Tensor-Field Visualization
Tensor fields carry directional information richer than a vector; visualizing them means showing magnitude and orientation in several directions at once.
Beyond vectors
A second-order tensor field assigns a matrix to each point, describing quantities like stress, strain, or diffusion that have magnitude along multiple directions. A single arrow cannot capture this; visualization must convey the principal directions and their strengths together.
Eigen-decomposition
A symmetric tensor is characterized by its eigenvalues and eigenvectors: the eigenvectors give principal directions and the eigenvalues give the magnitude along each. Most tensor visualization reduces to displaying this decomposition in a readable way. The shape of the eigenvalue set (isotropic, planar, linear) classifies the local behavior.
- Ellipsoid glyphs: axes aligned to eigenvectors, lengths to eigenvalues.
- Superquadric glyphs: sharpen ambiguous cases the ellipsoid blurs.
- Tensor lines (hyperstreamlines): trace a chosen principal direction like a streamline.
Derived scalars
Because full tensor displays are dense, scalar summaries help: an anisotropy measure, the largest eigenvalue, or a mode value can be shown as a field, with glyphs reserved for regions of interest.
Pitfalls
Ellipsoids from different viewpoints can look alike, hiding whether a region is planar or linear; superquadrics reduce this. Eigenvector sign is arbitrary, so orientation, not sense, is the meaningful part when tracing tensor lines.
Kronos use
Structural stress in simulated machine components is a tensor field; ellipsoid or superquadric glyphs on a scalar anisotropy background show where and how material is loaded.