Computational Methods for Algebraic Spline Surfaces: ESF Exploratory Workshop

Cover......Page 1
S Title......Page 2
Title......Page 3
ISBN 3-540-23274-5......Page 4
Preface......Page 6
Table of Contents......Page 8
1.1 Passive Observation Systems......Page 9
1.3 Measurement Error of the POS......Page 10
1.4 Confidence Sets......Page 11
2.1 Intersection of Quadric Surfaces of Revolution......Page 12
2.2 Example......Page 13
3.2 Implicit Representation......Page 14
3.4 Symbolic Computation of the Taylor Expansion......Page 15
3.5 Example......Page 16
4 Conclusion......Page 17
References......Page 18
1 Introduction......Page 19
3 Surface-Surface Intersections......Page 20
4 Recursive Subdivision......Page 23
5 Intersections Between one Parametric and one AlgebraicSurface......Page 24
6 Approximate Implicitization......Page 25
7 Interception Testing......Page 26
8 Simple case Situations......Page 27
9 Singular Surface-Surface Intersections......Page 28
10 Partial Coincidences and Tangential Intersections......Page 30
11 Conclusion......Page 32
References......Page 34
1 Introduction......Page 35
2.1 Resultant and Projection......Page 36
3 Topology of Algebraic Curves......Page 38
3.1 Critical Points and Generic Position......Page 39
3.2 The Projected Curves......Page 40
3.3 Lifting a Point of C′......Page 41
3.4 Computing Points of C at Critical Values......Page 42
3.5 Connecting the Branches......Page 44
4 Implementation and Experiments......Page 47
4.2 Examples of 3D Curves......Page 49
References......Page 51
1 Introduction......Page 53
2 Basic Notions and Preliminary Results......Page 55
3 ǫ–Roots......Page 60
4 ǫ–Points on Curves......Page 63
References......Page 69
1 Introduction and Background......Page 71
2 Distance Maps and Projected Distance Maps......Page 73
3 Examples......Page 75
4 Refining the IFA Tests......Page 78
References......Page 81
Appendix A: The Distance Functions for Two B´ezier Curves......Page 82
1 Introduction......Page 85
2 Projective Varieties, Degree, and Projection......Page 86
3 Varieties of Minimal Degree......Page 88
4 Curves of Almost Minimal Degree......Page 89
5 Surfaces of Almost Minimal Degree......Page 90
6 Classification of Del Pezzo Surfaces......Page 93
7 Del Pezzo Surfaces of Degree 2 and 1......Page 95
8 Real Del Pezzo Surfaces......Page 99
References......Page 101
2 Tangent Developables......Page 103
3 Local Properties of a Real Tangent Developable......Page 104
4 Illustrations......Page 106
References......Page 113
1 Introduction......Page 115
2.1 The Companion Resultant Matrix......Page 116
2.2 The Sylvester Resultant Matrix......Page 117
2.3 The B´ezout Resultant Matrix......Page 119
3 The Numerical Condition of a Resultant Matrix......Page 120
4 The Transformation of a Resultant Matrix Between the Powerand Bernstein Bases......Page 123
5 Summary......Page 125
References......Page 126
1 Introduction......Page 127
2 M-patches and their tensor-border......Page 128
2.1 Definition of M-Patches......Page 129
2.2 Tensor-border Structure......Page 130
2.3 Reparametrizations to Form a Tensor-Border......Page 131
3.1 Outline of the Construction......Page 132
3.2 The C2 Transition Between the Reparametrizations ρ2;s and a Bicubic C2Reparametrization......Page 133
3.3 Subsequent Annuli of Bidegree (5,5)......Page 134
3.4 Annuli of B´ezier Patches......Page 135
4 A Spline Cap of Bidegree (11,11)......Page 136
4.1 The 135-Reparametrizations ˜γs......Page 137
4.2 A Total Degree 7 C2 Approximation of the Geometry......Page 139
4.3 Capping Patches of Bidegree (11, 11)......Page 140
References......Page 142
1 Introduction......Page 143
2 The Subdivision Algorithm......Page 145
3 Taylor Method for Bounds......Page 146
4 Finding Bounds on Derivatives......Page 148
5.1 Algebraic Curves......Page 149
5.2 Algebraic Surfaces......Page 153
6 Why use Order Two Taylor Expansion?......Page 157
7 Theoretical Connection Between Taylor Method and MAA......Page 160
References......Page 161
1 Introduction......Page 163
2 Implicitization......Page 166
3 Finding Self-Intersections......Page 167
4 Surfaces......Page 169
5 Approximate Implicitization......Page 171
6 Two Open Problems......Page 172
6.2 How do we Find Self-Intersection Curves (as Opposed to Points)?......Page 173
References......Page 177
1 Introduction......Page 179
2 Numerical Characters of a Projected Surface......Page 180
3 The Veronese Surfaces......Page 181
4 The Segre Surfaces......Page 184
5 Del Pezzo Surfaces......Page 185
6 Rational Scrolls......Page 186
7 Monoid Surfaces......Page 188
References......Page 189
1 Introduction and Preliminaries......Page 191
2 Parabolic Loci......Page 192
3 Minimal Loci......Page 193
4 On the Influence of the Extrema of the Weight Function of theFree Control Point......Page 194
5 Numerical Experimentation with NURBS......Page 196
5.1 Cylinder......Page 197
5.2 Ellipsoid......Page 198
5.3 Torus......Page 199
References......Page 200
1 Introduction......Page 201
2 Invariants on Surfaces......Page 202
3 The Generators......Page 203
4 The Syzygies......Page 206
5 The Structure of the Invariant Ring......Page 209
6 Implicit Surfaces......Page 212
7 Applications......Page 213
7.1 Fairing......Page 215
7.2 Ridges and the Subparabolic Curve......Page 217
7.3 Darboux’s Classification of Umbilical Points......Page 218
References......Page 219
1 Introduction......Page 221
2.1 Examples of Universal Rational Parametrizations......Page 222
2.2 Complex Toric Surfaces......Page 224
2.3 Singular Case and Desingularization......Page 225
3 URP Theorem......Page 226
4.1 Real Structures on Quadrics......Page 228
4.3 The Torus Surface is Almost Toric!......Page 229
5.1 Interpolation on the Projective Line CP1......Page 231
5.2 Splines on CP1 and RP1......Page 232
5.4 Splines on Hirzebruch Surfaces......Page 234
5.5 Splines on a Hexagonal Toric B´ezier patch......Page 235
6 Application: Blending Natural Quadrics......Page 236
7 Conclusions......Page 238
References......Page 239
Panel Discussion......Page 241