We present our results on the visualization of nonlinear vector field topology. The underlying mathematics is done in Clifford algebra, a system describing geometry by extending the usual vector space by a multiplication of vectors. We started with the observation that all known algorithms for vector field topology are based on piecewise linear or bilinear approximation, and that these methods destroy the local topology if nonlinear behavior is present. Our algorithm looks for such situations, chooses an appropriate polynomial approximation in these areas, and, finally, visualizes the topology. This overcomes the problem, and the algorithm is still very fast because we are using linear approximation outside these small but important areas. The paper contains a detailed description of the algorithm and a basic introduction to Clifford algebra.

Presents an algorithm for the visualization of vector field topology based on Clifford algebra. It allows the detection of higher-order singularities. This is accomplished by first analysing the possible critical points and then choosing a suitable polynomial approximation, because conventional methods based on piecewise linear or bilinear approximation do not allow higher-order critical points and destroy the topology in such cases. The algorithm is still very fast, because of using linear approximation outside the areas with several critical points.