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Arrangements on Parametric Surfaces II: Concretizations and Applications

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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.

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Authors and Affiliations

  1. School of Computer Science, Tel Aviv University, Tel Aviv, 69978, Israel

    Eric Berberich, Efi Fogel, Dan Halperin & Ophir Setter

  2. D1: Algorithms and Complexity, Max-Planck-Institut für Informatik, 66123, Saarbrücken, Germany

    Michael Kerber

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  1. Eric Berberich
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  2. Efi Fogel
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  3. Dan Halperin
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  4. Michael Kerber
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  5. Ophir Setter
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Correspondence to Eric Berberich.

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This work has been supported in part by the Israel Science Foundation (grant no. 236/06), by the German-Israeli Foundation (grant no. 969/07), and by the Hermann Minkowski–Minerva Center for Geometry at Tel Aviv University.

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Berberich, E., Fogel, E., Halperin, D. et al. Arrangements on Parametric Surfaces II: Concretizations and Applications. Math.Comput.Sci. 4, 67–91 (2010). https://doi.org/10.1007/s11786-010-0043-4

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