Computational Line and Sphere Geometry
Methods from line geometry and sphere geometries turn out to be useful in a number of applications including robotics, geometric modeling and 3D shape understanding and reconstruction. However,the application of these geometries does not come without efforts. For example, approximation problems require the definition of appropriate, computationally tractable error metrics and recognition tasks require special models not treated in the classical literature. The present project not only investigates this computational aspect of the classical theory, but also extends the classical work. One example is line element geometry, which may be viewed as extension of both line geometry and Laguerre sphere geometry.

Supported by the Austrian Science Fund (FWF) under grant S92.



Conchoid surfaces of quadrics

D. Gruber and M. Peternell
Journal of Symbolic Computation 59: 36-53, 2013

Conchoid surfaces of spheres

M. Peternell, D. Gruber, and J. Sendra
Comput. Aided Geom. Design, 30: 35-44, 2013

Darboux Cyclides and Webs from Circles

H. Pottmann, L. Shi and M. Skopenkov
Comput. Aided Geom. Design, 29: 77-97, 2012

Generalized Dupin Cyclides with Rational Lines of Curvature

M. Peternell
Proc. Conf. Avignon 2010 (Springer 2012)

Conchoid surfaces of rational ruled surfaces

M. Peternell, D. Gruber and J. Sendra
Computer Aided Geometric Design, 28, 2011.
panels

Edge offset meshes in Laguerre geometry.

H. Pottmann, P. Grohs, and B. Blaschitz.
Adv. Comp. Math. (2010)

Rational Two-Parameter Families of Spheres
and Rational Offset Surfaces

M. Peternell
J. Symbolic Computation 45 , 1-18, 2010.

Rational offset surfaces and their modeling applications

Krasauskas, R. and Peternell, M.
IMA Volume 151: Nonlinear Computational Geometry, (eds.) I.Z. Emiris, F. Sottile, and Th. Theobald, p. 109-136, 2009.
panels

On Generalized LN-Surfaces in 4-Space.

M. Peternell and B. Odehnal.
Proceedings of ISSAC08, 223-230, 2008
envelope

On quadratic two-parameter families of spheres and their envelopes

M. Peternell, B. Odehnal, M. L. Sampoli
Comput. Aided Geom. Design 25, 342-355 (2008).
bezier

Convolution surfaces of quadratic triangular Bézier surfaces

M. Peternell and B. Odehnal
Comput. Aided Geom. Design (2008).
motion

Subdivision Schemes for the fair Discretization of the Spherical Motion Group

G. Nawratil and H. Pottmann
Journal of Computational and Applied Mathematics 222 (2) 574-591 (2008).
balls

Approximating boundary-triangulated objects with balls.

O. Aichholzer, F. Aurenhammer, T. Hackl, B. Kornberger, M. Peternell and H. Pottmann
In Proc. 23rd European Workshop on Computational Geometry.
voronoi

Voronoi diagrams for oriented spheres.

F. Aurenhammer, J. Wallner, M. Peternell and H. Pottmann
In Proc. ISVD'07: 4th Int. Conf. Voronoi Diagrams in Science and Engineering.IEEE Computer Society, 2007.
sphere

Sphere-geometric aspects of bisector surfaces

M. Peternell
Proceedings of the Conference Algebraic Geometry and Geometric Modeling 2006