On a Quaternionic Analogue of the Cross-Ratio

In this article we study an exact analogue of the cross-ratio for the algebra of quaternions and use it to derive several interesting properties of quaternionic fractional linear transformations. In particular, we show that there exists a fractional linear transformation T on H mapping four distinct quaternions q(1), q(2), q(3) and q(4) into q(1)', q(2)', q(3)' and q(4)' respectively if and only if the quadruples (q(1), q(2), q(3), q(4)) and (q(1)', q(2)', q(3)', q(4)') have the same cross-ratio. If such a fractional linear transformation T exists it is never unique. However, we prove that a fractional linear transformation on H is uniquely determined by specifying its values at five points in general position. We also prove some properties of the cross-ratio including criteria for four quaternions to lie on a single circle (or a line) and for five quaternions to lie on a single 2-sphere (or a 2-plane). As an application of the cross-ratio, we prove that fractional linear transformations on H map spheres (or affine subspaces) of dimension 1, 2 and 3 into spheres (or affine subspaces) of the same dimension.