Table of Contents
Abstract vi
Chapter 1 Introduction 1
1.1 Intersection detection of convex polyhedra . . . 1
1.2 Voronoi diagrams . . . 3
1.3 Facility location problems . . . 5
1.3.1 The k-center problem . . . 6
1.3.2 The geodesic 1-center . . . 7
Chapter 2 Related Work: State of the Art 8 2.1 Intersection detection of convex polyhedra . . . 8
2.2 Incremental Voronoi diagrams . . . 10
2.3 The k-center problem . . . 11
2.4 The geodesic 1-center . . . 12
Chapter 3 Preliminaries 13 3.1 Set systems . . . 14
3.1.1 VC-dimension and "-nets . . . 14
3.1.2 Geometric set systems . . . 15
3.1.3 Cuttings . . . 17
3.2 Decision to Optimization techniques . . . 18
3.3 Lower bound for membership problems . . . 20
3.4 Farthest-point Voronoi diagrams . . . 20
3.4.1 Properties of the farthest-point Voronoi diagram . . . 21
I
Intersection detection
24
4.1 Outline . . . 26
4.2 Algorithm in the plane . . . 28
4.2.1 The 2D algorithm . . . 29
4.2.2 Separation invariant. . . 29
4.2.3 Intersection invariant. . . 30
4.3 The polar transformation . . . 32
4.4 Polyhedra in 3D space . . . 36
4.4.1 Bounded hierarchies . . . 37
4.5 Detecting intersections in 3D . . . 40
4.5.1 Preprocessing . . . 40
4.5.2 Preliminaries of the algorithm . . . 40
4.5.3 The algorithm . . . 42
4.5.4 A walk in the primal space. . . 42
4.5.5 A walk in the polar space. . . 44
4.5.6 Analysis of the algorithm . . . 45
4.6 Detecting intersections in higher dimensions . . . 46
4.6.1 Hierarchical trees . . . 46
4.6.2 Testing intersection in higher dimensions . . . 48
4.6.3 Preprocessing . . . 48
4.6.4 Preliminaries of the algorithm . . . 48
4.6.5 Separation invariant. . . 50
4.6.6 Inverse separation invariant. . . 50
II
Voronoi diagrams and Facility location
52
Chapter 5 Incremental Voronoi 53 5.1 Flarbs . . . 545.2 Results . . . 55
5.3 Outline . . . 56
5.5 The combinatorial upper bound . . . 61
5.5.1 Flarbable sub-curves . . . 61
5.5.2 How much do faces shrink in a flarb? . . . 64
5.5.3 Flarbable sequences . . . 66
5.6 The lower bound . . . 69
5.7 Computing the flarb . . . 70
5.7.1 Grappa trees . . . 70
5.7.2 The Voronoi diagram . . . 73
5.7.3 Heavy paths in Voronoi diagrams . . . 74
5.7.4 Finding non-preserved edges. . . 76
5.7.5 The compressed tree . . . 79
5.7.6 Completing the flarb . . . 81
Chapter 6 Constrained minimum enclosing circle 83 6.1 Outline . . . 85
6.2 Preliminaries . . . 86
6.2.1 P -circles . . . 86
6.3 Solving the decision problem on a set of segments . . . 87
6.3.1 The algorithm. . . 88
6.3.2 Constructing slices . . . 89
6.3.3 Divide and conquer . . . 92
6.4 Converting decision to optimization . . . 94
6.5 Constraining to a simple polygon . . . 98
6.5.1 Star-shaped regions . . . 98
6.5.2 The decision problem on a simple polygon . . . 99
6.6 Lower bounds . . . 101
6.6.1 Lower bounds when constraining to a set of points . . . 101
6.6.2 Lower bound when constraining to a simple polygon . . . 103
6.6.3 Another lower bound when constraining to sets of points . . . 106
6.6.5 The A-B-subset problem . . . 107
Chapter 7 Geodesic Center 112 7.1 Outline . . . 112
7.2 Decomposing the boundary . . . 114
7.3 Hourglasses . . . 116
7.3.1 Building hourglasses . . . 122
7.4 Funnels . . . 123
7.4.1 Funnels of marked vertices . . . 124
7.5 Covering the polygon with apexed triangles . . . 125
7.5.1 Inside a transition hourglass . . . 126
7.5.2 Inside the funnels of marked vertices . . . 129
7.6 Prune and search . . . 130
7.7 Finding the center within a triangle . . . 134
7.7.1 Optimization problem in a convex domain . . . 136
7.8 Bounding the VC dimension . . . 138
Bibliography 141
