Coxeter Triangulations Have Good Quality

Good triangulations yield good tours

Consider the following heuristic for planar Euclidean instances of the traveling salesman problem (TSP): select a subset of the edges which induces a planar graph, and solve either the TSP or its graphical relaxation on that graph. In this paper, we give several motivations for considering this heuristic, along with extensive computational results. It turns out that the Delaunay and greedy tria...

Good Triangulations in Plane

Quality triangulations made smaller

We study alternative types of Steiner points (to circumcenters) for computing quality guaranteed Delaunay triangulations in three dimensions. We show through experiments that their effective use results in smaller (in the number of tetrahedra) triangulations than the output of the traditional circumcenter refinement methods.

Dense point sets have sparse Delaunay triangulations

The spread of a finite set of points is the ratio between the longest and shortest pairwise distances. We prove that the Delaunay triangulation of any set of n points in R3 with spread ∆ has complexity O(∆3). This bound is tight in the worst case for all ∆ = O( √ n). In particular, the Delaunay triangulation of any dense point set has linear complexity. We also generalize this upper bound to re...

Quality Improvement of Surface Triangulations

This paper presents a new procedure to improve the quality of triangular meshes defined on surfaces. The improvement is obtained by an iterative process in which each node of the mesh is moved to a new position that minimizes certain objective function. This objective function is derived from an algebraic quality measures of the local mesh (the set of triangles connected to the adjustable or fr...