Rounding Meshes in 3D

Abstract

Let ${\mathcal {P}}$ be a set of n polygons in ${\mathbb {R}}^3$, each of constant complexity and with pairwise disjoint interiors. We propose a rounding algorithm that maps ${\mathcal {P}}$ to a simplicial complex ${\mathcal {Q}}$ whose vertices have integer coordinates. Every face of ${\mathcal {P}}$ is mapped to a set of faces (or edges or vertices) of ${\mathcal {Q}}$ and the mapping from ${\mathcal {P}}$ to ${\mathcal {Q}}$ can be done through a continuous motion of the faces such that: (i) the $L_\infty$ Hausdorff distance between a face and its image during the motion is at most 3/2, and (ii) if two points become equal during the motion, they remain equal through the rest of the motion. In the worst case the size of ${\mathcal {Q}}$ is $O(n^{13})$ and the time complexity of the algorithm is $O(n^{15})$ but, under reasonable assumptions, these complexities decrease to $O(n^{4}\sqrt{n})$ and $O(n^{5})$. Furthermore, these complexities are likely not tight and we expect, in practice on non-pathological data, $O(n\sqrt{n})$ space and time complexities.