Enhanced descriptions from Syndetics:

The geometry of lines occurs naturally in such different areas as sculptured surface machining, computation of offsets and medial axes, surface reconstruction for reverse engineering, geometrical optics, kinematics and motion design, and modeling of developable surfaces. This book covers line geometry from various viewpoints and aims towards computation and visualization. Besides applications, it contains a tutorial on projective geometry and an introduction into the theory of smooth and algebraic manifolds of lines. It will be useful to researchers, graduate students, and anyone interested either in the theory or in computational aspects in general, or in applications in particular.

Table of contents provided by Syndetics

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