A tool for visualizing the topology of three-dimensional vector fields

Abstract

A description is given of a software system, TOPO, that numerically analyzes and graphically displays topological aspects of a three-dimensional vector field, v, to produce a single, relatively simple picture that characterizes v. The topology of v considered consists of its critical points (where v=0), their invariant manifolds, and the integral curves connecting these invariant manifolds. The field in the neighborhood of each critical point is approximated by the Taylor expansion. The coefficients of the first nonzero term of the Taylor expansion around a critical point are the 3*3 matrix Delta v. Critical points are classified by examining Delta v's eigenvalues. The eigenvectors of Delta v span the invariant manifolds of the linearized field around a critical point. Curves integrated from initial points on the eigenvectors a small distance from a critical point connect with other critical points (or the boundary) to complete the topology. One class of critical surfaces that is important in computational fluid dynamics is analyzed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>