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| Format: | Preprint |
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2026
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| Accès en ligne: | https://arxiv.org/abs/2603.19826 |
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| _version_ | 1866914411060396032 |
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| author | Shewchuk, Jonathan Richard |
| author_facet | Shewchuk, Jonathan Richard |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_19826 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Better Sampling Bounds for Restricted Delaunay Triangulations and a Star-Shaped Property for Restricted Voronoi Cells Shewchuk, Jonathan Richard Computational Geometry F.2.2 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. |
| title | Better Sampling Bounds for Restricted Delaunay Triangulations and a Star-Shaped Property for Restricted Voronoi Cells |
| topic | Computational Geometry F.2.2 |
| url | https://arxiv.org/abs/2603.19826 |