Computational Line Geometry

1. Front Matter

   Pages I-IX

   PDF

2. Fundamentals

   • Helmut Pottmann, Johannes Wallner

   Pages 1-132

3. Models of Line Space

   • Helmut Pottmann, Johannes Wallner

   Pages 133-158

4. Linear Complexes

   • Helmut Pottmann, Johannes Wallner

   Pages 159-194

5. Approximation in Line Space

   • Helmut Pottmann, Johannes Wallner

   Pages 195-222

6. Ruled Surfaces

   • Helmut Pottmann, Johannes Wallner

   Pages 223-326

7. Developable Surfaces

   • Helmut Pottmann, Johannes Wallner

   Pages 327-422

8. Line Congruences and Line Complexes

   • Helmut Pottmann, Johannes Wallner

   Pages 423-496

9. Linear Line Mappings — Computational Kinematics

   • Helmut Pottmann, Johannes Wallner

   Pages 497-546

10. Back Matter

    Pages 547-563

    PDF

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.

• Area
• animation
• clipping
• geometric design
• kinematics
• line geometry
• scientific computing
• visualization

From the reviews of the first edition:

"... The depth of the presentation is one of the striking things about this work. It does not only scratch the surface of the numerous fields but delves into the matter meticulously. But all the same there is enough room left for applications which keep the book exciting, not only for students and mathematicians but also for ambitious engineers. ... I can well imagine that this book will render a tremendous service to many of its readers. I wholeheartedly recommend this outstanding monograph."

J.Lang (Graz), IMN - Internationale Mathematische Nachrichten 190, 2002, pp. 65-66

"...There is a vast amount of fascinating geometry of all sorts in this book. The topics are perhaps somewhat eclectic - they mirror the primary interests of the authors - but, because the motivation is to develop the geometry that applies to real world problems, the subject is far from monolithic and is open to interpretation. The ideas here build up layer upon layer. In the end, the authors have been mostly successful in sustaining their central theme, despite the need to weave together projective, differential, algebraic and metric geometry. They have also presented the mathematics in a predominantly modern way. That is important because there exist in the engineering literature archaelogical remnants of outdated notation and concepts. [....] The large number (264) of line diagrams are of very good quality and considerably enhance one's understanding. [...] a book which is without doubt an important contribution to this growing branch of geometrical research."

P. Donelan (Wellington) - New Zealand Matheamtical Society Newsletter 87, April 2003.

"The authors have combined results from the classical parts of geometry with computational methods. This results in a unique and fascinating blend, which is shown to be useful for a variety of applications, includingrobotics, geometrical optics, computer animation, and geometric design. The contents of the book are visualized by a wealth of carefully chosen illustrations, making the book a sheer pleasure to read, or even just browse in. The book will help to bring the concepts and techniques of line geometry, which have been shown to be useful for various applications in geometric design and engineering, to the attention of a wider audience."—

MATHEMATICAL REVIEWS

"The didactically well-considered and marvelously illustrated book is self-contained at least for readers acquainted with basic concepts of (classical) differential geometry and CAGD, and, of course, with (advanced) linear algebra and some calculus. The reader of this demanding textbook is richly rewarded by applicable knowledge, and he surely will enjoy the elegance of geometric reasoning and the effectiveness of line-geometric calculations."

(Gunter Weiß, Zentralblatt MATH, Vol. 1006, 2003)

"... In der Einleitung schreiben die Autoren, dass ihnen von verschiedenen Kollegen naheglegt wurde, ein Buch über "klassische" Geometrie und deren Anwendung auf geometrisches Rechnen zu schreiben. Da ihnen diese Aufgabe als zu umfangreich erschien, haben sie sich hier auf die Geraden-Geometrie beschränkt. Sie haben damit eine sehr glückliche Wahl getroffen: Einserseits ist diese Darstellung thematisch einheitlich und geschlossen, andererseits wird eine schier unglaubliche Fülle von Material detailliert beschrieben. ... Das Buch enthält 237 sehr informative und sorgfältig ausgeführte Illustrationen und 17 Farbtafeln. Damit ist das Buch auch ästhetisch sehr beeindruckend. Das Buch wird wohl ein Standard-Werk für die sogenannte "Angewandte Geometrie" werden, eine Weiterentwicklung der traditionellen Ingenieurs-Geometrie mit modernsten Methoden."

W.Barth (Erlangen), Jahresbericht der DMV 105, Band (2003), Heft 2

• Technology (KAUST), King Abdullah University of Science &, Jeddah, Saudi Arabia

  Helmut Pottmann

• Institut für Geometrie, TU Graz, Graz, Austria

  Johannes Wallner

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