Polar Decomposition - Alternative Planar Decompositions

Alternative Planar Decompositions

In the Cartesian plane, alternative planar ring decompositions arise as follows:

• If x ≠ 0, z = x ( 1 + (y/x) ε) is a polar decomposition of a dual number z = x + y ε, where ε ε = 0. In this polar decomposition, the unit circle has been replaced by the line x = 1, the polar angle by the slope y/x, and the radius x is negative in the left half-plane.

• If x2 ≠ y2, then the unit hyperbola x2 − y2 = 1 and its conjugate x2 − y2 = −1 can be used to form a polar decomposition based on the branch of the unit hyperbola through (1,0). This branch is parametrized by the hyperbolic angle a and is written

where j 2 = +1 and the arithmetic of split-complex numbers is used. The branch through (−1,0) is traced by −ea j. Since the operation of multiplying by j reflects a point across the line y = x, the second hyperbola has branches traced by jea j or −jea j. Therefore a point in one of the quadrants has a polar decomposition in one of the forms:

The set { 1, −1, j, −j } has products that make it isomorphic to the Klein four-group. Evidently polar decomposition in this case involves an element from that group.