Section: New Results
Classical Computational Geometry and Graph Drawing
Participants : Vincent Despré, Olivier Devillers, Sylvain Lazard.
Delaunay Triangulations of Points on Circles
Delaunay triangulations of a point set in the Euclidean plane are ubiquitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon as 4 or more points are concyclic but since it is not a generic situation, this difficulty is usually handled by using a (symbolic or explicit) perturbation. As an alternative, we proposed to define a canonical triangulation for a set of concyclic points by using a max-min angle characterization of Delaunay triangulations. This point of view leads to a well defined and unique triangulation as long as there are no symmetric quadruples of points. This unique triangulation can be computed in quasi-linear time by a very simple algorithm [18].
In collaboration with Hugo Parlier and Jean-Marc Schlenker (University of Luxembourg).
Improved Routing on the Delaunay Triangulation
A geometric graph
In collaboration with Nicolas Bonichon (Labri), Prosenjit Bose, Jean-Lou De Carufel, Michiel Smid and Daryl Hill (Carleton University)
Limits of Order Types
We completed an extended version of a work published at SoCG 2015, in
which we apply ideas from the theory of limits of dense combinatorial
structures to study order types, which are combinatorial encodings of
finite point sets. Using flag algebras we obtain new numerical results
on the Erdös problem of finding the minimal density of 5-or
6-tuples in convex position in an arbitrary point set, and also an
inequality expressing the difficulty of sampling order types
uniformly. Next we establish results on the analytic representation of
limits of order types by planar measures. Our main result is a
rigidity theorem: we show that if sampling two measures induce the
same probability distribution on order types, then these measures are
projectively equivalent provided the support of at least one of them
has non-empty interior. We also show that some condition on the
Hausdorff dimension of the support is necessary to obtain projective
rigidity and we construct limits of order types that cannot be
represented by a planar measure. Returning to combinatorial geometry
we relate the regularity of this analytic representation to the
aforementioned problem of Erdös on the density of
In collaboration with Alfredo Hubard (Laboratoire d'Informatique Gaspard-Monge) Rémi De Joannis de Verclos (Radboud university, Nijmegen) Jean-Sébastien Sereni (CNRS) Jan Volec (Department of Mathematics and Computer Science, Emory University)
Snap rounding polyhedral subdivisions
Let
In collaboration with William J. Lenhart (Williams College, USA).
On the Edge-length Ratio of Outerplanar Graphs
We show that any outerplanar graph admits a planar straight-line drawing such that the length ratio
of the longest to the shortest edges is strictly less than 2.
This result is tight in the sense that for any
In collaboration with William J. Lenhart (Williams College, USA) and Giuseppe Liotta (Università di Perugia, Italy).