Minimum Distance Queries for Haptic Rendering

Finding the closest points between two modeled objects is a fundamental operation in robotics, computer graphics, and computational geometry. This dissertation is motivated by the use of distance functions in haptic interfaces for virtual prototyping, where distance measures provide the basis for forces that are applied to a human user. The requirements for haptic interfaces mean that these distances must be computed both quickly and robustly. This dissertation begins by exploring the robustness of simple numerical methods for finding the minimum distance between a point and a curve. A geometric analysis of the convergence conditions yields an algorithm for precomputing a set of starting values with robustness guarantees. Embedding this simple local method within a geometric convergence test then provides some guarantees of global convergence. The requirements of haptic interfaces motivate another approach, based on normal cones, for global search of local minima. This technique extends to surfaces and mixed with numerical methods allows a haptic rendering system for NURBS models. Finally, the normal cone approach is applied to polygonal models, which provides the basis for a general 6DOF haptic interface for virtual prototyping. These methods provide significant performance and reliability benefits over existing haptic rendering techniques.