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: we disprove two long-standing conjectures on the crossing number of the complete and complete bipartite graphs, we present the first exponential algorithm for planar minimum Steiner tree, and the first subexponential algorithms for planar traveling salesman tour and optimum triangulation, and we present an algorithm for generating all non-isomorphic $V$-vertex planar graphs, in O($V\sp3$) time per graph, using O($V$) total workspace. (2) Mesh generation, and the triangulation of polytopes: We have 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 suggests 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 data structures are used to devise fast algorithms for constructing many geometrical graphs. (4) Useful new results concerning the mensuration and structure of $d$-polytopes. In particular we extensively generalize the famous formula of Heron and Alexandria (75 AD), for the area of a triangle, and we present the first linear time congruence algorithm for 3-dimensional polyhedra. There are few areas in the field that we leave untouched. (One of them is Haussler, Welzl, and others' recently developed theory of "$\varepsilon$-nets" (haus87) (chaz88).) Therefore, this thesis may profitably be thought of as "Part II" of the text by Preparata and Shamos (prep85). ($\geq$3 dimensions, and more advanced and more recent material.) We close with the largest bibliography of the field, containing over 3000 references.