Finite Time Steady Vector Field Topology - Theoretical Foundation and 3D Case

Abstract

Vector Field Topology is the standard approach for the analysis of asymptotic particle behavior in a vector field flow: A topological skeleton is separating the flow into regions by the movement of massless particles for an integration time converging to infinity. In some use cases however only a finite integration time is feasible. To this end, the idea of a topological skeleton with an augmented finite-time separation measure was introduced for 2D vector fields. We lay the theoretical foundation for that method and extend it to 3D vector fields. From the observation of steady vector fields in a temporal context we show the Galilean invariance of Vector Field Topology. In addition, we present a set of possible visualizations for finite-time topology on 3D topological skeletons.