Theoretically-based algorithms for robustly tracking intersection curves of deforming surfaces
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This paper applies singularity theory of mappings of surfaces to 3-space and the generic transitions occurring in their deformations to develop algorithms for contin- uously and robustly tracking the intersection curves of two deforming parametric spline surfaces, when the deformation is represented as a family of generalized offset surfaces. The set of intersection curves of 2 deforming surfaces over all time is for- mulated as an implicit 2-manifold I in an augmented (by time domain) parametric space R5. Hyper-planes corresponding to some fixed time instants may touch I at some isolated transition points, which delineate transition events, i.e., the topologi- cal 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 androbustly with a rational constraint solver using subdivision and hyper- tangent bounding cones. The actual transition events are computed by contouring 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 in which the surfaces are embedded.