TU Graz Logo

Institute of Geometry

Contact   People   Courses & Exams   Events   Research   Lehramtsstudium  
Research project: Computational Line Geometry

This web page and the list of publications below is a snapshot of research done at TU Wien in the years 1998-2005, mostly by Helmut Pottmann and his coauthors.


Helmut Pottmann, Johannes Wallner, Computational Line Geometry, Springer Verlag, Heidelberg, Berlin u.a. 2001. (ISBN 3-540-42058-4, 565 pp. 264 figs., 17 in color)

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.

Supported by project P13648-MAT of the Austrian Science Fund (FWF).


H. Pottmann, M. Hofer, B. Odehnal, and J. Wallner. Line geometry for 3D shape understanding and reconstruction. In T. Pajdla and J. Matas, editors, Computer Vision - ECCV 2004, Part I, volume 3021 of Lecture Notes in Computer Science, pages 297-309. Springer, 2004.

We understand and reconstruct special surfaces from 3D data with line geometry methods. Based on estimated surface normals we use approximation techniques in line space to recognize and reconstruct rotational, helical, developable and other surfaces, which are characterized by the configuration of locally intersecting surface normals.

For the computational solution we use a modified version of the Klein model of line space. Obvious applications of these methods lie in Reverse Engineering. We have tested our algorithms on real world data obtained from objects such as antique pottery, gear wheels, and a surface of the ankle joint.