On a Quaternionic Analogue of the Cross-Ratio

Abstract

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 q ₁, q ₂, q ₃ and q ₄ into ${q^{\prime}_{1}, q^{\prime}_{2}, q^{\prime}_{3}}$ and ${q^{\prime}_{4}}$ respectively if and only if the quadruples (q ₁, q ₂, q ₃, q ₄) 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.