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. Affine space is the setting for affine geometry. As in Euclidean space, the fundamental objects in an affine space are called points, which can be thought of as locations in the space without any size or shape: zero-dimensional. Through any pair of points an infinite straight line can be drawn, a one-dimensional set of points; through any three points that are not collinear, a two-dimensional plane can be drawn; and, in general, through k + 1 points in general position, a k-dimensional flat or affine subspace can be drawn. Affine space is characterized by a notion of pairs of parallel lines that lie within the same plane but never meet each-other (non-parallel lines within the same plane intersect in a point). Given any line, a line parallel to it can be drawn through any point in the space, and the equivalence class of parallel lines are said to share a direction. Unlike for vectors in a vector space, in an affine space there is no distinguished point that serves as an origin. There is no predefined concept of adding or multiplying points together, or multiplying a point by a scalar number. However, for any affine space, an associated vector space can be constructed from the differences between start and end points, which are called free vectors, displacement vectors, translation vectors or simply translations. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point (of the same affine space) translated from the starting point by that vector. While points cannot be arbitrarily added together, it is meaningful to take affine combinations of points: weighted sums with numerical coefficients summing to 1, resulting in another point. These coefficients define a barycentric coordinate system for the flat through the points. Any vector space may be viewed as an affine space; this amounts to "forgetting" the special role played by the zero vector. In this case, elements of the vector space may be viewed either as points of the affine space or as displacement vectors or translations. When considered as a point, the zero vector is called the origin. Adding a fixed vector to the elements of a linear subspace (vector subspace) of a vector space produces an affine subspace of the vector space. One commonly says that this affine subspace has been obtained by translating (away from the origin) the linear subspace by the translation vector (the vector added to all the elements of the linear subspace). In finite dimensions, such an affine subspace is the solution set of an inhomogeneous linear system. The displacement vectors for that affine space are the solutions of the corresponding homogeneous linear system, which is a linear subspace. Linear subspaces, in contrast, always contain the origin of the vector space. The dimension of an affine space is defined as the dimension of the vector space of its translations. An affine space of dimension one is an affine line. An affine space of dimension 2 is an affine plane. An affine subspace of dimension n – 1 in an affine space or a vector space of dimension n is an affine hyperplane.
Informal description
The following characterization may be easier to understand than the usual formal definition: an affine space is what is left of a vector space after one has forgotten which point is the origin (or, in the words of the French mathematician Marcel Berger, "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps"). Imagine that Alice knows that a certain point is the actual origin, but Bob believes that another point—call it p—is the origin. Two vectors, a and b, are to be added. Bob draws an arrow from point p to point a and another arrow from point p to point b, and completes the parallelogram to find what Bob thinks is a + b, but Alice knows that he has actually computed
p + (a − p) + (b − p). Similarly, Alice and Bob may evaluate any linear combination of a and b, or of any finite set of vectors, and will generally get different answers. However, if the sum of the coefficients in a linear combination is 1, then Alice and Bob will arrive at the same answer. If Alice travels to
λa + (1 − λ)b then Bob can similarly travel to
p + λ(a − p) + (1 − λ)(b − p) = λa + (1 − λ)b. Under this condition, for all coefficients λ + (1 − λ) = 1, Alice and Bob describe the same point with the same linear combination, despite using different origins. While only Alice knows the "linear structure", both Alice and Bob know the "affine structure"—i.e. the values of affine combinations, defined as linear combinations in which the sum of the coefficients is 1. A set with an affine structure is an affine space.
Definition While affine space can be defined axiomatically (see § Axioms below), analogously to the definition of Euclidean space implied by Euclid's Elements, for convenience most modern sources define affine spaces in terms of the well developed vector space theory. An affine space is a set A together with a vector space A → {\displaystyle {\overrightarrow {A}}} , and a transitive and free action of the additive group of A → {\displaystyle {\overrightarrow {A}}} on the set A. The elements of the affine space A are called points. The vector space A → {\displaystyle {\overrightarrow {A}}} is said to be associated to the affine space, and its elements are called vectors, translations, or sometimes free vectors. Explicitly, the definition above means that the action is a mapping, generally denoted as an addition,
A × A → → A ( a , v ) ↦ a + v , {\displaystyle {\begin{aligned}A\times {\overrightarrow {A}}&\to A\\(a,v)\;&\mapsto a+v,\end{aligned}}}
that has the following properties.
