The topology of symmetric, second-order tensor fields
Thierry Delmarcelle, Lambertus Hesselink · 1994
We study the topology of symmetric, second-order tensor elds. The goal is to represent their complex structure by a simple set of carefully chosen points and lines analogous to vector eld topology. We extract topological skeletons of the eigenvector elds, and we track their evolution over time. We study tensor topological transitions and correlate tensor and vector data. The basic constituents of tensor topology are the degen-erate points, or points where eigenvalues are equal to each other. Degenerate points play a similar role as critical points in vector elds. We identify two kinds of elementary degen-erate points, which we call wedges and trisectors. They can combine to form more familiar singularities|such as sad-dles, nodes, centers, or foci. However, these are generally unstable structures in tensor elds. Finally, we show a topological rule that puts a constraint on the topology of tensor elds dened across surfaces, ex-tending to tensor elds the Poincare-Hopf theorem for vector elds. 1