Symbolic and numerical techniques for constraint solving
Abstract
This work investigates 3D geometric constraint solving for a representative class of basic problems that appear in practice as building blocks of more complex designs. It shows that combining symbolic manipulation with homotopy continuation can be effective in solving certain families of problems. In particular, polyhedral and parameter homotopies are shown to improve significantly the computation of the solutions by reducing considerably the number of paths to be tracked. The choice of representation for the primitives and the order in which they are placed are shown to play a major role on the solution process. The generic solutions for all tetrahedral problems are computed, and a systematic framework to solve octahedral problems involving points and planes is presented. Two non trivial one-step constructions involving lines and spheres are also solved. They correspond to the problems of placing a line tangent to 4 given spheres and the problem of Apollonius in 3D. A family of problems involving 6 primitives, which can be points or lines is exhaustively studied. They are shown to be intrinsically more complex and cannot be solved interactively with the current technological limitations. This thesis also introduces a new terminology in geometric constraint solving which clearly distinguishes among the concepts of Abstract Constraint System, Realization, and Constraint Problem.
Degree
Ph.D.
Advisors
Hoffmann, Purdue University.
Subject Area
Computer science|Mathematics
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