Affine space

The properties of the group action allows for the definition of subtraction for any given ordered pair (b, a) of points in A, producing a vector of ${\displaystyle {\overrightarrow {A}}}$. This vector, denoted ${\displaystyle b-a}$ or ${\displaystyle {\overrightarrow {ab}}}$, is defined to be the unique vector in ${\displaystyle {\overrightarrow {A}}}$ such that

${\displaystyle a+(b-a)=b.}$

Existence follows from the transitivity of the action, and uniqueness follows because the action is free.

This subtraction has the two following properties, called Weyl's axioms:[7]

1. ${\displaystyle \forall a\in A,\;\forall v\in {\overrightarrow {A}}}$, there is a unique point ${\displaystyle b\in A}$ such that ${\displaystyle b-a=v.}$
2. ${\displaystyle \forall a,b,c\in A,\;(c-b)+(b-a)=c-a.}$

The parallelogram property is satisfied in affine spaces, where it is expressed as: given four points ${\displaystyle a,b,c,d,}$ the equalities ${\displaystyle b-a=d-c}$ and ${\displaystyle c-a=d-b}$ are equivalent. This results from the second Weyl's axiom, since ${\displaystyle d-a=(d-b)+(b-a)=(d-c)+(c-a).}$

Affine spaces can be equivalently defined as a point set A, together with a vector space ${\displaystyle {\overrightarrow {A}}}$, and a subtraction satisfying Weyl's axioms. In this case, the addition of a vector to a point is defined from the first of Weyl's axioms.

An affine transformation or endomorphism of an affine space ${\displaystyle A}$ is an affine map from that space to itself. One important family of examples is the translations: given a vector ${\displaystyle {\overrightarrow {v}}}$, the translation map ${\displaystyle T_{\overrightarrow {v}}:A\rightarrow A}$ that sends ${\displaystyle a\mapsto a+{\overrightarrow {v}}}$ for every ${\displaystyle a}$ in ${\displaystyle A}$ is an affine map. Another important family of examples are the linear maps centred at an origin: given a point ${\displaystyle b}$ and a linear map ${\displaystyle M}$, one may define an affine map ${\displaystyle L_{M,b}:A\rightarrow A}$ by

$${\displaystyle L_{M,b}(a)=b+M(a-b)}$$

for every ${\displaystyle a}$ in ${\displaystyle A}$.

After making a choice of origin ${\displaystyle b}$, any affine map may be written uniquely as a combination of a translation and a linear map centred at ${\displaystyle b}$.

Euclidean spaces (including the one-dimensional line, two-dimensional plane, and three-dimensional space commonly studied in elementary geometry, as well as higher-dimensional analogues) are affine spaces.

Indeed, in most modern definitions, a Euclidean space is defined to be an affine space, such that the associated vector space is a real inner product space of finite dimension, that is a vector space over the reals with a positive-definite quadratic form q(x). The inner product of two vectors x and y is the value of the symmetric bilinear form

${\displaystyle x\cdot y={\frac {1}{2}}(q(x+y)-q(x)-q(y)).}$

The usual Euclidean distance between two points A and B is

${\displaystyle d(A,B)={\sqrt {q(B-A)}}.}$

In older definition of Euclidean spaces through synthetic geometry, vectors are defined as equivalence classes of ordered pairs of points under equipollence (the pairs (A, B) and (C, D) are equipollent if the points A, B, D, C (in this order) form a parallelogram). It is straightforward to verify that the vectors form a vector space, the square of the Euclidean distance is a quadratic form on the space of vectors, and the two definitions of Euclidean spaces are equivalent.

In Euclidean geometry, the common phrase "affine property" refers to a property that can be proved in affine spaces, that is, it can be proved without using the quadratic form and its associated inner product. In other words, an affine property is a property that does not involve lengths and angles. Typical examples are parallelism, and the definition of a tangent. A non-example is the definition of a normal.

Equivalently, an affine property is a property that is invariant under affine transformations of the Euclidean space.

Let A be an affine space of dimension n over a field k, and ${\displaystyle \{x_{0},\dots ,x_{n}\}}$ be an affine basis of A. The properties of an affine basis imply that for every x in A there is a unique (n + 1)-tuple ${\displaystyle (\lambda _{0},\dots ,\lambda _{n})}$ of elements of k such that

${\displaystyle \lambda _{0}+\dots +\lambda _{n}=1}$

and

${\displaystyle x=\lambda _{0}x_{0}+\dots +\lambda _{n}x_{n}.}$

The ${\displaystyle \lambda _{i}}$ are called the barycentric coordinates of x over the affine basis ${\displaystyle \{x_{0},\dots ,x_{n}\}}$. If the x_i are viewed as bodies that have weights (or masses) ${\displaystyle \lambda _{i}}$, the point x is thus the barycenter of the x_i, and this explains the origin of the term barycentric coordinates.

