On a Quaternionic Analogue of the Cross-Ratio

TL;DR

This paper develops a quaternionic analogue of the cross-ratio, exploring its properties and applications to quaternionic fractional linear transformations, including criteria for geometric configurations and mapping behaviors of spheres and affine subspaces.

Contribution

It introduces a quaternionic cross-ratio, establishes its properties, and demonstrates its use in characterizing quaternionic fractional linear transformations and geometric configurations.

Findings

Quaternionic cross-ratio characterizes fractional linear transformations.

Transformations are uniquely determined by five points in general position.

Mappings preserve dimensions of spheres and affine subspaces.

Abstract

In this article we study an exact analogue of the cross-ratio for the algebra of quaternions 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 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.