In affine geometry, one uses Playfair's axiom
to find the line through C1 and parallel to B1B2, and to find the line through B2 and parallel to B1C1: their intersection C2 is the result of the indicated translation.
In mathematics, affine geometry is what remains of Euclidean geometry when not using (mathematicians often say "when forgetting") the metric notions of distance and angle.
As the notion of parallel lines is one of the main properties that is independent of any metric, affine geometry is often considered as the study of parallel lines. Therefore, Playfair's axiom (given a line L and a point P not on L, there is exactly one line parallel to L that passes through P) is fundamental in affine geometry. Comparisons of figures in affine geometry are made with affine transformations, which are mappings that preserve alignment of points and parallelism of lines.
Affine geometry can be developed in two ways that are essentially equivalent.
In synthetic geometry, an affine space is a set of points to which is associated a set of lines, which satisfy some axioms (such as Playfair's axiom).
Affine geometry can also be developed on the basis of linear algebra. In this context an affine space is a set of points equipped with a set of transformations (that is bijective mappings), the translations, which forms a vector space (over a given field, commonly the real numbers), and such that for any given ordered pair of points there is a unique translation sending the first point to the second; the composition of two translations is their sum in the vector space of the translations.
In more concrete terms, this amounts to having an operation that associates to any ordered pair of 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 an affine space may be viewed as obtained from its associated vector space by "forgetting" the origin (zero vector).
Although this article only discusses affine spaces, the notion of "forgetting the metric" is much more general, and can be applied to arbitrary manifolds, in general. This extension of the notion of affine spaces to manifolds in general is developed in the article on the affine connection.
In 1748, Leonhard Euler introduced the term affine (Latin affinis, "related") in his book Introductio in analysin infinitorum (volume 2, chapter XVIII). In 1827, August Möbius wrote on affine geometry in his Der barycentrische Calcul (chapter 3).
After Felix Klein's Erlangen program, affine geometry was recognized as a generalization of Euclidean geometry.
In 1912, Edwin B. Wilson and Gilbert N. Lewis developed an affine geometry to express the special theory of relativity.
In 1918, Hermann Weyl referred to affine geometry for his text Space, Time, Matter. He used affine geometry to introduce vector addition and subtraction at the earliest stages of his development of mathematical physics. Later, E. T. Whittaker wrote:
- Weyl's geometry is interesting historically as having been the first of the affine geometries to be worked out in detail: it is based on a special type of parallel transport [...using] worldlines of light-signals in four-dimensional space-time. A short element of one of these world-lines may be called a null-vector; then the parallel transport in question is such that it carries any null-vector at one point into the position of a null-vector at a neighboring point.
In 1984, "the affine plane associated to the Lorentzian vector space L2" was described by Graciela Birman and Katsumi Nomizu in an article entitled "Trigonometry in Lorentzian geometry".