As with other affine spatial transformation models, the matrix logarithm and matrix exponential functions can be used to compute intermediate positions in a transformation. For example, dividing the matrix logarithm by two and then computing the matrix exponential will compute an intermediate transformation halfway along the most direct path from the initial position to the final position.
Transformations interpolated in this way will respect the geometry associated with the special reference frame computed by eigenvector analysis and singular-value decomposition.
The situation in three dimensions is similar to that in two dimensions. Barring special cases (e.g., rigid-body transformations), a three-dimensional affine transformation will have at least one real eigenvector with a fourth term that is nonzero. Consequently, some point will be unchanged by the transformation. Singular-value decomposition can be used to identify a special reference frame that has its origin at this unchanged point. If the object being transformed is viewed as being embedded in this special reference frame, the transformation can be interpreted as a simple anisotropic rescaling along these special coordinate axes. Meanwhile, the special reference frame itself can be viewed as undergoing a three-dimensional rotation in space around its origin.
Consider the three-dimensional example
Was this article helpful?