Right identity:
∀ a ∈ A , a + 0 = a {\displaystyle \forall a\in A,\;a+0=a} , where 0 is the zero vector in A → {\displaystyle {\overrightarrow {A}}}
Associativity:
∀ v , w ∈ A → , ∀ a ∈ A , ( a + v ) + w = a + ( v + w ) {\displaystyle \forall v,w\in {\overrightarrow {A}},\forall a\in A,\;(a+v)+w=a+(v+w)} (here the last + is the addition in A → {\displaystyle {\overrightarrow {A}}} ) Free and transitive action: For every a ∈ A {\displaystyle a\in A} , the mapping A → → A : v ↦ a + v {\displaystyle {\overrightarrow {A}}\to A\colon v\mapsto a+v} is a bijection. The first two properties are simply defining properties of a (right) group action. The third property characterizes free and transitive actions, the onto character coming from transitivity, and then the injective character follows from the action being free. There is a fourth property that follows from 1, 2 above:
Existence of one-to-one translations For all v ∈ A → {\displaystyle v\in {\overrightarrow {A}}} , the mapping A → A : a ↦ a + v {\displaystyle A\to A\colon a\mapsto a+v} is a bijection. Property 3 is often used in the following equivalent form (the 5th property).
Subtraction: For every a, b in A, there exists a unique v ∈ A → {\displaystyle v\in {\overrightarrow {A}}} , denoted b – a, such that b = a + v {\displaystyle b=a+v} . Another way to express the definition is that an affine space is a principal homogeneous space for the action of the additive group of a vector space. Homogeneous spaces are, by definition, endowed with a transitive group action, and for a principal homogeneous space, such a transitive action is, by definition, free.
Subtraction and Weyl's axioms 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 A → {\displaystyle {\overrightarrow {A}}} . This vector, denoted b − a {\displaystyle b-a} or a b → {\displaystyle {\overrightarrow {ab}}} , is defined to be the unique vector in A → {\displaystyle {\overrightarrow {A}}} such that
a + ( b − a ) = b . {\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:
∀ a ∈ A , ∀ v ∈ A → {\displaystyle \forall a\in A,\;\forall v\in {\overrightarrow {A}}} , there is a unique point b ∈ A {\displaystyle b\in A} such that b − a = v . {\displaystyle b-a=v.}
∀ a , b , c ∈ A , ( c − b ) + ( b − a ) = c − a . {\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 a , b , c , d , {\displaystyle a,b,c,d,} the equalities b − a = d − c {\displaystyle b-a=d-c} and c − a = d − b {\displaystyle c-a=d-b} are equivalent. This results from the second Weyl's axiom, since d − a = ( d − b ) + ( b − a ) = ( d − c ) + ( c − a ) . {\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 A → {\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.
Affine subspaces and parallelism An affine subspace (also called, in some contexts, a linear variety, a flat, or, over the real numbers, a linear manifold) B of an affine space A is a subset of A for which there exists a point a ∈ B {\displaystyle a\in B} such that the set of vectors B → = { b − a ∣ b ∈ B } {\displaystyle {\overrightarrow {B}}=\{b-a\mid b\in B\}} is a linear subspace of A → {\displaystyle {\overrightarrow {A}}} . If B {\displaystyle B} is an affine subspace then the set B → = { b − a ∣ b ∈ B } {\displaystyle {\overrightarrow {B}}=\{b-a\mid b\in B\}} is a linear subspace for all a ∈ B {\displaystyle a\in B} (that is, the choice of the point a {\displaystyle a} is irrelevant). An affine subspace B is an affine space which has B → {\displaystyle {\overrightarrow {B}}} as its associated vector space. The affine subspaces of A are the subsets of A of the form
a + V = { a + w : w ∈ V } , {\displaystyle a+V=\{a+w:w\in V\},}
where a is a point of A, and V a linear subspace of A → {\displaystyle {\overrightarrow {A}}} . The linear subspace associated with an affine subspace is often called its direction, and two subspaces that share the same direction are said to be parallel. This implies the following generalization of Playfair's axiom: Given a direction V, for any point a of A there is one and only one affine subspace of direction V, which passes through a, namely the subspace a + V. Every translation A → A : a ↦ a + v {\displaystyle A\to A:a\mapsto a+v} maps any affine subspace to a parallel subspace. The term parallel is also used for two affine subspaces such that the direction of one is included in the direction of the other.