The barycentric coordinates define an affine isomorphism between the affine space A and the affine subspace of k^{n + 1} defined by the equation ${\displaystyle \lambda _{0}+\dots +\lambda _{n}=1}$.

For affine spaces of infinite dimension, the same definition applies, using only finite sums. This means that for each point, only a finite number of coordinates are non-zero.

An affine frame of an affine space consists of a point, called the origin, and a linear basis of the associated vector space. More precisely, for an affine space A with associated vector space ${\displaystyle {\overrightarrow {A}}}$, the origin o belongs to A, and the linear basis is a basis (v₁, ..., v_n) of ${\displaystyle {\overrightarrow {A}}}$ (for simplicity of the notation, we consider only the case of finite dimension, the general case is similar).

For each point p of A, there is a unique sequence ${\displaystyle \lambda _{1},\dots ,\lambda _{n}}$ of elements of the ground field such that

${\displaystyle p=o+\lambda _{1}v_{1}+\dots +\lambda _{n}v_{n},}$

or equivalently

${\displaystyle {\overrightarrow {op}}=\lambda _{1}v_{1}+\dots +\lambda _{n}v_{n}.}$

The ${\displaystyle \lambda _{i}}$ are called the affine coordinates of p over the affine frame (o, v₁, ..., v_n).

Example: In Euclidean geometry, Cartesian coordinates are affine coordinates relative to an orthonormal frame, that is an affine frame (o, v₁, ..., v_n) such that (v₁, ..., v_n) is an orthonormal basis.

Barycentric coordinates and affine coordinates are strongly related, and may be considered as equivalent.

In fact, given a barycentric frame

${\displaystyle (x_{0},\dots ,x_{n}),}$

one deduces immediately the affine frame

${\displaystyle (x_{0},{\overrightarrow {x_{0}x_{1}}},\dots ,{\overrightarrow {x_{0}x_{n}}})=\left(x_{0},x_{1}-x_{0},\dots ,x_{n}-x_{0}\right),}$

and, if

${\displaystyle \left(\lambda _{0},\lambda _{1},\dots ,\lambda _{n}\right)}$

are the barycentric coordinates of a point over the barycentric frame, then the affine coordinates of the same point over the affine frame are

${\displaystyle \left(\lambda _{1},\dots ,\lambda _{n}\right).}$

Conversely, if

${\displaystyle \left(o,v_{1},\dots ,v_{n}\right)}$

is an affine frame, then

${\displaystyle \left(o,o+v_{1},\dots ,o+v_{n}\right)}$

is a barycentric frame. If

${\displaystyle \left(\lambda _{1},\dots ,\lambda _{n}\right)}$

are the affine coordinates of a point over the affine frame, then its barycentric coordinates over the barycentric frame are

${\displaystyle \left(1-\lambda _{1}-\dots -\lambda _{n},\lambda _{1},\dots ,\lambda _{n}\right).}$

Therefore, barycentric and affine coordinates are almost equivalent. In most applications, affine coordinates are preferred, as involving less coordinates that are independent. However, in the situations where the important points of the studied problem are affinely independent, barycentric coordinates may lead to simpler computation, as in the following example.

The vertices of a non-flat triangle form an affine basis of the Euclidean plane. The barycentric coordinates allows easy characterization of the elements of the triangle that do not involve angles or distances:

The vertices are the points of barycentric coordinates (1, 0, 0), (0, 1, 0) and (0, 0, 1). The lines supporting the edges are the points that have a zero coordinate. The edges themselves are the points that have one zero coordinate and two nonnegative coordinates. The interior of the triangle are the points whose coordinates are all positive. The medians are the points that have two equal coordinates, and the centroid is the point of coordinates (1/3, 1/3, 1/3).

Let

${\displaystyle f\colon E\to F}$

be an affine homomorphism, with

${\displaystyle {\overrightarrow {f}}\colon {\overrightarrow {E}}\to {\overrightarrow {F}}}$

its associated linear map. The image of f is the affine subspace ${\displaystyle f(E)=\{f(a)\mid a\in E\}}$ of F, which has ${\displaystyle {\overrightarrow {f}}({\overrightarrow {E}})}$ as associated vector space. As an affine space does not have a zero element, an affine homomorphism does not have a kernel. However, the linear map ${\displaystyle {\overrightarrow {f}}}$ does, and if we denote by ${\displaystyle K=\{v\in {\overrightarrow {E}}\mid {\overrightarrow {f}}(v)=0\}}$ its kernel, then for any point x of ${\displaystyle f(E)}$, the inverse image ${\displaystyle f^{-1}(x)}$ of x is an affine subspace of E whose direction is ${\displaystyle K}$. This affine subspace is called the fiber of x.

An important example is the projection parallel to some direction onto an affine subspace. The importance of this example lies in the fact that Euclidean spaces are affine spaces, and that these kinds of projections are fundamental in Euclidean geometry.

