Affine space

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. One-dimensional affine space is the affine line.

Physical space (in pre-relativistic conceptions) is not only an affine space. It also has a metric structure and in particular a conformal structure.

Informal descriptions
The following characterization may be easier to understand than a precise definition: an affine space is what is left of a vector space after you've forgotten which point is the origin (or, in the words of mathematical physicist John Baez, "An affine space is a vector space that's forgotten its origin"). Imagine that Smith knows that a certain point is the true origin, and Jones believes that another point&mdash;call it p&mdash;is the origin. Two vectors, a and b, are to be added. Jones draws an arrow from p to a and another arrow from p to b, and completes the parallelogram to find what Jones thinks is a + b, but Smith knows that it is actually p + (a − p) + (b − p). Similarly, Jones and Smith may evaluate any linear combination of a and b, or of any finite set of vectors, and will generally get different answers. However&mdash;and note this well:


 * If the sum of the coefficients in a linear combination is 1, then Smith and Jones will agree on the answer!

The proof is a routine exercise. Here is the punch line: Smith knows the "linear structure", but both Smith and Jones know the "affine structure"&mdash;i.e. the values of affine combinations, defined as linear combinations in which the sum of the coefficients is 1. An underlying set with an affine structure is an affine space.

Precise definition
An affine space is a set with a faithful transitive vector space action, a principal homogeneous space with a vector space action.

Alternatively an affine space is a set S, together with a vector space V, and a map


 * $$\Theta : S^2 \to V : (a, b) \mapsto \Theta(a, b) =: a - b\,$$

such that


 * 1. for every b in S the map


 * $$\Theta_b : S \to V : a \mapsto a - b\,$$


 * is a bijection, and


 * 2. for every a, b and c in S we have


 * $$(a-b) + (b-c) = a-c.\,$$

Consequences
We can define addition of vectors and points as follows


 * $$\Phi : S \times V \to S : (a, v) \mapsto a + v := \Theta_a^{-1}v.$$

By choosing an origin a we can thus identify S with V, hence change S into a vector space.

Conversely, any vector space V is an affine space for vector subtraction.

If O, a and b are points in S and $$\ell$$ is a real number, then


 * $$\oplus_O : S^2 \to S : (a, b) \mapsto a \oplus_O b := O+\ell(a-O)+(1-\ell)(b-O)\,$$

is independent of O. Instead of arbitrary linear combinations, only such affine combinations of points have meaning.

Affine subspaces
An affine subspace of a vector space V is a subset closed under affine combinations of vectors in the space. For example, the set
 * $$S=\left \{\left. \sum^N_i \alpha_i \mathbf{v}_i \right\vert \sum^N_i\alpha_i=1\right\}$$

is an affine space, where {vi}i is a family of vectors in V. To see that this is indeed an affine space, observe that this set carries a transitive action of the vector subspace W of V
 * $$W=\left\{\left. \sum^N_i \beta_i\mathbf{v}_i \right\vert \sum^N_i \beta_i=0\right\}.$$

This vector subspace, and therefore also the affine subspace, is of dimension N–1. This affine subspace can be equivalently described as the coset of the W-action
 * $$S=\mathbf{p}+W,$$

where p is any element of S.

One might like to define an affine subspace of an affine space as a set closed under affine combinations. However, affine combinations are only defined in vector spaces; one cannot add points of an affine space. Allowing a slightly more abstract definition, one may define an affine subspace of an affine space as a subset which is left invariant under an affine transformation.

In affine geometry there is not only no notion of origin, but neither a notion of length or angle.

An affine transformation between two vector spaces is a combination of a linear transformation and a translation. For specifying one the origins are used, but the set of affine transformations does not depend on the origins.