Mathematics in Computer Science

, Volume 4, Issue 1, pp 67–91 | Cite as

Arrangements on Parametric Surfaces II: Concretizations and Applications

  • Eric Berberich
  • Efi Fogel
  • Dan Halperin
  • Michael Kerber
  • Ophir Setter
Article

Abstract

We describe the algorithms and implementation details involved in the concretizations of a generic framework that enables exact construction, maintenance, and manipulation of arrangements embedded on certain two-dimensional orientable parametric surfaces in three-dimensional space. The fundamentals of the framework are described in a companion paper. Our work covers arrangements embedded on elliptic quadrics and cyclides induced by intersections with other algebraic surfaces, and a specialized case of arrangements induced by arcs of great circles embedded on the sphere. We also demonstrate how such arrangements can be used to accomplish various geometric tasks efficiently, such as computing the Minkowski sums of polytopes, the envelope of surfaces, and Voronoi diagrams embedded on parametric surfaces. We do not assume general position. Namely, we handle degenerate input, and produce exact results in all cases. Our implementation is realized using Cgal and, in particular, the package that provides the underlying framework. We have conducted experiments on various data sets, and documented the practical efficiency of our approach.

Keywords

Computational geometry Arrangement of curves Parametric surface Cgal Robust geometric computing Voronoi diagram Lower envelope Gaussian map Quadric Ring Dupin cyclide 

Mathematics Subject Classification (2010)

Primary 68U05 Secondary 14Q10 

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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Eric Berberich
    • 1
  • Efi Fogel
    • 1
  • Dan Halperin
    • 1
  • Michael Kerber
    • 2
  • Ophir Setter
    • 1
  1. 1.School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  2. 2.D1: Algorithms and ComplexityMax-Planck-Institut für InformatikSaarbrückenGermany

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