## Affine geometry |

In **affine geometry** is what remains of ^{[1]}^{[2]}) the

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,

Affine geometry can be developed in two ways that are essentially equivalent.^{[3]}

In *points* to which is associated a set of lines, which satisfy some

Affine geometry can also be developed on the basis of *points* equipped with a set of *transformations* (that is

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

Although this article only discusses

- history
- systems of axioms
- affine transformations
- affine space
- projective view
- see also
- references
- further reading
- external links

In 1748, *affine*^{[4]}^{[5]} (Latin *affinis*, "related") in his book * Introductio in analysin infinitorum* (volume 2, chapter XVIII). In 1827,

After ^{[6]}

In 1912, ^{[7]}^{[8]} to express the

In 1918, *Space, Time, Matter*. He used affine geometry to introduce vector addition and subtraction^{[9]} at the earliest stages of his development of ^{[10]}

- 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 *L*^{2}" was described by Graciela Birman and ^{[11]}