In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "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 and a point not on , there is exactly one line parallel to that passes through .) 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).
The idea of forgetting the metric can be applied in the theory of manifolds. That is developed in the article on the affine connection.
History
In 1748, Leonhard Euler introduced the term affine () in his book (volume 2, chapter XVIII). In 1827, August Möbius wrote on affine geometry in his (chapter 3).
After Felix Klein's Erlangen program, affine geometry was recognized as a generalization of Euclidean geometry.
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.
Systems of axioms
Several axiomatic approaches to affine geometry have been put forward:
Pappus' law
As affine geometry deals with parallel lines, one of the properties of parallels noted by Pappus of Alexandria has been taken as a premise: Suppose are on one line and on another. If the lines and are parallel and the lines and are parallel, then the lines and are parallel.
The full axiom system proposed has point, line, and line containing point as primitive notions: Two points are contained in just one line. For any line and any point , not on , there is just one line containing and not containing any point of . This line is said to be parallel to . Every line contains at least two points. There are at least three points not belonging to one line.
According to H. S. M. Coxeter: The interest of these five axioms is enhanced by the fact that they can be developed into a vast body of propositions, holding not only in Euclidean geometry but also in Minkowski's geometry of time and space (in the simple case of 1 + 1 dimensions, whereas the special theory of relativity needs 1 + 3). The extension to either Euclidean or Minkowskian geometry is achieved by adding various further axioms of orthogonality, etc.