Subdividing Alpha Complex

Abstract

Given two simplicial complexes ${\mathcal C}_{\rm 1}$ and ${\mathcal C}_{\rm 2}$ embedded in Euclidean space ${\mathbb R}^{d}$, ${\mathcal C}_{\rm 1}$ subdivides ${\mathcal C}_{\rm 2}$ if (i) ${\mathcal C}_{\rm 1}$ and ${\mathcal C}_{\rm 2}$ have the same underlying space, and (ii) every simplex in ${\mathcal C}_{\rm 1}$ is contained in a simplex in ${\mathcal C}_{\rm 2}$. In this paper we present a method to compute a set of weighted points whose alpha complex subdivides a given simplicial complex.

Following this, we also show a simple method to approximate a given polygonal object with a set of balls via computing the subdividing alpha complex of the boundary of the object. The approximation is robust and is able to achieve a union of balls whose Hausdorff distance to the object is less than a given positive real number ε.