More precisely, given an affine space E with associated vector space ${\displaystyle {\overrightarrow {E}}}$, let F be an affine subspace of direction ${\displaystyle {\overrightarrow {F}}}$, and D be a complementary subspace of ${\displaystyle {\overrightarrow {F}}}$ in ${\displaystyle {\overrightarrow {E}}}$ (this means that every vector of ${\displaystyle {\overrightarrow {E}}}$ may be decomposed in a unique way as the sum of an element of ${\displaystyle {\overrightarrow {F}}}$ and an element of D). For every point x of E, its projection to F parallel to D is the unique point p(x) in F such that

${\displaystyle p(x)-x\in D.}$

This is an affine homomorphism whose associated linear map ${\displaystyle {\overrightarrow {p}}}$ is defined by

${\displaystyle {\overrightarrow {p}}(x-y)=p(x)-p(y),}$

for x and y in E.

The image of this projection is F, and its fibers are the subspaces of direction D.

Although kernels are not defined for affine spaces, quotient spaces are defined. This results from the fact that "belonging to the same fiber of an affine homomorphism" is an equivalence relation.

Let E be an affine space, and D be a linear subspace of the associated vector space ${\displaystyle {\overrightarrow {E}}}$. The quotient E/D of E by D is the quotient of E by the equivalence relation such that x and y are equivalent if

${\displaystyle x-y\in D.}$

This quotient is an affine space, which has ${\displaystyle {\overrightarrow {E}}/D}$ as associated vector space.

For every affine homomorphism ${\displaystyle E\to F}$, the image is isomorphic to the quotient of E by the kernel of the associated linear map. This is the first isomorphism theorem for affine spaces.

By the definition above, the choice of an affine frame of an affine space ${\displaystyle \mathbb {A} _{k}^{n}}$ allows one to identify the polynomial functions on ${\displaystyle \mathbb {A} _{k}^{n}}$ with polynomials in n variables, the ith variable representing the function that maps a point to its ith coordinate. It follows that the set of polynomial functions over ${\displaystyle \mathbb {A} _{k}^{n}}$ is a k-algebra, denoted ${\displaystyle k\left[\mathbb {A} _{k}^{n}\right]}$, which is isomorphic to the polynomial ring ${\displaystyle k\left[X_{1},\dots ,X_{n}\right]}$.

When one changes coordinates, the isomorphism between ${\displaystyle k\left[\mathbb {A} _{k}^{n}\right]}$ and ${\displaystyle k[X_{1},\dots ,X_{n}]}$ changes accordingly, and this induces an automorphism of ${\displaystyle k\left[X_{1},\dots ,X_{n}\right]}$, which maps each indeterminate to a polynomial of degree one. It follows that the total degree defines a filtration of ${\displaystyle k\left[\mathbb {A} _{k}^{n}\right]}$, which is independent from the choice of coordinates. The total degree defines also a graduation, but it depends on the choice of coordinates, as a change of affine coordinates may map indeterminates on non-homogeneous polynomials.

Affine spaces over topological fields, such as the real or the complex numbers, have a natural topology. The Zariski topology, which is defined for affine spaces over any field, allows use of topological methods in any case. Zariski topology is the unique topology on an affine space whose closed sets are affine algebraic sets (that is sets of the common zeros of polynomial functions over the affine set). As, over a topological field, polynomial functions are continuous, every Zariski closed set is closed for the usual topology, if any. In other words, over a topological field, Zariski topology is coarser than the natural topology.

There is a natural injective function from an affine space into the set of prime ideals (that is the spectrum) of its ring of polynomial functions. When affine coordinates have been chosen, this function maps the point of coordinates ${\displaystyle \left(a_{1},\dots ,a_{n}\right)}$ to the maximal ideal ${\displaystyle \left\langle X_{1}-a_{1},\dots ,X_{n}-a_{n}\right\rangle }$. This function is a homeomorphism (for the Zariski topology of the affine space and of the spectrum of the ring of polynomial functions) of the affine space onto the image of the function.

The case of an algebraically closed ground field is especially important in algebraic geometry, because, in this case, the homeomorphism above is a map between the affine space and the set of all maximal ideals of the ring of functions (this is Hilbert's Nullstellensatz).

This is the starting idea of scheme theory of Grothendieck, which consists, for studying algebraic varieties, of considering as "points", not only the points of the affine space, but also all the prime ideals of the spectrum. This allows gluing together algebraic varieties in a similar way as, for manifolds, charts are glued together for building a manifold.

Like all affine varieties, local data on an affine space can always be patched together globally: the cohomology of affine space is trivial. More precisely, ${\displaystyle H^{i}\left(\mathbb {A} _{k}^{n},\mathbf {F} \right)=0}$ for all coherent sheaves F, and integers ${\displaystyle i>0}$. This property is also enjoyed by all other affine varieties. But also all of the étale cohomology groups on affine space are trivial. In particular, every line bundle is trivial. More generally, the Quillen–Suslin theorem implies that every algebraic vector bundle over an affine space is trivial.