Let P be a set of n polygons in R^3, each of constant complexity and with pairwise disjoint interiors. We propose a rounding algorithm that maps P to a simplicial complex Q whose vertices have integer coordinates. Every face of P is mapped to a set of faces (or edges or vertices) of Q and the mapping from P to 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 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})

Devillers, Olivier

Lazard, Sylvain

Lenhart, William J.

English

International audienceLet $\mathcal{P}$ 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 $O(n^{6}\sqrt{n})$

3D Snap Rounding

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