The Visualization of Second-Order Tensor Fields

The field of scientific visualization covers the study of visual representations of scientific data. This dissertation is the first extensive study of visualization techniques for second-order tensor fields defined across a n-dimensional space, $n\geq1$. We study the geometry and the topological structure of tensor fields, designing appropriate icons to represent the information. Tensor fields are multivariate; they can embed as much information as $n\sp2$ scalar fields, or equivalently, n vector fields. Visualizing continuous tensor data is, therefore, difficult mainly because the underlying continuity must be rendered while visual clutter has to be avoided. First, we design icons that decrease visual clutter by emphasizing the continuity of the tensor data. A n-dimensional, symmetric tensor field is equivalent to n orthogonal families of smooth and continuous curves that are tangent to the eigenvector fields. For $n=2$ we generate textures to render these trajectories, and for $n=3$ we use numerical integration. To fully represent the tensor data, we surround the resulting trajectories by tubular surfaces that represent the transverse eigenvectors--we call these surfaces hyperstreamlines. We also define the concept of a solenoidal tensor field, and we show that its hyperstreamlines possess geometric properties similar to the streamlines of solenoidal vector fields. Then, we analyze the topology of tensor fields. We derive a set of points and trajectories that represent the collective behavior of a continuous distribution of hyperstreamlines. We call these points degenerate points, because they occur where eigenvalues are equal to each other. Degenerate points can be of various types--trisectors, wedges, nodes, foci, saddles, dipoles, cusps, saddle-nodes, as well as other, more exotic patterns. We call the trajectories that underlie the topology of tensor fields separatrices. They are specific hyperstreamlines that are either limit cycles, or trajectories emanating from degenerate points and lying on the border of hyperbolic sectors. When a tensor field is defined across a surface, the topology of the surface limits the number and the nature of the degenerate points. We explore this relation, extending to tensor fields the Poincare-Hopf Theorem for vector fields.