## Section: New Results

### Classical Computational Geometry and Graph Drawing

Participants : Olivier Devillers, Sylvain Lazard.

#### Celestial Walk: A Terminating Oblivious Walk for Convex Subdivisions

We present a new oblivious walking strategy for convex subdivisions. Our walk is faster than the straight walk and more generally applicable than the visiblity walk. To prove termination of our walk we use a novel monotonically decreasing distance measure [10].

This work was done in collaboration with Wouter Kuijper and Victor Ermolaev (Nedap Security Management).

#### Snap rounding polyhedral subdivisions

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 build 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 worse, the size of $\mathcal{Q}$ is $O\left({n}^{15}\right)$, but, under reasonable hypotheses, this complexities decreases to $O\left({n}^{5}\right)$.

This work was done in collaboration with William J. Lenhart (Williams College, USA).

#### Explicit array-based compact data structures for triangulations

We consider the problem of designing space efficient solutions for representing triangle meshes. Our main result is a new explicit data structure for compactly representing planar triangulations: if one is allowed to permute input vertices, then a triangulation with $n$ vertices requires at most $4n$ references ($5n$ references if vertex permutations are not allowed). Our solution combines existing techniques from mesh encoding with a novel use of maximal Schnyder woods. Our approach extends to higher genus triangulations and could be applied to other families of meshes (such as quadrangular or polygonal meshes). As far as we know, our solution provides the most parsimonious data structures for triangulations, allowing constant time navigation. Our data structures require linear construction time, and are fast decodable from a standard compressed format without using additional memory allocation. All bounds, concerning storage requirements and navigation performances, hold in the worst case. We have implemented and tested our results, and experiments confirm the practical interest of compact data structures.

This work was done in collaboration with Luca Castelli Aleardi (LIX).