Delaunay Triangulation of Manifolds

  • Jean-Daniel Boissonnat
  • Ramsay Dyer
  • Arijit Ghosh
Article

Abstract

We present an algorithm for producing Delaunay triangulations of manifolds. The algorithm can accommodate abstract manifolds that are not presented as submanifolds of Euclidean space. Given a set of sample points and an atlas on a compact manifold, a manifold Delaunay complex is produced for a perturbed point set provided the transition functions are bi-Lipschitz with a constant close to 1, and the original sample points meet a local density requirement; no smoothness assumptions are required. If the transition functions are smooth, the output is a triangulation of the manifold. The output complex is naturally endowed with a piecewise-flat metric which, when the original manifold is Riemannian, is a close approximation of the original Riemannian metric. In this case the output complex is also a Delaunay triangulation of its vertices with respect to this piecewise-flat metric.

Keywords

Delaunay complex Triangulation Manifold Protection Perturbation 

Mathematics Subject Classification

Primary 57R05 Secondary 52B70 54B15 

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Copyright information

© SFoCM 2017

Authors and Affiliations

  • Jean-Daniel Boissonnat
    • 1
  • Ramsay Dyer
    • 1
  • Arijit Ghosh
    • 2
  1. 1.INRIA, DataShapeSophia-AntipolisFrance
  2. 2.ACM UnitIndian Statistical InstituteKolkataIndia

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