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.