Projective transformations (homographies) can be classified according to the algebraic and geometric multiplicity of their eigenvalues. The algebraic multiplicity of an eigenvalue is the number of times the root is repeated in the characteristic equation. The geometric multiplicity may be determined from the rank of the matrix (H − λ I), where H is the homography and λ the eigenvalue. A complete classification is given in projective geometry textbooks such as [Springer-64]. Here we mention several special cases which are important in practical situations, and occur at several points throughout this book. The description will be for plane transformations where H is a 3 × 3 matrix, but the generalization to 3-space transformations is straightforward.

The special forms are significant because H satisfies a number of relationships (remember the only restriction a general projective transformation is that it has full rank). Since H satisfies constraints it has fewer degrees of freedom, and consequently can be computed from fewer correspondences than a general projective transformation. The special transformations also have richer geometry and invariants than in the general case.

Note that unlike the special forms (affine etc.) discussed in chapter 2, which form subgroups, the following special projectivities do not form subgroups in general since they are not closed under multiplication. They do form a subgroup if all the elements have coincident fixed points and lines (i.e. differing only in their eigenvalues).