The Stretch Factor of Hexagon-Delaunay Triangulations

Abstract

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 (Chew, 1989) and a square (Bonichon et al., 2015}. 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 contributions of this paper are that 1) we successfully extended the overall approach in (Bonichon et al., 2015) to Hexagon-Delaunay triangulations and 2) two techniques that we developed to deal with the increased complexity of Hexagon-Delaunay triangulations, that we use to obtain tight upper bounds for their stretch factor, and that may be generalizable to other regular polygon Delaunay triangulations.