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Title:  Optimal TwoDimensional Triangulations 
Author(s):  Tan, TiowSeng 
Doctoral Committee Chair(s):  Edelsbrunner, Herbert 
Department / Program:  Computer Science 
Discipline:  Computer Science 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Engineering, General
Computer Science 
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 minmax 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 edgeinsertion paradigm. It computes for a set of n vertices an optimal triangulation defined by some generic criterion. We then deduce that a minmax angle and a maxmin height triangulation can be computed in O($n\sp2\ \log n$) time, and a minmax slope and a minmax eccentricity triangulation in cubic time. Second, we show that a minmax 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 noncrossing 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. 
Issue Date:  1993 
Type:  Text 
Description:  122 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1993. 
URI:  http://hdl.handle.net/2142/72082 
Other Identifier(s):  (UMI)AAI9314949 
Date Available in IDEALS:  20141217 
Date Deposited:  1993 
This item appears in the following Collection(s)

Dissertations and Theses  Computer Science
Dissertations and Theses from the Dept. of Computer Science 
Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois