Dynamic computational topology for piecewise linear curves

Abstract

A piecewise linear (PL) approximation often serves as the graphics representation for a parametric curve. Algorithms for preserving correct topology for a single static image are available, but significant challenges remain to ensure correct topology when the PL curve is changing shape during synchronized visualization with an ongoing simulation, such as a molecule writhing over time. A tubular neighborhood of the curve is defined to preserve topology under perturbation, but as the perturbed geometry approaches the boundary of that tubular neighborhood, any required update of the neighborhood should maintain the synchronization. The algorithimic performance of these updates is directly dependent upon the number of approximating edges and the techniques presented here decrease that data volume versus previous methods, as shown by a comprehensive comparative analysis and a representative example.