Affine Geometry

In mathematics, affine geometry is the study of parallel lines. Its use of Playfair's axiom is fundamental since comparative measures of angle size are foreign to affine geometry so that Euclid's parallel postulate is beyond the scope of pure affine geometry. In affine geometry, the relation of parallelism may be adapted so as to be an equivalence relation. Comparisons of figures in affine geometry are made with affinities which are mappings comprising the affine group A. Since A lies between the Euclidean group E and the group of projectivities P, affine geometry is sometimes mentioned in connection with the Erlangen program, which is concerned with group inclusions such as E ⊂ A ⊂ P.

Affine geometry can be developed on the basis of linear algebra. One can define an affine space as a set of points equipped with a set of transformations, the translations, which forms (the additive group of) a vector space (over a given field), and such that for any given ordered pair of points there is a unique translation sending the first point to the second. In more concrete terms, this amounts to having an operation that associates to any two points a vector and another operation that allows translation of a point by a vector to give another point; these operations are required to satisfy a number of axioms (notably that two successive translations have the effect of translation by the sum vector). By choosing any point as "origin", the points are in one-to-one correspondence with the vectors, but there is no preferred choice for the origin; thus this approach can be characterized as obtaining the affine space from its associated vector space by "forgetting" the origin (zero vector).