An Application of Singularity Theory to Robust Geometric Calculation of Interactions Among Dynamically Deforming Geometric Objects

This dissertation presents a general mathematical framework for geometric inquiries and geometric operations on dynamically deforming models, performing the efficient and robust computation of these events: (1) Evolution of an existing solution to a nearby one under small perturbation; (2) Detection of transition events; (3) Identification of the set of transition points where a structural change of solutions, i.e., a transition event, occurs; (4) Classification of transition type such as creation, annihilation etc.; (5) Computation of the actual geometry that realizes the structural change of a detected and classified transition event. The general mathematical framework starts with formulating the problem in a solution space that is the product of the parametric space in which curves or surfaces are defined and the control space in which deformation is defined. The entire solution is encoded as a manifold in this solution space. Then, independent tangent vector fields are constructed on this solution manifold, and are used to define the differential of the projection map from the solution manifold to the control space. If this projection map is locally nonsingular, it can be used to associate the perturbation in the control space to the incremental change in the solution space that is required to evolve the existing solution to the nearby one under the specific perturbation. On the other hand, at a critical point of the projection map, a Morse function is constructed and its Hessian function is computed from covariant derivatives of the tangent vector fields. Consequently, the transition event, i.e., the structural change between nearby solutions, is computed by the local shape approximation to the manifold at the critical point. This dissertation also significantly advances the state-of-the-art regarding efficient symbolic computation on B-spline curves, which is one of the most fundamental operations in geometric modeling and is also essential to an efficient and robust tracking of point-curve distances.

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