On a Quaternionic Analogue of the Cross-Ratio

Details On a Quaternionic Analogue of the Cross-Ratio On a Quaternionic Analogue of the Cross-Ratio

2011-12-03

arxiv.org/abs/1112.0612v2

Mathematics

Complex Variables

11 pages, no figures, references added, to appear in Advances in Applied Clifford Algebras

Scientific paper

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.

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