Combining Computation with Geometry

Abstract

This thesis seeks to establish mathematical principles and to provide efficient solutions to various time consuming operations in computer-aided geometric design. It contains a discussion of three major topics: (1) design validation by means of object interference detection, (2) object reconstruction through the union, intersection, and subtraction of two polyhedra, and (3) calculation of basic engineering properties such as volume, center of mass, or moments of inertia. Two criteria are presented for solving the problems of point-polygon enclosure and point polyhedron enclosure in object interference detection. An algorithm for efficient point-polyhedron-enclosure detection is presented. Singularities encountered in point-polyhedron-enclosure detection are categorized and simple methods for for resolving them are also included. A new scheme for representing solid objects, called skeletal polyhedron representation, is proposed. Also included are algorithms for performing set operations on polyhedra (or polygons) represented in skeletal polyhedron representation, algorithms for performing edge-edge intersection and face-face intersection in a set operation, and a perturbation method which can be used to resolve singularities for an easy execution of edge--edge intersection and face-face interaction. A symbolic method for calculating basic engineering properties (such as volume, center of mass, moments of inertia, and similar integral properties of geometrically complex solids) is proposed. The same method is generalized for computing the integral properties of a set combined polyhedron and for computing the integral properties of an arbitrary polyhedron inc m-dimensional (Rm) space.

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