Current research in geometric modeling has focused on extending the coverage of geometric modeling systems. The most difficult obstacle in extending the coverage has been integrating freeform surfaces into a system that allows Boolean set operations such as intersection and union. This project is concerned with the design and implementation of a small-scale geometric modeling system based on algebraic volumes, those sets that can be described as the intersection, union, and difference of algebraic halfspaces (solutions to polynomial inequalities). In particular, the project involves: Design and implementation of algorithms for creating freeform algebraic volumes (the 3D analog of freeform algebraic surfaces). These algorithms will be based on recent advances in the theory of polynomial spline spaces. Design and implementation of algorithms for tessellating algebraic volumes. Application tasks such as rendering and finite element analysis will operate on the resulting tessellation. Design and implementation of algorithms for performing finite element analysis on algebraic volumes. These algorithms would analyze the structural and thermal properties of the modeled object. The resulting system would allow the design and analysis of a range of geometric objects not covered under current geometric modeling systems.
