Theoretically Based Robust Algorithms for Tracking Intersection Curves of Two Deforming Parametric Surfaces
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This paper presents the mathematical framework, and de- velops algorithms accordingly, to continuously and robustly track the intersection curves of two deforming parametric surfaces, with the de- formation represented as generalized offset vector fields. The set of in- tersection curves of 2 deforming surfaces, over all time, is formulated as an implicit 2-manifold I in the augmented (by time domain) paramet- ric space R5. Hyper-planes corresponding to some fixed time instants may touch I at some isolated transition points, which delineate transi- tion events, i.e., the topological changes to the intersection curves. These transition points are the 0-dimensional solution to a rational system of 5 constraints in 5 variables, and can be computed efficiently and robustly with a rational constraint solver using subdivision and hyper tangent bounding cones. The actual transition events are computed by contour- ing the local osculating paraboloids. Away from any transition points, the intersection curves do not change topology and evolve according to a simple evolution vector field that is constructed in the Euclidean space where the surfaces are embedded.