- Page 1: Medial Spheres for Shape Representa
- Page 5 and 6: ABSTRACT This thesis presents a par
- Page 7 and 8: sphères médianes permet d’effec
- Page 9 and 10: TABLE OF CONTENTS ACKNOWLEDGEMENTS
- Page 11 and 12: II Quality of Shape Description 99
- Page 13 and 14: LIST OF TABLES Table page 2-1 Stati
- Page 15 and 16: 2-2 Medial axes, in black, for two
- Page 17 and 18: 4-6 The angles of interest in Case
- Page 19 and 20: 8-4 A fat triangle, or the convex h
- Page 21 and 22: The choice of an appropriate shape
- Page 23 and 24: 1.1.1 Parametric/Boundary Represent
- Page 25 and 26: epresent a shape. Octrees use axis-
- Page 27 and 28: the shape. Blinn [15] proposed usin
- Page 29 and 30: Definition 1.4. The vectors from a
- Page 31 and 32: Figure 1-5: Left: Both the interior
- Page 33 and 34: Not only does the medial surface tr
- Page 35 and 36: Figure 1-7: The edges of the Vorono
- Page 37 and 38: The number of simplices of DC(B) is
- Page 39 and 40: points to medial points that are no
- Page 41 and 42: PUBLICATIONS [A] S. Stolpner, P. Kr
- Page 43 and 44: Chapter 2 Medial Surface Approximat
- Page 45 and 46: Figure 2-2: Medial axes, in black,
- Page 47 and 48: three categories: Voronoi methods,
- Page 49 and 50: algorithm to compute a cell complex
- Page 51 and 52: 2.2.2 Tracing Methods Computing the
- Page 53 and 54:
angle is estimated using the angle
- Page 55 and 56:
Consider a sphere S interior to B w
- Page 57 and 58:
Theorem 2.1. Consider a sphere S in
- Page 59 and 60:
To ensure the efficiency of this al
- Page 61 and 62:
where the second inequality follows
- Page 63 and 64:
B(p) = B(q). By Lemma 2.3, Algorith
- Page 65 and 66:
quadrilateral Q ′ pqr. By constru
- Page 67 and 68:
We described in Section 2.3 how vox
- Page 69 and 70:
Using Algorithm 2, at most one appr
- Page 71 and 72:
Figure 2-8: A dragon polyhedron (le
- Page 73 and 74:
most 12 locations on a given sphere
- Page 75 and 76:
We will group neighbouring approxim
- Page 77 and 78:
to organize medial points into dist
- Page 79 and 80:
Figure 2-13: Left: A bumpy sphere m
- Page 81 and 82:
availability of grid neighbourhoods
- Page 83 and 84:
Algorithm 3 DECIDEMS(B, S, Φ) Inpu
- Page 85 and 86:
Figure 3-2: Objects of interest in
- Page 87 and 88:
Proof. 1 Let x be the intersection
- Page 89 and 90:
occurs first. Let Bm ′ = B(m′ ,
- Page 91 and 92:
Summary of balls in F A f : In this
- Page 93 and 94:
a small value of w implies a small
- Page 95 and 96:
Now suppose that at least one ball
- Page 97 and 98:
with additional assumptions about t
- Page 99 and 100:
terms of practical implementations,
- Page 101 and 102:
Figure 4-1: ∇D(A), ∇D(B) point
- Page 103 and 104:
4.2 Algorithm Using the tools devel
- Page 105 and 106:
eturns ‘Undetermined’, when sam
- Page 107 and 108:
Let d(p, A) = d(p, B) be d. When d
- Page 109 and 110:
(a) (b) (c) (d) Figure 4-4: Illustr
- Page 111 and 112:
The angle β = ∠uwv gives an uppe
- Page 113 and 114:
Figure 4-6: The angles of interest
- Page 115 and 116:
R be the radius of the circle C. Le
- Page 117 and 118:
Nonetheless, the error bounds we ha
- Page 119 and 120:
Chapter 5 Boundary Differential Geo
- Page 121 and 122:
curvatures are the called the princ
- Page 123 and 124:
estimation of boundary differential
- Page 125 and 126:
Denote ψ a (x0) by x a 0 and ψ b
- Page 127 and 128:
Surfaces can be rendered using surf
- Page 129 and 130:
5.4.2 Derivatives on Medial Sheets
- Page 131 and 132:
points we have computed does not in
- Page 133 and 134:
2. The polyhedron’s boundary is o
- Page 135 and 136:
works at a fixed spatial resolution
- Page 137 and 138:
Figure 5-5: The principal curvature
- Page 139 and 140:
this collection of medial points is
- Page 141 and 142:
κ1 κ1 κ ′ 1 µ(|κ1 − κ ′
- Page 143 and 144:
solid [109]. However, this operatio
- Page 145 and 146:
(a) (b) Figure 6-1: (a) A union of
- Page 147 and 148:
2 sheets 4 sheets 6 sheets 60 sheet
- Page 149 and 150:
Part III Fast and Tight Shape Appro
- Page 151 and 152:
method. When a polyhedron undergoes
- Page 153 and 154:
such methods require an optimizatio
- Page 155 and 156:
Figure 7-2: Consider the solid that
- Page 157 and 158:
|SV | = 512 506 474 497 512 496 510
- Page 159 and 160:
The AMAA construction proceeds top-
- Page 161 and 162:
7.4 Fast Updates under Deformation
- Page 163 and 164:
Also, c1 − c0 = |r0 − r1| =
- Page 165 and 166:
Following [6], the local feature si
- Page 167 and 168:
16857 triangles 4307 spheres rel. s
- Page 169 and 170:
In collision detection, one looks t
- Page 171 and 172:
Consider the set of balls B corresp
- Page 173 and 174:
0.4 0.2 0 −0.5 0 0.5 1 0.5 0 −0
- Page 175 and 176:
Figure 8-3: A rectangle swept spher
- Page 177 and 178:
The timings for the AMAA sphere con
- Page 179 and 180:
Error Timings Size Ave. Max. SH RSS
- Page 181 and 182:
conservative approximations. This i
- Page 183 and 184:
Construct a bounding volume hierarc
- Page 185 and 186:
Given a set of spheres, generate an
- Page 187 and 188:
Chapter 9 Thesis Summary and Conclu
- Page 189 and 190:
e used to generate sphere set appro
- Page 191 and 192:
spheres, where the choice of sheet
- Page 193 and 194:
174 [11] D. Attali, J.D. Boissonnat
- Page 195 and 196:
[38] J. Damon. Global geometry of r
- Page 197 and 198:
[64] M. Gross and H. Pfister. Point
- Page 199 and 200:
[91] C. O’Sullivan and J. Dinglia
- Page 201:
182 [116] M. Teichmann and S. Telle