Absolute Polarities and Central Inversions

Abstract

There are two different geometric transformations that go by the name of “inversion.” One is inversion in a point, also called “central” inversion, and the other is inversion in a circle or a sphere, which is the basis of inversive geometry. Other than the fact that both of these transformations are of period two, they seem to have little in common except for the name. However, when we extend the concept of inversion in a circle to include inversion in imaginary circles, we find that inversion in an ordinary or ideal point of hyperbolic space can be identified with inversion in an imaginary or real circle at infinity, thus uniting the two meanings of the term. Such a correspondence is possible because the group of isometries of a hyperbolic space of two or more dimensions is isomorphic to the group of circle-preserving transformations of an inversive space of one dimension less. This isomorphism, noted in 1905 by Liebmann [13, p. 54] and subsequently by many others, has been dealt with extensively in two papers by Coxeter [6; 10J.