Affine map Given two affine spaces A and B whose associated vector spaces are A → {\displaystyle {\overrightarrow {A}}} and B → {\displaystyle {\overrightarrow {B}}} , an affine map or affine homomorphism from A to B is a map
f : A → B {\displaystyle f:A\to B}
such that
f → : A → → B → b − a ↦ f ( b ) − f ( a ) {\displaystyle {\begin{aligned}{\overrightarrow {f}}:{\overrightarrow {A}}&\to {\overrightarrow {B}}\\b-a&\mapsto f(b)-f(a)\end{aligned}}}
is a well defined linear map. By f {\displaystyle f} being well defined is meant that b – a = d – c implies f(b) – f(a) = f(d) – f(c). This implies that, for a point a ∈ A {\displaystyle a\in A} and a vector v ∈ A → {\displaystyle v\in {\overrightarrow {A}}} , one has
f ( a + v ) = f ( a ) + f → ( v ) . {\displaystyle f(a+v)=f(a)+{\overrightarrow {f}}(v).}
Therefore, since for any given b in A, b = a + v for a unique v, f is completely defined by its value on a single point and the associated linear map f → {\displaystyle {\overrightarrow {f}}} .
Endomorphisms
An affine transformation or endomorphism of an affine space A {\displaystyle A} is an affine map from that space to itself. One important family of examples is the translations: given a vector v → {\displaystyle {\overrightarrow {v}}} , the translation map T v → : A → A {\displaystyle T_{\overrightarrow {v}}:A\rightarrow A} that sends a ↦ a + v → {\displaystyle a\mapsto a+{\overrightarrow {v}}} for every a {\displaystyle a} in A {\displaystyle A} is an affine map. Another important family of examples are the linear maps centred at an origin: given a point b {\displaystyle b} and a linear map M {\displaystyle M} , one may define an affine map L M , b : A → A {\displaystyle L_{M,b}:A\rightarrow A} by
L M , b ( a ) = b + M ( a − b ) {\displaystyle L_{M,b}(a)=b+M(a-b)}
for every a {\displaystyle a} in A {\displaystyle A} . After making a choice of origin b {\displaystyle b} , any affine map may be written uniquely as a combination of a translation and a linear map centred at b {\displaystyle b} .
Vector spaces as affine spaces Every vector space V may be considered as an affine space over itself. This means that every element of V may be considered either as a point or as a vector. This affine space is sometimes denoted (V, V) for emphasizing the double role of the elements of V. When considered as a point, the zero vector is commonly denoted o (or O, when upper-case letters are used for points) and called the origin. If A is another affine space over the same vector space (that is V = A → {\displaystyle V={\overrightarrow {A}}} ) the choice of any point a in A defines a unique affine isomorphism, which is the identity of V and maps a to o. In other words, the choice of an origin a in A allows us to identify A and (V, V) up to a canonical isomorphism. The counterpart of this property is that the affine space A may be identified with the vector space V in which "the place of the origin has been forgotten".
Relation to Euclidean spaces
Definition of Euclidean spaces 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
x ⋅ y = 1 2 ( q ( x + y ) − q ( x ) − q ( y ) ) . {\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
d ( A , B ) = q ( B − A ) . {\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.
Affine properties 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.
