Optimal Two � Dimensional Triangulations

A (geometric) 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 xed set of vertices, there are, in general, exponentially many w ays to form a triangulation. Various criteria related to the geometry of triangles are used to deene 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 nite set S of vertices, possibly with some prescribed edges, how c a n w e choose the rest of the edges to obtain an optimal triangulation? Just to mention an example, we a r e i n terested in computing a min-max angle triangulation of S, that is, 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 and scientiic computations, such as nite element analysis and surface interpolation. All algorithms are the rst and, currently, the only ones that construct the deened optimal triangulations in time polynomial in the input size. These main 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 deened by some generic criterion. From this, we deduce that a min-max angle and a max-min height triangulation can be computed in O(n 2 log n) time and linear storage, and a min-max slope and a min-max eccentricity triangu-lation in cubic time and quadratic storage. Second, we s h o w that a min-max length triangulation for a set of n vertices can be computed in quadratic time and storage. Length refers to edge length and is measured by some normed metric such as the Euclidean or any o t h e r l p metric. Third, for a given plane graph of n vertices and m non-crossing edges, we p r o ve that there is a set of O(m 2 n) p o i n ts 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 eecient w ay to construct a so-called conforming Delaunay triangulation, which is a Delaunay triangulation that subdivides the …