## Section: New Results

### Non-Euclidean Computational Geometry

Participants : Vincent Despré, Yan Garito, Elies Harington, Benedikt Kolbe, Georg Osang, Monique Teillaud, Gert Vegter.

#### Flipping Geometric Triangulations on Hyperbolic Surfaces

We consider geometric triangulations of surfaces, i.e., triangulations whose edges can be realized by disjoint locally geodesic segments. We prove that the flip graph of geometric triangulations with fixed vertices of a flat torus or a closed hyperbolic surface is connected. We give upper bounds on the number of edge flips that are necessary to transform any geometric triangulation on such a surface into a Delaunay triangulation [28].

*In collaboration with Jean-Marc Schlenker (University of Luxembourg).*

#### Computing the Geometric Intersection Number of Curves

The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve $c$ represented by a closed walk of length at most $\ell $ on a combinatorial surface of complexity $n$ we describe simple algorithms to compute the geometric intersection number of $c$ in $O(n+{\ell}^{2})$ time, construct a curve homotopic to $c$ that realizes this geometric intersection number in $O(n+{\ell}^{4})$ time, decide if the geometric intersection number of $c$ is zero, i.e. if c is homotopic to a simple curve, in $O(n+\ell log(\ell \left)\right)$ time [14].

*In collaboration with Francis Lazarus (University of Grenoble).*