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 ∈ R 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 R = R × R, and the totality of point ps’s (when p moves over the whole plane R) as an implicit surface I in R. 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, when the perturbation crosses the evolute of γ, there will be a transition event when a pair of p’s current critical distances is annihilated, or a new pair is created and added to the set of p’s critical distances. To safely eliminate any global search for critical distances, we develop robust and efficient algorithm to perform the detection and computation of transition events. Extra transition events due to various curve discontinuities are also investigated. Our implementation assumes B-spline representation for the curve and has interactive speed even on a lower end laptop computer.