Optimal Two-Dimensional Triangulations

Abstract

A triangulation in the plane is a maximal connected plane graph with straight edges. It is thus a plane graph whose bounded faces are triangles. For a fixed set of vertices, there are, in general, exponentially many ways to form a triangulation. Various criteria related to the geometry of triangles are used to define what one could mean by a triangulation that is optimal over all possibilities. The general problem studied in this thesis is the following: given a finite set S of vertices, possibly with some prescribed edges, how can we choose the rest of the edges to obtain an optimal triangulation? For example, we want to compute a min-max angle triangulation of S, i.e., a triangulation whose maximum angle over all its triangles is the smallest among all triangulations of S.

This thesis presents a number of new algorithms to construct optimal triangulations useful in engineering the scientific computations, such as finite element and surface interpolation. All algorithms are currently the only ones that construct the defined optimal triangulations in time polynomial in the input size. These results are described in three parts.

First, we develop a new algorithmic technique called the edge-insertion paradigm. It computes for a set of n vertices an optimal triangulation defined by some generic criterion. We then deduce that a min-max angle and a max-min height triangulation can be computed in O($n\sp2\ \log n$) time, and a min-max slope and a min-max eccentricity triangulation in cubic time.

Second, we show that a min-max length triangulation for a set of n vertices can be computed in quadratic time. Length refers to edge length and is measured by some normed metric such as any $l\sb{p}$ metric.

Third, for a given plane graph of n vertices and m non-crossing edges, we prove that there is a set of O($m\sp2n$) points so that, for each adjacent pair of points on an edge, there exists a circle passing through the two points that encloses no other points. This implies an efficient way to construct a Delaunay triangulation that subdivides the plane graph.