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.

翻译：摘要：闭合曲面 $Σ$ 和有限点集 $V \subset Σ$ 的限制性Delaunay三角剖分是 $V$ 的Delaunay四面体剖分的一个子复形，其三角形近似于 $Σ$。众所周知，如果 $V$ 是光滑曲面 $Σ$ 的足够稠密样本，则限制性Delaunay三角形的并集与 $Σ$ 同胚。我们证明，当 $ε \leq 0.3245$ 的 $ε$-样本即满足此条件。相比之下，Dey证明了 $0.18$-样本的结论；我们改进的采样界将所需样本点数减少了 $3.25$ 倍。更重要的是，我们改进了Cheng等人在Delaunay曲面网格划分中相关的采样界，将所需样本点数减少了 $21$ 倍。我们同胚证明的第一步尤为有趣：我们证明对于 $0.44$-样本，每个站点 $v \in V$ 的限制性Voronoi细胞与圆盘同胚，且该细胞到 $T_vΣ$（$Σ$ 在 $v$ 处的切平面）的正交投影是星形的。