Visualizing Nonlinear Vector Field Topology

We present our results on the visualization of non-linear 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 non-linear 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. Keywords— vector field, topology, Clifford algebra, visualization