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 $p \in \mathbb R^2$ to a static parametrized piecewise rational curve γ(s) ($s \in \mathbb R$). 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. Unlike many techniques, it does not use any global search strategy except at initialization. Implementing the mathematical idea from singularity community, we encode the critical distance surface as an implicit surface $\mathcal{I}$ in the augmented parameter space. A point ps = (p,s) is in the augmented parametric space $\mathbb R^3 = \mathbb R^2 \times \mathbb R$, where p varies over $\mathbb R^2$. In most situations, when p is perturbed, its corresponding critical distances can be evolved without structural change by marching along a sectional curve on $\mathcal{I}$. However, occasionally, when the perturbation crosses the evolute of γ, there is a transition event at which 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. Additional transition events caused by various curve discontinuities are also investigated. Our implementation assumes a B-spline representation for the curve and has interactive speed even on a lower end laptop computer.