Optimal two-dimensional triangulations
Tiow-Seng Tan · 1992
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 ways to form a triangulation. Various criteria related to the geometry of triangles are used to de ne 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 canwe choose the rest of the edges to obtain an optimal triangulation? Just to mention an example, we areinterested 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 scienti c computations, such as nite element analysis and surface interpolation. All algorithms are the rst and, currently, the only ones that construct the de ned optimal triangulations in time polynomial in the input size. These main results are described in three parts.