This web page and the list of publications below
is a snapshot of research done at TU Wien in the years
19982005, mostly by Helmut Pottmann and his coauthors.
Helmut Pottmann, Johannes Wallner,
Computational Line Geometry,
Springer Verlag, Heidelberg,
Berlin u.a. 2001. (ISBN 3540420584, 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 P13648MAT 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 297309.
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.


M. Hofer, B. Odehnal, H. Pottmann, T. Steiner, and J. Wallner.
3D shape
recognition and reconstruction based on line element geometry.
In Tenth IEEE International Conference on Computer Vision, volume 2,
pages 15321538. IEEE Computer Society, 2005, ISBN 076952334X.

B. Odehnal, H. Pottmann, and J. Wallner.
Equiform kinematics
and the geometry of line elements.
Beitr. Algebra Geom. 47/2 (2006), 567582.
[Zbl], [MR].

B. Odehnal and H. Stachel.
The upper
talocalcanean join.
Technical Report 127, Geometry Preprint Series, Vienna Univ. of Technology,
October 2004.

B. Odehnal.
Zur geometrischen
Erzeugung linearer Geradenabbildungen.
Österreich. Akad. Wiss. Math.Naturw. Kl. S.B. II 213
(2004), 4369.

B. Odehnal.
On isotropic
congruences of lines in elliptic threespace.
Math. Pannon. 16 (2005), 119135.

B. Odehnal.
On rational
isotropic congruences of lines.
J. Geom. 81 (2005), 126138.

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
297309. Springer, 2004, ISBN 3540219846.
[Zbl], [doi].

H. Pottmann and J. Wallner.
Computational
Line Geometry.
Mathematics + Visualization. Springer, Heidelberg, 2001.
ISBN 3540420584.
[Zbl], [MR].

M. Peternell.
G^{1}Hermite interpolation of ruled surfaces.
In T. Lyche and L. L. Schumaker, editors, Mathematical Methods in CAGD:
Oslo 2000, Innov. Appl. Math, pages 413422. Vanderbilt Univ. Press,
Nashville, TN, 2001.

B. Odehnal and H. Pottmann.
Computing
with discrete models of ruled surfaces and line congruences.
In F. C. Park and C. C. Iurascu, editors, Computational Kinematics,
pages 211226, 2001.
Proceedings of the workshop in Seoul, May 1922, 2001.

M. Peternell and H. Pottmann.
Interpolating functions on lines in 3space.
In A. Cohen, C. Rabut, and L. L. Schumaker, editors, Curve and Surface
Fitting: Saint Malo 1999, pages 351358. Vanderbilt University Press,
Nashville, TN, 2000.

H.Y. Chen and H. Pottmann.
Approximation by
ruled surfaces.
J. Comput. Appl. Math. 102 (1999), 143156.

H. Pottmann, M. Peternell, and B. Ravani.
An
introduction to line geometry with applications.
ComputerAided Design 31 (1999), 316.

M. Peternell, H. Pottmann, and B. Ravani.
On the
computational geometry of ruled surfaces.
ComputerAided Design 31 (1999), 1732.

H. Pottmann, M. Peternell, and B. Ravani.
Approximation in
line space: applications in robot kinematics and surface reconstruction.
In J. Lenarčič and M. Husty, editors, Advances in Robot
Kinematics: Analysis and Control, pages 403412. Kluwer, 1998.

H. Pottmann, M. Peternell, and B. Ravani.
Contributions to computational line geometry.
In D. P. Chi, H. I. Choi, M.S. Kim, and R. Martin, editors,
Differential/Topological Techniques in Geometric Modeling and Processing
'98, pages 4381. Bookplus Press, 1998, ISBN 8986518100.
Proceedings of the Workshop in Pohang, Korea, April 78,1998.
