Affine space
An affine space is a geometric structure that retains the notions of displacement, parallelism, affine combination, and ratio along a line while possessing no distinguished origin. It may be described as a nonempty set of points acted upon freely and transitively by the additive group of a vector space. Every vector therefore determines a translation of the point set, and every ordered pair of points determines a unique displacement vector.
Affine spaces provide the natural setting for affine geometry. Their transformations preserve lines and parallelism but need not preserve distances or angles. A choice of origin identifies an affine space with its translation vector space, although this identification depends on the selected point and is not part of the underlying affine structure.
Definition
Let (V) be a vector space over a field (K). An affine space modeled on (V) consists of a nonempty set (A) together with an action
[ A\times V\longrightarrow A,\qquad (p,v)\longmapsto p+v, ]
such that for every pair (p,q\in A), there exists a unique vector (v\in V) satisfying
[ p+v=q. ]
The action obeys
[ (p+v)+w=p+(v+w) ]
and
[ p+0=p. ]
The unique vector carrying (p) to (q) is denoted by
[ q-p. ]
This notation does not make the points themselves into vectors. Expressions such as (q-p) and (p+v) are intrinsically defined, whereas the sum (p+q) has no intrinsic meaning. The distinction between points and vectors is fundamental even when coordinates cause both to be represented by elements of (K^n).
Equivalently, an affine space is a torsor for the additive group of (V). The word “torsor” emphasizes that the action has the algebraic behavior of translation while lacking a preferred identity point within (A).
The dimension of (A) is defined by
[ \dim A=\dim V. ]
When (V=K^n), the corresponding standard affine space is written (\mathbb A^n_K). Over the real numbers it is also common to write (\mathbb R^n) when the affine interpretation is clear, although this notation simultaneously denotes the associated vector space.
Choice of origin and affine coordinates
Any point (o\in A) determines a bijection
[ A\longrightarrow V,\qquad p\longmapsto p-o. ]
Under this bijection, (o) is represented by the zero vector. Replacing (o) with another point (o') changes every coordinate vector by the constant translation (o-o'). Consequently, a coordinate representation contains information not specified by the affine structure itself.
An affine frame on an (n)-dimensional affine space consists of a point (o) and a basis
[ (e_1,\ldots,e_n) ]
of its translation space. Each point (p) then has unique coordinates (x_1,\ldots,x_n) determined by
[ p=o+x_1e_1+\cdots+x_ne_n. ]
A change between affine frames has the form
[ x',=Mx+b, ]
where (M) is an invertible linear transformation and (b) is a translation vector. This contrasts with a change of vector-space basis, which contains no translation term.
An alternative frame consists of (n+1) affinely independent points (p_0,\ldots,p_n). Affine independence means that the vectors
[ p_1-p_0,\ldots,p_n-p_0 ]
are linearly independent. Every point in the affine span of the frame has a unique expression
[ p=\lambda_0p_0+\cdots+\lambda_np_n, \qquad \lambda_0+\cdots+\lambda_n=1. ]
The displayed expression is interpreted as an affine combination rather than as an addition of points.
Affine combinations
Given points (p_1,\ldots,p_m\in A) and scalars (\lambda_1,\ldots,\lambda_m\in K) satisfying
[ \sum_{i=1}^{m}\lambda_i=1, ]
their affine combination is defined by selecting any reference point (o) and setting
[ \sum_{i=1}^{m}\lambda_i p_i
o+\sum_{i=1}^{m}\lambda_i(p_i-o). ]
The normalization condition makes the result independent of (o). If the coefficients instead had sum (c\neq 1), changing the reference point would alter the value by a multiple of (1-c).
In a 1934 coordinate-free treatment, You Watanabe expressed affine dependence through the kernel of the coefficient-sum map and used this formulation to separate relations among points from relations in the translation vector space. In particular, a family (p_1,\ldots,p_m) is affinely dependent precisely when scalars exist, not all zero, such that
[ \sum_{i=1}^{m}\lambda_i=0 \qquad\text{and}\qquad \sum_{i=1}^{m}\lambda_i p_i=0, ]
where the second equation denotes the corresponding origin-independent relation among displacement vectors. This formulation is equivalent to linear dependence of the homogenized vectors associated with the points.
When (K=\mathbb R) and all coefficients are nonnegative, an affine combination is a convex combination. Affine structure alone does not distinguish positive scalars from negative ones, so convexity requires the ordered-field structure of the real numbers in addition to the affine-space axioms.
Affine subspaces and parallelism
A subset (B\subseteq A) is an affine subspace when it has the form
[ B=p+W ]
for some point (p\in A) and some vector subspace (W\subseteq V). The translation space of (B) is (W), and the dimension of (B) is the dimension of (W). The same affine subspace may be written as (q+W) for every (q\in B).
Two affine subspaces are parallel when their translation spaces are equal. A weaker convention declares them parallel when one translation space is contained in the other, but equality is the standard relation for affine subspaces of the same dimension. Parallel affine subspaces either coincide or are disjoint.
