Medial axis and contact types

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Table of contents

I Presentation of general notions 
1 Geometry 
1.1 Geometry of plane curves
1.1.1 The notion of curvature
1.1.2 The medial axis of a curve
1.2 Geometry of surfaces
1.2.1 Basic notions on surfaces
1.2.2 Analytic description of a surface
1.3 Generic surface
1.3.1 Maxima of curvature
1.3.2 Medial axis and contact types
1.3.3 Summary of the cases
2 Combinatorics 
2.1 Graph theory
2.1.1 Generalities on graphs
2.1.2 Counting the edges of a planar graph
2.2 The Delaunay triangulation
2.2.1 The 2D-Delaunay triangulation
2.2.2 In three dimensions (and higher)
3 Probabilities 
3.1 Poisson point process
3.2 Slivnyak-Mecke’s theorem
4 State of the art 
4.1 Expected size of the dD-Delaunay triangulation of a uniform set of points
4.2 Analysis of the 3D-Delaunay triangulation of points on a polyhedral surface
4.2.1 Probabilistic analysis
4.2.2 Deterministic analysis
4.2.3 On polyhedral surfaces of any dimension
4.3 Evaluating the size of the Delaunay triangulation with respect to the spread of the points
4.4 Another probabilistic approach
4.5 Deterministic nice sample on generic surfaces
5 A Poisson sample on a surface is a good sample 
5.1 Notation, definitions, previous results
5.2 Is a Poisson sample a good sample?
II Stochastic analysis of empty region graphs 
6 A sub-graph and a super-graph of the 2D-Delaunay triangulation 
6.1 The Gabriel and half-moon graphs
7 General method 
7.1 Empty region graphs
7.2 Combination and Partition lemmas
7.3 Alternative proof of the linear complexity of the Delaunay triangulation
7.4 Formalized method
7.5 Expected degree in some empty singleton-region graphs
8 Empty axis-aligned ellipse graphs 
8.1 Some features with axis-aligned ellipses
8.2 Empty axis-aligned ellipse graph
8.2.1 Upper bound on the expected degree
8.2.2 Lower bound on the expected degree
8.3 Ellipses with bounded aspect ratio, the rhombus graph
8.3.1 An upper bound on the expected degree
8.3.2 A lower bound on the expected degree
8.4 On empty axis-aligned ellipse graphs with a single aspect ratio
9 On the probability of the existence of far neighbors 
9.1 In the Delaunay triangulation
9.2 In the empty axis-aligned graph with bounded aspect ratio
10 Analysis of two additional empty region graphs 
10.1 Empty ellipse graph with bounded aspect ratio
10.2 Empty 4/2-ball graph
11 On nearest-neighbor-like graphs, a way to compute some integrals 
11.1 The nearest-neighbor graph
11.2 Formalization
11.3 Application of nearest-neighbor-like graphs
III 3D-Delaunay triangulation for two specific surfaces 
12 Delaunay triangulation of a Poisson process on a cylinder 
12.1 The right circular cylinder
12.2 Description of the fundamental regions on the cylinder
12.3 Proof of the graph inclusion
12.4 Computation of an upper bound on E [] Del(X)]
12.5 A lower bound on E [] Del(X)]
12.6 Conjecture on two classes of surface
13 Delaunay triangulation of a Poisson process on an oblate spheroid 
13.1 The oblate spheroid
13.1.1 Some generalities on the oblate spheroid
13.2 Overview of the proof
13.3 On the probability of existence of neighbors far from the medial sphere
13.4 Expected number of close neighbors
13.4.1 General scheme
13.4.2 Description of the regions of F1 on the spheroid
13.4.3 Choice of specific spheres for q on the side of p with respect to PMed
13.4.4 Proof of the graph inclusion
13.4.5 When q is on the side of p with respect to PMed
13.4.6 Computation of an upper bound on the expected number of close neighbors
13.4.7 On some geometric quantities close to Z
13.4.8 On the probability of existence Delaunay neighbors outside CN(p)
13.5 Expected degree of a point close to Z
13.5.1 Description of fundamental regions on the spheroid
13.5.2 Choice of specific spheres
13.5.3 Proof of the graphs inclusion
13.5.4 Computation of an upper bound on the expected degree
13.5.5 On the probability of existence Delaunay neighbors outside MRN(p)
14 Experimental results 
14.1 Simulation
14.2 Experimental results
IV Expected size of the 3D-Delaunay triangulation of a Poisson point process distributed on a generic surface 
15 Generic surfaces 
15.1 What is generic or not in an oblate spheroid
15.1.1 Common points between an oblate spheroid and a generic surface
15.1.2 Differences between an oblate spheroid and a generic surface
15.2 Sketch of proof
15.2.1 Decomposition of the generic surface
15.2.2 Approach of the proof
16 Expected local degree of a point 
16.1 Local degree of a point on the convex hull and a little beyond
16.1.1 Choice of the specific spheres
16.1.2 Proof of the graph inclusion
16.1.3 Computation of the expected local degree
16.2 Local degree of a point far from the convex hull and from Z
16.2.1 Choice of the specific spheres
16.2.2 Proof of the graph inclusion
16.2.3 Computation of the expected degree
16.3 Local degree of a point close to Z or Y
16.3.1 On the position of p and the value 1 􀀀 1(p)r(p)
16.3.2 Local degree of a point at distance hp from Z
16.3.3 Local degree of a point close to Y
Bibliography