Table of Contents
Preface v
Table of Contents vii
1 Fundamentals 1
1.1 Real Projective Geometry 1
1.1.1 The Real Projective Plane 1
1.1.2 n-dimensional Projective Space 6
1.1.3 Projective Mappings 11
1.1.4 Projectivities, Cross Ratio and Harmonic Position 20
1.1.5 Polarities and Quadrics 28
1.1.6 Complex Extension and the Way from Projective to Euclidean Geometry 54
1.2 Basic Projective Differential Geometry 68
1.2.1 Curves 68
1.2.2 Surfaces 78
1.2.3 Duality 82
1.3 Elementary Concepts of Algebraic Geometry 86
1.3.1 Definitions and Algorithms 86
1.3.2 Geometric Properties of Varieties in Projective Space 99
1.3.3 Duality 104
1.4 Rational Curves and Surfaces in Geometric Design 105
1.4.1 Rational Bézier Curves 105
1.4.2 Dual Bézier Curves 121
1.4.3 Rational Bézier Surfaces 126
2 Models of Line Space 133
2.1 The Klein Model 133
2.1.1 Plücker Coordinates 133
2.1.2 Computing with Plücker Coordinates 137
2.1.3 The Klein Quadric 141
2.2 The Grassmann Algebra 144
2.3 The Study Sphere 154
3 Linear Complexes 159
3.1 The Structure of a Linear Complex 159
3.1.1 Linear Complexes and Null Polarities in Projective Space 159
3.1.2 Linear Complexes and Helical Motions in Euclidean Space 163
3.1.3 Linear Complexes in the Klein Model 168
3.2 Linear Manifolds of Complexes 171
3.2.1 Pencils of Linear Line Complexes 172
3.2.2 Euclidean Properties of Pencils of Linear Complexes 178
3.3 Reguli and Bundles of Linear Complexes 181
3.4 Applications 185
3.4.1 Spatial Kinematics 185
3.4.2 Statics and Screw Theory 191
4 Approximation in Line Space 195
4.1 Fitting Linear Complexes 195
4.2 Kinematic Surfaces 202
4.3 Approximation via Local Mappings into Euclidean 4-Space 211
4.4 Approximation in the Set of Line Segments 221
5 Ruled Surfaces 223
5.1 Projective Differential Geometry of Ruled Surfaces 223
5.1.1 Infinitesimal Properties of First Order 225
5.1.2 Infinitesimal Properties of Higher Order 234
5.2 Algebraic Ruled Surfaces 238
5.2.1 Rational Ruled Surfaces 242
5.2.2 The Bézier Representation of Rational Ruled Surfaces 247
5.2.3 Skew Cubic Surfaces 252
5.3 Euclidean Geometry of Ruled Surfaces 261
5.3.1 First Order Properties 263
5.3.2 A Complete System of Euclidean Invariants 270
5.4 Numerical Geometry of Ruled Surfaces 282
5.4.1 Discrete Models and Difference Geometry 282
5.4.2 Interpolation and Approximation Algorithms 291
5.4.3 Variational Design 296
5.4.4 Offset Surfaces and their Applications 303
5.4.5 Intersection of Ruled Surfaces 309
Color Plates 311
6 Developable Surfaces 327
6.1 Differential Geometry of Developable Surfaces 327
6.2 Dual Representation 334
6.2.1 Differential Geometry of the Dual Surface 334
6.2.2 Developable Bézier and B-Spline Surfaces 343
6.2.3 Interpolation and Approximation Algorithms with Developable Surfaces 352
6.3 Developable Surfaces of Constant Slope and Applications 358
6.3.1 Basics 359
6.3.2 The Cyclographic Mapping and its Applications 366
6.3.3 Rational Developables Surfaces of Constant Slope and Rational Pythagorean-Hodograph Curves 383
6.4 Connecting Developables and Applications 396
6.4.1 Basics 396
6.4.2 Convex Hulls and Binder Surfaces 400
6.4.3 Geometric Tolerancing 405
6.4.4 Two-Dimensional Normed Spaces and Minkowski Offsets 410
6.5 Developable Surfaces with Creases 416
7 Line Congruences and Line Complexes 423
7.1 Line Congruences 423
7.1.1 Projective Differential Geometry of Congruences 423
7.1.2 Rational Congruences and Trivariate Bézier Representations 428
7.1.3 Euclidean Differential Geometry of Line Congruences 434
7.1.4 Normal Congruences and Geometrical Optics 446
7.1.5 Singularities of Motions Constrained by Contacting Surfaces and Applications in Sculptured Surface Machining 452
7.1.6 Numerical Geometry of Line Congruences 465
7.1.7 Projection via Line Congruences 469
7.2 Line Complexes 474
7.2.1 Differential Geometry of Line Complexes 474
7.2.2 Algebraic Complexes and Congruences 480
7.2.3 Special Quadratic Complexes 487
8 Linear Line Mappings — Computational Kinematics 497
8.1 Linear Line Mappings and Visualization of the Klein Model 497
8.1.1 Linear Line Mappings into P2 498
8.1.2 Linear Line Mappings into P3 511
8.1.3 Visualization of the Klein Image 519
8.2 Kinematic Mappings 522
8.2.1 Quaternions 523
8.2.2 The Spherical Kinematic Mapping 527
8.2.3 Other Kinematic Mappings 535
8.3 Motion Design 538
References 547
List of Symbols 556
Index 557
