Tracking Point-Curve Critical Distances

This paper presents a novel approach to continuously and robustly tracking critical (geometrically, perpendicular and/or extremal) distances from a moving plane point p ∈ R2 to a static parametrized piecewise rational curve γ(s) (s ∈ R). The approach is a combination of local marching, and the detection and computation of global topological change, both based on the differential properties of a constructed implicit surface; it does not use any global search strategy except the initialization. Implementing the mathematical idea from singularity community, we encode a particular critical distance as a point ps = (p, s) in the so-called augmented parametric space R3 = R2 × R, and the totality of point ps’s (when p moves over the whole plane R2 ) as an implicit surface I in R3 . In most situations, when p is perturbed in the plane, all of its corresponding critical distances, are only evolved, without structural change, by marching on a sectional curve on I. However, occasionally,...

• Type: Conference
• Year: 2006
• Where: GMP
• Authors: Xianming Chen, Elaine Cohen, Richard F. Riesenfeld
• Keywords: Critical Distances, GMP 2006, Implicit Surface, Transition Events