Topology of a Tensor Field

Copyright (c) 1994 Institute of Electrical and Electronics Engineers. Reprinted from IEEE Visualization '94, October 17-21, 1994, Washington D.C., pp. 140-147.

Lambertus Hesselink and Thierry Delmarcelle visualize a 2D symmetric second-order tensor field by extracting its topological sceleton. The topological analysis of a tensor field is build from the concept of degenerated points, which play the role of critical points in vector fields (see image 4).
A point is degenerated if the tensor field in the point has two identical eigenvalues. In degenerated points hyperstreamlines (image 1) do cross. The authors use the tensor gradient in the degenerated points to classify the points into trisectors and widgets and connect them by hyperstreamlines.
Above image visualizes a 2D stress tensor in a periodic flow past a cylinder. The texture represents the most compressive eigenvector and color encodes the magnitude of the compressive force (red most, blue least), The topological sceleton of the tensor field is superimposed, where W symbolises a wedge and T a trisector. And degenerated points (\ie trisectors and widgets) the viscous stresses vanish and both eigenvalues are equal to the pressure; degenerated points in above image are hence points of pure pressure.