3D Snap Rounding

Olivier Devillers (1) , Sylvain Lazard (1) , William Lenhart (2)

1 GAMBLE - Geometric Algorithms and Models Beyond the Linear and Euclidean realm
2 Computer Science Department [Williamstown MA]

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 $O(n^{15})$ and the time complexity of the algorithm is $O(n^{19})$ but, under reasonable hypotheses, these complexities decrease to $O(n^{5})$ and $O(n^{6}\sqrt{n})$.

Mots clés: Geometric algorithms; Robustness; Fixed-precision computations

Domaines: Géométrie algorithmique [cs.CG]

Olivier Devillers, Sylvain Lazard, William Lenhart. 3D Snap Rounding. Proceedings of the 34th International Symposium on Computational Geometry, Jun 2018, Budapest, Hungary. pp.30:1 - 30:14, ⟨10.4230/LIPIcs.SoCG.2018.30⟩. ⟨hal-01727375⟩