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.

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

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. SkopenkovComput. Aided Geom. Design, 29: 77-97, 2012 |

Generalized Dupin Cyclides with Rational Lines of Curvature M. PeternellProc. Conf. Avignon 2010 (Springer 2012) |

Conchoid surfaces of rational ruled surfaces M. Peternell, D. Gruber and J.
Sendra
Computer Aided Geometric Design, 28,
2011. |

Edge offset meshes in
Laguerre
geometry. H. Pottmann, P.
Grohs, and B. Blaschitz.
Adv. Comp. Math. (2010) |

Rational
Two-Parameter
Families of Spheres 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. |

On Generalized
LN-Surfaces in
4-Space. M. Peternell and
B. Odehnal.
Proceedings of ISSAC08, 223-230, 2008 |

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). |

Convolution
surfaces of quadratic triangular Bézier surfaces M. Peternell and
B. Odehnal
Comput. Aided Geom. Design (2008). |

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) . |

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
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. |