Application Domains
Bilateral Contracts and Grants with Industry
Partnerships and Cooperations
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## Section: New Results

### Classical Computational Geometry

Participants : Olivier Devillers, Sylvain Lazard, Leo Valque.

#### Rounding Meshes

Let $𝒫$ be a set of $n$ polygons in ${ℝ}^{3}$, each of constant complexity and with pairwise disjoint interiors. We previously proposed [5] a rounding algorithm that maps $𝒫$ to a simplicial complex $𝒬$ whose vertices have integer coordinates such that 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 built 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. We developed [30] the first implementation of this algorithm, which is also the first implementation for rounding a mesh on a grid (whose fineness is independent of the input size) while preserving reasonable geometric and topological properties. We also provided some insight that this algorithm and implementation have practical average complexity in $O\left(n\sqrt{n}\right)$ on “real data", which has to be compared to its $O\left({n}^{15}\right)$ worst-case time complexity. Our implementation is still too slow to be used in practice but it provides a good proof of concept.

#### Hardness results on Voronoi, Laguerre and Apollonius diagrams

In collaboration with Kevin Buchin (TU Eindhoven), Pedro de Castro (University Pernanbuco), and Menelaos Karavelas (University Heraklion).