Report on Relaxed Jordan Canonical Form for Computer Animation and Visualization

Relaxed Jordan canonical form (RJCF) is introduced for three-dimensional linear transformations, and applied to various problems in computer animation and visualization, especially problems involving matrix exponentials and matrix logarithms. In all cases our procedures use closed-form expressions, so they are not iterative, whereas the corresponding procedures that have been published in the graphics literature are iterative. New numerically robust procedures are presented for Alexa’s “linear” interpolation between linear transformations, based on a matrix Lie product, for polar decomposition, and other techniques that require matrix exponentials and matrix logarithms. Extensions to cases not covered by the published procedures are also discussed, particularly for matrices with less than full rank. There are also applications to analysis of vector fields and tensor fields in three dimensions. Besides providing a basis for numerical procedures, the geometric interpretations of RJCF often provide useful analytical insights. RJCF is based on new numerically stable procedures to derive a canonical coordinate system for a two or three-dimensional linear transformation. The coordinate transformation has unit determinant and is composed of meaningful building blocks: shear, rotation, nonuniform scale. The derivation uses the new concept of “outer root” of a cubic polynomial, and properties of the characteristic and minimal polynomials. New closed-form procedures to compute eigenvectors are developed, which use fewer operations than eigenvector routines found in texts. Accuracy and stability are examined empirically. Tests indicate that the new method is an order of magnitude faster than general methods, as well as being more accurate. The three-dimensional problem is separated into independent one-dimensional and two-dimensional problems. The outer root’s eigenspace is one-dimensional, collinear with the outer eigenvector. A simple calculation determines the two-dimensional invariant subspace of the other eigenvalues, which need not be orthogonal to the outer eigenvector and may be complex.