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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 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—call it ''p''—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—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"—i.e. the values of Affine Combination s, 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 o V : (a, b) \mapsto \Theta(a, b) =: a - b\,

such that

:1. for every ''b'' in ''S'' the map

:::\Theta_b : S o 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 imes V o 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 o S : (a, b) \mapsto a \oplus_O b := O+\ell(a-O)+(1-\ell)(b-O)\,

is independent of ''O''. Instead of arbitrary Linear Combination s, only such Affine Combination s of points have meaning.


AFFINE SUBSPACES


An affine subspace of a vector space ''V'' is a subset closed under Affine Combination s of vectors in the space. For example, the set
:S=\left \{\left. \sum^N_i \alpha_i \mathbf{v}_i ight ert \sum^N_i\alpha_i=1 ight\}
is an affine space, where {v''i''}''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 ight ert \sum^N_i \beta_i=0 ight\}.
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.


SEE ALSO