Affine Space

One dimensional affine space is the affine line.

Affine space. If x and y are vectors in an affine space then 1 λ x λ y is also a vector in that space for any real λ but you don t necessarily have that a x b y is in the space for arbitrary a b. In mathematics an affine space is a geometric structure that generalizes some of the properties of euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles keeping only the properties related to parallelism and ratio of lengths for parallel line segments. The simplest example of an affine space is just the image of a vector space under an affine map x a x b.

A finite dimensional affine space can be provided with the structure of an affine variety with the zariski topology cf. In an affine space it is possible to fix a point and coordinate axis such that every point in the space can be represented as an tuple of its coordinates. Consequently sets of parallel affine subspaces remain parallel after an affine transformation.

It behaves a lot like a vector space except that it isn t closed under arbitrary linear combinations. Affine spaces associated with a vector space over a skew field k are constructed in a similar manner. In an affine space however this zero origin is no longer necessarily the single origin and translation of the elements of an affine space called points will result in different points whence the idea of multiple origins originates.

You can read the definition yourself but here s a little intuition. In mathematics an affine space is an abstract structure that generalises the affine geometric properties of euclidean space. In an affine space one can subtract points to get vectors or add a vector to a point to get another point but one cannot add points since there is no origin.

A coordinate system for the dimensional affine space is determined by any basis of vectors which are not necessarily orthonormal. More generally an affine transformation is an automorphism of an affine space that is a function which maps an affine space onto itself while preserving both the dimension of any affine subspaces and the ratios of the lengths of parallel line segments. In euclidean geometry an affine transformation or an affinity is a geometric transformation that preserves lines and parallelism.

Affine the adjective affine indicates everything that is related to the geometry of affine spaces.