51 research outputs found
The Stretch Factor of Hexagon-Delaunay Triangulations
The problem of computing the exact stretch factor (i.e., the tight bound on the worst case stretch factor) of a Delaunay triangulation is one of the longstanding open problems in computational geometry. Over the years, a series of upper and lower bounds on the exact stretch factor have been obtained but the gap between them is still large. An alternative approach to solving the problem is to develop techniques for computing the exact stretch factor of "easier" types of Delaunay triangulations, in particular those defined using regular-polygons instead of a circle. Tight bounds exist for Delaunay triangulations defined using an equilateral triangle and a square. In this paper, we determine the exact stretch factor of Delaunay triangulations defined using a regular hexagon: It is 2. We think that the main contribution of this paper are the two techniques we have developed to compute tight upper bounds for the stretch factor of Hexagon-Delaunay triangulations
Algorithmic and Combinatorial Results on Fence Patrolling, Polygon Cutting and Geometric Spanners
The purpose of this dissertation is to study problems that lie at the intersection of geometry and computer science. We have studied and obtained several results from three different areas, namelyâgeometric spanners, polygon cutting, and fence patrolling. Specifically, we have designed and analyzed algorithms along with various combinatorial results in these three areas. For geometric spanners, we have obtained combinatorial results regarding lower bounds on worst case dilation of plane spanners. We also have studied low degree plane lattice spanners, both square and hexagonal, of low dilation. Next, for polygon cutting, we have designed and analyzed algorithms for cutting out polygon collections drawn on a piece of planar material
using the three geometric models of saw, namely, line, ray and segment cuts. For fence patrolling, we have designed several strategies for robots patrolling both open and closed fences
Courbure discrÚte : théorie et applications
International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Collection of abstracts of the 24th European Workshop on Computational Geometry
International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop
Large bichromatic point sets admit empty monochromatic 4-gons
We consider a variation of a problem stated by ErdËos
and Szekeres in 1935 about the existence of a number
fES(k) such that any set S of at least fES(k) points in
general position in the plane has a subset of k points
that are the vertices of a convex k-gon. In our setting
the points of S are colored, and we say that a (not necessarily
convex) spanned polygon is monochromatic if
all its vertices have the same color. Moreover, a polygon
is called empty if it does not contain any points of
S in its interior. We show that any bichromatic set of
n â„ 5044 points in R2 in general position determines
at least one empty, monochromatic quadrilateral (and
thus linearly many).Postprint (published version
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