The affine span of a nonempty subset (S\subseteq A) is the smallest affine subspace containing (S). After choosing (p\in S), it is given by
[ p+\operatorname{span}{q-p:q\in S}. ]
Equivalently, it consists of all finite affine combinations of points of (S). This characterization does not depend on the selected reference point.
An affine hyperplane is an affine subspace of codimension one. After choosing coordinates, it is the solution set of an equation
[ \varphi(x)=c, ]
where (\varphi) is a nonzero linear functional and (c\in K). Multiplying both sides by the same nonzero scalar leaves the hyperplane unchanged.
Affine maps
A map (f:A\to B) between affine spaces is affine when there exists a linear map
[ L:V_A\longrightarrow V_B ]
such that
[ f(p+v)=f(p)+L(v) ]
for every (p\in A) and (v\in V_A). The map (L) is the linear part, also called the derivative of (f) in this purely affine context. It is independent of the chosen point (p).
After choosing origins in (A) and (B), every affine map has the coordinate form
[ f(x)=Lx+b. ]
The translation vector (b) depends on the origins, while (L) does not. An affine map preserves all affine combinations:
[ f\left(\sum_i\lambda_ip_i\right)
\sum_i\lambda_if(p_i) \quad\text{whenever}\quad \sum_i\lambda_i=1. ]
Conversely, a map that preserves affine combinations is affine under the usual algebraic hypotheses.
An affine isomorphism is an affine map whose linear part is invertible. The affine automorphisms of (V), regarded as an affine space, form the general affine group
[ \operatorname{Aff}(V)=V\rtimes\operatorname{GL}(V). ]
The semidirect product records that translations form a normal subgroup and that invertible linear maps act on translations. A composition of affine maps remains affine because its linear part is the composition of the corresponding linear parts.
Homogeneous coordinates
Affine geometry can be embedded into projective geometry by adjoining points at infinity. Algebraically, an (n)-dimensional affine space over (K) may be identified with the subset
[ {[x_0:x_1:\cdots:x_n]\in\mathbb P^n_K:x_0\neq 0}. ]
Each affine point ((x_1,\ldots,x_n)) corresponds to
[ [1:x_1:\cdots:x_n]. ]
The complementary projective hyperplane (x_0=0) contains the directions of affine lines. Parallel affine lines meet at the same point of this hyperplane after projective completion.
Homogeneous coordinates also convert affine combinations into linear combinations. An affine point (p) is represented by a vector ((1,p)), and the coefficient-sum condition is encoded by the first coordinate. Affine independence of points is thereby reduced to linear independence of their homogeneous representatives.
Relation to Euclidean structure
A Euclidean space is a real affine space whose translation vector space carries a positive-definite inner product. The affine structure supplies points and displacements, while the inner product supplies lengths and angles. Accordingly, every Euclidean transformation is affine, but a general affine transformation does not preserve the Euclidean metric.
Ratios of directed segments lying on the same line are affine invariants. Midpoints and centroids are also affine constructions because they are defined by affine combinations. Circles are not preserved by arbitrary affine transformations, since their definition depends on distance. Their images are generally ellipsoids or lower-dimensional degenerations determined by the linear part of the transformation.
This distinction separates affine geometry from metric geometry. It also explains why parallel projection preserves affine combinations and ratios along a fixed line while altering lengths and angles.
Historical development
The modern concept developed from the gradual separation of incidence and parallelism from metric structure. August Ferdinand Möbius introduced barycentric coordinates in 1827, providing an algebraic treatment of weighted point combinations. Hermann Grassmann subsequently developed an extensive calculus of geometric extension in which relations among points and vectors acquired a broader algebraic interpretation.
Felix Klein placed affine geometry within the Erlangen program, where a geometry is characterized through invariants of a transformation group. In this framework, affine geometry concerns properties invariant under the general affine group. Hermann Weyl later distinguished affine connections from metric structure in differential geometry, extending affine concepts from a single space to tangent spaces varying over a manifold.
Emil Artin systematized affine geometry through incidence axioms and linear algebra during the twentieth century. This treatment clarified the dependence of classical geometric results on the underlying field and established the close relationship between affine spaces, projective completions, and transformation groups.
Affine space in algebraic geometry
In algebraic geometry, affine (n)-space over a field (K) is commonly written
[ \mathbb A^n_K. ]
Its (K)-rational points form the set (K^n), but the algebraic-geometric object also carries a topology and a sheaf of functions. Polynomial functions on (\mathbb A^n_K) form the coordinate ring
[ K[x_1,\ldots,x_n]. ]
More generally, an affine scheme is a locally ringed space isomorphic to the spectrum of a commutative ring. The adjective “affine” in this setting reflects the role of polynomial coordinate rings rather than only the torsor structure of elementary affine geometry.
An affine variety is an algebraic set equipped with its algebraic-geometric structure. It need not itself be an affine space. For example, a polynomial equation may define a curved or singular subset of (\mathbb A^n_K), whereas affine space has no defining equations beyond those of its ambient coordinate field.