In the complex plane, an even number of reflection through lines or circles can be expressed in complex coordinates as a linear fractional transformation w=(az+b)/(cz+d) with Image and ad−bc≠0. This also holds in Image : an even number of reflections through spheres or planes correspond to transformations k=(ah+b)(ch+d)−1 with Image . A theorem by Poincaré about direct isometries of hyperbolic spaces may therefore be rephrased: direct isometries of H5 correspond to quaternionic linear fractional transformations.
