The types of transformations that are used to produce paradoxes in Euclidean spaces and on spheres are usually the Euclidean isometries, but occasionally more general affine maps arise. Since the affine group is useful in studying and classifying isometries, we summarize the relevant facts about affine transformations. The book by Hausner [92] is a good reference for a more detailed presentation.

Definition A.1. A bijection f: Rn → Rnis called affine if for all P, Q ∈ Rnand reals α, β with α + β = 1, f(αP + βQ) = αf(P) + βf(Q). The affine transformations ofRnform a group, which is denoted by An.

Geometrically, a bijection is affine if and only if it carries lines to lines and preserves the ratio of distances along a line. Any nonsingular linear transformation is affine, since a linear transformation satisfies Definition A.1 for all α, β, not just pairs summing to one. The group of nonsingular linear transformations of Rn is denoted by GLn (general linear group). Linear maps leave the origin fixed, but affine maps need not do so; all translations of Rn are affine. Let Tn denote the group of translations of Rn. Tn is isomorphic to the additive group of Rn because composition of translations corresponds to addition of the translation vectors. It is an extremely useful fact that every affine map has a canonical representation in terms of linear maps and translations.