Theoretically Based Robust Algorithms for Tracking Intersection Curves of Two Deforming Parametric Surfaces Xianming Chen 1 , Richard F. Riesenfeld 1 , Elaine Cohen 1 , and James Damon 2 1 School of Computing, University of Utah, Salt Lake City, UT 84112 2 Department of Mathematics, University of North Carolina Abstract. 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 R 5 . 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. 1 Introduction and Related Work In this paper, we consider the dynamic intersection of two deforming parametric surfaces. The surface deformation is represented by generalized offset surfaces, an extension [6, 7] to the traditional unit normal vector offset surfaces. Specifi- cally, let ς (s) be a regular primary surface with parameterization s ∈ R 2 , the generalized offset surface is defined as, σ(s,t)= ς (s)+ tU (s), (1) where t is the offset time, and U (s) is another (vector) surface parameterized again in s. The offset vector U needs not be orthogonal to the tangent plane nor of unit-length. Both surfaces are assumed to be free of local and global self-intersection during the whole deformation process. The research in this paper is, in some extent, related to the traditional static surface-surface intersection [3, 29, 41, 16, 36], and to the unit normal offset surfaces [30, 23, 39, 21, 11, 18, 8, 10]. We emphasize the topological robustness