polygon
1.7.5 Molecular Surfaces
1.7.6 Additional Applications
1.8 Further Reading and Open problems
2 Curved Voronoi Diagrams
Jean-Daniel Boissonnat, Camille Wormser, Mariette Yvinec
2.1 Introduction
2.2 Lower Envelopes and Minimization Diagrams
2.3 Affine Voronoi Diagrams
2.3.1 Euclidean Voronoi Diagrams of Points
2.3.2 Delaunay Triangulation
2.3.3 PowerDiagrams
2.4 Voronoi Diagrams with Algebraic Bisectors
2.4.1 Möbius Diagrams
2.4.2 Anisotropic Diagrams
2.4.3Apollonius Diagrams
2.5 Linearization
2.5.1Abstract Diagrams
2.5.2 Inverse Problem
2.6 Incremental Voronoi Algorithms
2.6.1 Planar Euclidean diagrams
2.6.2 Incremental Construction
2.6.3 The Voronoi Hierarchy
2.7 Medial Axis
2.7.1 Medial Axis and Lower Envelope
2.7.2 Approximation of the Medial Axis
2.8 Voronoi Diagrams in Cgal
2.9 Applications
3 Algebraic Issues in Computational Geometry
Bernard Mourrain, Sylvain Pion, Susanne Schmitt, Jean-Pierre Técourt, Elias Tsigaridas, Nicola Wolpert
3.1 Introduction
3.2 Computers and Numbers
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 Effective Real Numbers .......................................123
3.3.1 Algebraic Numbers ......................................124
3.3.2 Isolating Interval Representation of Real Algebraic Numbers
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
3.4.3Algebraic 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
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 CriticalValues.........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 Mo