Studies in computational geometry motivated by mesh generation
This thesis sprawls over most of discrete and computational geometry. There are four loose bodies of theory developed. (1) A quantitative and algorithmic theory of crossing number and crossing-free line segment graphs in the plane. As five applications of this theory: the author disproves two long - standing conjectures on the crossing number of the complete and complete bipartite graphs, he presents the first exponential algorithm for planar minimum Steiner tree, and the first subexponential algorithms for planar traveling salesman tour and optimum triangulation, and he presents an algorithm for generating all non-isomorphic V-vertex planar graphs, in O(V{sup 3)}time per graph, using O(V) total workspace. (2) Mesh generation, and the triangulation of polytopes: He has the strongest bounds on the number of d-simplices required to triangulate the d-cube, and new triangulation methods in the plane. A quantitative and qualitative - and practical - theory of finite element mesh quality suggest a new, simple strategy for generating good meshes. (3) The theory of geometrical graphs on N point sites in d-space. This subsumes many new results in: geometrical probability, sphere packing, and extremal configurations. An array of new multidimensional search date structures are used to devise fast algorithms for construction many geometrical graphs. (4) Useful new results concerning the mensuration and structure of d-polytopes. In particular he extensively generalizes the famous formula of Heron and Alexandria (75 AD), for the area of a triangle, and he presents the first linear time congruence algorithm for 3 -dimensional polyhedra. He closes with the largest bibliography of the field, containing over 3000 references.
- Research Organization:
- Princeton Univ., NJ (USA)
- OSTI ID:
- 5923035
- Resource Relation:
- Other Information: Thesis (Ph. D.)
- Country of Publication:
- United States
- Language:
- English
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