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

Jonathan Richard Shewchuk

Abstract

The restricted Delaunay triangulation of a closed surface $Σ$ and a finite point set $V \subset Σ$ 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 $ε\leq 0.3245$ 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 $v \in V$ 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.

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

Abstract

The restricted Delaunay triangulation of a closed surface and a finite point set is a subcomplex of the Delaunay tetrahedralization of whose triangles approximate . It is well known that if is a sufficiently dense sample of a smooth , then the union of the restricted Delaunay triangles is homeomorphic to . We show that an -sample with suffices. By comparison, Dey proves it for a -sample; our improved sampling bound reduces the number of sample points required by a factor of . 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 . The first step of our homeomorphism proof is particularly interesting: we show that for a -sample, the restricted Voronoi cell of each site is homeomorphic to a disk, and the orthogonal projection of the cell onto (the plane tangent to at ) is star-shaped.
Paper Structure (8 sections, 28 theorems, 3 equations, 2 figures)

This paper contains 8 sections, 28 theorems, 3 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Restricted Delaunay triangulations and ${\bf \epsilon}$-samples
  3. A relationship between surface points and nearby tangent planes
  4. Restricted Voronoi cells are topological disks
  5. Restricted Voronoi vertices are lone points
  6. Restricted Voronoi edges are topological intervals
  7. The restricted Delaunay triangulation is homeomorphic to ${\bf \Sigma}$
  8. Restricted Voronoi cells in higher dimensions

Key Result

Lemma 1

Consider two points $v, x \in \Sigma$ such that $|vx| < \xi \, \mathrm{lfs}(v)$, where $\xi = \sqrt{(\sqrt{5} - 1) / 2} \doteq 0.786151$. Then $o$ and $o'$ lie on strictly opposite sides of $T_x\Sigma$.

Figures (2)

  • Figure 1: (a) A two-dimensional view of restricted Delaunay triangulations. The input is a smooth, closed curve $\Sigma$ and a sample $V \subset \Sigma$. (b) The restricted Voronoi diagram is the restriction of the (classic) Voronoi diagram to $\Sigma$. (c) The restricted Delaunay triangulation (bold) is the dual of the restricted Voronoi diagram and a subcomplex of the (classic) Delaunay triangulation. (d) A restricted Voronoi diagram in three dimensions. (e) Its dual restricted Delaunay triangulation.
  • Figure 2: Left: A $1$-manifold $\Sigma$ and its medial axis $M$. Center: Some of the medial balls that define $M$. Balls with black centers touch two points on $\Sigma$. The white points are in the closure of the black centers. Right: A $0.5$-sample of $\Sigma$ (black points). The ball with center $x$ and radius $0.5 \, \mathrm{lfs}(x)$ contains a site.

Theorems & Definitions (28)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 6
  • Lemma 7
  • Lemma 8
  • Lemma 9: Feature Translation Lemma amenta02dey07
  • Corollary 10
  • ...and 18 more