Geometric Modeling and Algebraic Geometry (eBook)

Preface 5
Contents 7
Part I Survey of the European project GAIA II 9
1 The GAIA Project on Intersection and Implicitization 11
1.1 Introduction 11
1.2 Project background and facts 12
1.3 Why are CAD-type intersections still a problem for industry? 14
1.4 Challenges of CAD-type intersections 14
1.5 Extend the use of algebraic geometry within CAD 18
1.6 Exact and approximate implicitization 20
1.7 Intersection algorithms 22
1.8 New applications of the approach of approximate implicitization 24
1.9 Potential impact of the GAIA project 25
References 28
Part II Some special algebraic surfaces 32
2 Some Covariants Related to Steiner Surfaces 34
2.1 Introduction 34
2.2 Orbits of quadratic parameterizations of quartics 36
2.3 Preliminaries on classical invariant theory 37
2.4 Some elements of the geometry of the Steiner surface 38
2.5 A collection of covariants 39
2.6 Application: Equations and inequalities defining the types of Steiner surfaces 46
Conclusion 48
References 48
3 Real Line Arrangements and Surfaces with Many Real Nodes 50
3.1 Introduction 50
3.2 Variants of Chmutov’s Surfaces with Many Real Nodes 51
3.3 On Two-Colorings of Real Simple Line Arrangements 54
3.4 Concluding Remarks 55
References 56
4 Monoid Hypersurfaces 58
4.1 Introduction 58
4.2 Basic properties 59
4.3 Monoid surfaces 61
4.4 Quartic monoid surfaces 67
References 79
5 Canal Surfaces De.ned by Quadratic Families of Spheres 81
5.1 Introduction 81
5.2 Elements of Laguerre geometry 82
5.3 Conics in R4 1 83
5.4 Rational parametrizations 86
5.5 Bezier representations 89
5.6 Implicit equations and double points 90
5.7 Conclusions 93
References 93
6 General Classi.cation of (1,2) Parametric Surfaces in P3 95
6.1 Introduction 95
6.2 Rational ruled surfaces 97
6.3 Dual scroll 98
6.4 The generic case 102
6.5 Non generic cases 105
6.6 Detection of the singularities of a patch 109
6.7 Conclusion 114
References 114
Part III Algorithms for geometric computing 116
7 Curve Parametrization over Optimal Field Extensions Exploiting the Newton Polygon 118
7.1 Introduction 118
7.2 Toric geometry 119
7.3 Divisors 124
7.4 Rational curves 127
7.5 An anticanoncial divisor 128
7.6 The algorithm 132
7.7 Example 132
References 138
8 Ridges and Umbilics of Polynomial Parametric Surfaces 140
8.1 Introduction 140
8.2 Relevant equations for ridges and its singularities 144
8.3 Certified topological approximation 146
8.4 Illustration 153
8.5 Conclusion 156
References 157
9 Intersecting Biquadratic Bezier Surface Patches 159
9.1 Introduction 159
9.2 Intersection and self–intersection curves 160
9.3 A resultant–based approach 161
9.4 Approximate implicitization by a quartic surface 163
9.5 Tracing intersections of parameter lines 166
9.6 Self–intersections of biquadratic surface patches 170
9.7 Examples 172
9.8 Conclusion 176
References 177
10 Cube Decompositions by Eigenvectors of Quadratic Multivariate Splines 179
10.1 Introduction 179
10.2 Circulant matrices and polygonal decompositions 181
10.3 G-circulant matrices 185
10.4 Cube decompositions 187
10.5 Prism decomposition 191
10.6 Conclusion - Future Work 193
10.7 Appendix 194
References 194
11 Subdivision Methods for the Topology of 2d and 3d Implicit Curves 196
11.1 Introduction 196
11.2 Fundamental ingredients 198
11.3 Algorithmic ingredients 200
11.4 Experiments 206
11.5 Discussion 210
References 210
12 Approximate Implicitization of Space Curves and of Surfaces of Revolution 212
12.1 Introduction 212
12.2 Simultaneous approximation of points and normals 213
12.3 Approximate implicitization of space curves 215
12.4 Approximate Implicitization of Surfaces of Revolution 219
12.5 Conclusion 222
References 224
Index 225