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Better Sampling Bounds for Restricted Delaunay Triangulations and a Star-Shaped Property for Restricted Voronoi Cells

Author Jonathan Richard Shewchuk

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Restricted Delaunay triangulation
  • restricted Voronoi diagram
  • surface sampling
  • surface mesh generation
  • surface reconstruction
  • ε-sample
  • homeomorphism

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Abstract

The restricted Delaunay triangulation of a closed surface Σ and a finite point set V ⊂ Σ is a subcomplex of the Delaunay tetrahedralization of V whose triangles approximate Σ. It is well known that if V is a sufficiently dense sample of a smooth Σ, then the union of the restricted Delaunay triangles is homeomorphic to Σ. We show that an ε-sample with ε ≤ 0.3245 suffices. By comparison, Dey proves it for a 0.18-sample; our improved sampling bound reduces the number of sample points required by a factor of 3.25. More importantly, we improve a related sampling bound of Cheng et al. for Delaunay surface meshing, reducing the number of sample points required by a factor of 21. The first step of our homeomorphism proof is particularly interesting: we show that for a 0.44-sample, the restricted Voronoi cell of each site v ∈ V is homeomorphic to a disk, and the orthogonal projection of the cell onto T_vΣ (the plane tangent to Σ at v) is star-shaped.

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Jonathan Richard Shewchuk. Better Sampling Bounds for Restricted Delaunay Triangulations and a Star-Shaped Property for Restricted Voronoi Cells. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 90:1-90:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.90

Author Details

Jonathan Richard Shewchuk
  • University of California, Berkeley, CA, USA

Funding

Supported by the National Science Foundation under Award CCF-1909204.

Acknowledgements

I thank Nina Amenta, Jean-Daniel Boissonnat, Siu-Wing Cheng, Tamal Dey, Arijit Ghosh, and Marc Khoury for discussions about surface sampling; and INRIA Sophia-Antipolis and the Geometrica Group, where this work began, for their kind reception during my 2010 sabbatical.

References

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