Computational Line Geometry

H. Pottmann and J. Wallner

Computational Line Geometry by H. Pottmann and J. Wallner, Springer Verlag, Berlin, 2001, ISBN 3-540-42058-4, 565 pp. 264 figs., 17 in color. New softcover edition to appear in November 2009. Springer-Verlag, Berlin 2010. ISBN 978-3-642-04017-7. See here for information from Springer Verlag.

Abstract (First edition, 2001) This book for the first time studies line geometry from the viewpoint of scientific computation and shows the interplay between theory and numerous applications. On the one hand, the reader will find a modern presentation of `classical' material. On the other hand we show how the methods of line geometry enable an elegant approach to many problems whose connection to line geometry is not obvious at first sight.

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

1. Fundamentals 1.1 – Real Projective Geometry / 1.2 – Basic Projective Differential Geometry / 1.3 – Elementary Concepts of Algebraic Geometry / 1.4 – Rational Curves and Surfaces in Geometric Design 2. Models of Line Space 2.1 – The Klein Model / 2.2 – The Grassmann Algebra / 2.3 – The Study Sphere 3. Linear Complexes 3.1 – The Structure of a Linear Complex / 3.2 – Linear Manifolds of Complexes / 3.3 – Reguli and Bundles of Linear Complexes / 3.4 – Applications 4. Approximation in Line Space 4.1 – Fitting Linear Complexes / 4.2 – Kinematic Surfaces / 4.3 – Approximation via Local Mappings into Euclidean 4-Space / 4.4 – Approximation in the Set of Line Segments 5. Ruled Surfaces 5.1 – Projective Differential Geometry of Ruled Surfaces / 5.2 – Algebraic Ruled Surfaces / 5.3 – Euclidean Geometry of Ruled Surfaces / 5.4 – Numerical Geometry of Ruled Surfaces 6. Developable Surfaces 6.1 – Differential Geometry of Developable Surfaces / 6.2 – Dual Representation / 6.3 – Developable Surfaces of Constant Slope and Applications / 6.4 – Connecting Developables and Applications / 6.5 – Developable Surfaces with Creases 7. Line Congruences and Line Complexes 7.1 – Line Congruences / 7.2 – Line Complexes 8. Linear Line Mappings — Computational Kinematics 8.1 – Linear Line Mappings and Visualization of the Klein Model 8.2 – Kinematic Mappings 8.3 – Motion Design