## Section: New Results

### Classical Computational Geometry

Participants : Olivier Devillers, Sylvain Lazard, Leo Valque.

#### Rounding Meshes

Let $\mathcal{P}$ be a set of $n$ polygons in ${\mathbb{R}}^{3}$, each of constant complexity and with pairwise disjoint interiors. We previously proposed [5] a rounding algorithm that maps $\mathcal{P}$ to a simplicial complex $\mathcal{Q}$ whose vertices have integer coordinates such that 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 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

We show that converting Apollonius and Laguerre diagrams from an already built Voronoi diagram of a set of n points in 2D requires at least $\Omega (nlogn)$ computation time. We also show that converting an Apollonius diagram of a set of $n$ weighted points in 2D from a Laguerre diagram and vice-versa requires at least $\Omega (nlogn)$ computation time as well. Furthermore , we present a very simple randomized incremental construction algorithm that takes expected $O(nlogn)$ computation time to build an Apollonius diagram of non-overlapping circles in 2D [17].

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