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 $${\mathbb{H}}$$ 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 $${\mathbb{H}}$$ mapping four distinct quaternions q1, q2, q3 and q4 into $${q^{\prime}_{1}, q^{\prime}_{2}, q^{\prime}_{3}}$$ and $${q^{\prime}_{4}}$$ respectively if and only if the quadruples (q1, q2, q3, q4) and ($${q^{\prime}_{1}, q^{\prime}_{2}, q^{\prime}_{3}, q^{\prime}_{4}}$$) 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 $${\mathbb{H}}$$ 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 $${\mathbb{H}}$$ map spheres (or affine subspaces) of dimension 1, 2 and 3 into spheres (or affine subspaces) of the same dimension.