Arrangements on Parametric Surfaces II: Concretizations and Applications

Eric Berberich, Efi Fogel, Dan Halperin, Michael Kerber, Ophir Setter

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)67-91
Number of pages25
JournalMathematics in Computer Science
Issue number1
StatePublished - Nov 2010


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


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