Computational Topology for Surface Approximation

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This research examines broad issues of guaranteeing that the essential shape of an object is preserved during algorithms used in engineering design, animation and molecular modeling. While geometric characteristics such as surface area and volume can be objectively measured and compared for two objects, other shape characterizations, such as the distinction between a garden hose and the same hose tied into a knot, are more subtle and require the methods of this research. Surprisingly, these distinctions are not easy for present day algorithms and difficulties in detecting these differences has been estimated to cost billions of dollars annually in lost productivity in the aeronautical and automotive industries. This research is an interdisciplinary investigation between computer scientists and mathematicians with a corresponding educational emphasis upon introducing the next generation of scientists to a holistic perspective that combines theory with novel engineering and life science applications. The applications to molecular modeling are expected to be attractive to the many women in the biological sciences and thereby increase the participation of women in the computational and mathematical sciences.This research integrates new topological constraints and numerical error bounds into geometric approximation algorithms, defining when a surface approximation is in the same topological equivalence class as the original surface. While surface approximation is a classical mathematical topic, the issue of topological equivalence has typically been left to human inspection. The novel approach of this research is the development of the theory and practice to create computationally tractable algorithms that ensure topological equivalence for a rich class of surface approximation techniques. The creation of appropriate applied mathematics to eliminate any direct algorithmic dependence upon computation of the medial axis significantly enriches the class of robust approximation algorithms delivering verifiable topology. Another major innovation is the comprehensive consideration of geometric models composed from bounded surface patches. This perspective eliminates a prevailing, but unrealistic theoretical hypothesis that geometric models have only a single bounding surface. The additional subtleties rely upon new numerical approximations along the boundaries of each constituent patch.