Affine combinations and barycenter Let a1, ..., an be a collection of n points in an affine space, and λ 1 , … , λ n {\displaystyle \lambda _{1},\dots ,\lambda _{n}} be n elements of the ground field. Suppose that λ 1 + ⋯ + λ n = 0 {\displaystyle \lambda _{1}+\dots +\lambda _{n}=0} . For any two points o and o' one has
λ 1 o a 1 → + ⋯ + λ n o a n → = λ 1 o ′ a 1 → + ⋯ + λ n o ′ a n → . {\displaystyle \lambda _{1}{\overrightarrow {oa_{1}}}+\dots +\lambda _{n}{\overrightarrow {oa_{n}}}=\lambda _{1}{\overrightarrow {o'a_{1}}}+\dots +\lambda _{n}{\overrightarrow {o'a_{n}}}.}
Thus, this sum is independent of the choice of the origin, and the resulting vector may be denoted
λ 1 a 1 + ⋯ + λ n a n . {\displaystyle \lambda _{1}a_{1}+\dots +\lambda _{n}a_{n}.}
When n = 2 , λ 1 = 1 , λ 2 = − 1 {\displaystyle n=2,\lambda _{1}=1,\lambda _{2}=-1} , one retrieves the definition of the subtraction of points. Now suppose instead that the field elements satisfy λ 1 + ⋯ + λ n = 1 {\displaystyle \lambda _{1}+\dots +\lambda _{n}=1} . For some choice of an origin o, denote by g {\displaystyle g} the unique point such that
λ 1 o a 1 → + ⋯ + λ n o a n → = o g → . {\displaystyle \lambda _{1}{\overrightarrow {oa_{1}}}+\dots +\lambda _{n}{\overrightarrow {oa_{n}}}={\overrightarrow {og}}.}
One can show that g {\displaystyle g} is independent from the choice of o. Therefore, if
λ 1 + ⋯ + λ n = 1 , {\displaystyle \lambda _{1}+\dots +\lambda _{n}=1,}
one may write
g = λ 1 a 1 + ⋯ + λ n a n . {\displaystyle g=\lambda _{1}a_{1}+\dots +\lambda _{n}a_{n}.}
The point g {\displaystyle g} is called the barycenter of the a i {\displaystyle a_{i}} for the weights λ i {\displaystyle \lambda _{i}} . One says also that g {\displaystyle g} is an affine combination of the a i {\displaystyle a_{i}} with coefficients λ i {\displaystyle \lambda _{i}} .
Examples The number line is a one-dimensional affine space: each real number is identified with a point on the line. Addition or subtraction by a number corresponds to translation of the line. Time can be modelled as a one-dimensional affine space. Specific points in time (such as a date on the calendar) are points in the affine space, while durations (such as a number of days) are displacements. The space of energies is an affine space for R {\displaystyle \mathbb {R} } , since it is often not meaningful to talk about absolute energy, but it is meaningful to talk about energy differences. The vacuum energy when it is defined picks out a canonical origin. Physical space is often modelled as an affine space for R 3 {\displaystyle \mathbb {R} ^{3}} in non-relativistic settings and R 1 , 3 {\displaystyle \mathbb {R} ^{1,3}} in the relativistic setting. To distinguish them from the vector space these are sometimes called Euclidean spaces E ( 3 ) {\displaystyle {\text{E}}(3)} and E ( 1 , 3 ) {\displaystyle {\text{E}}(1,3)} . Any coset of a subspace V of a vector space is an affine space over that subspace. In particular, a line in the plane that doesn't pass through the origin is an affine space that is not a vector space relative to the operations it inherits from R 2 {\displaystyle \mathbb {R} ^{2}} , although it can be given a canonical vector space structure by picking the point closest to the origin as the zero vector; likewise in higher dimensions and for any normed vector space If T is a matrix and b lies in its column space, the set of solutions of the equation Tx = b is an affine space over the subspace of solutions of Tx = 0. The solutions of an inhomogeneous linear differential equation form an affine space over the solutions of the corresponding homogeneous linear equation. Generalizing all of the above, if T : V → W is a linear map and y lies in its image, the set of solutions x ∈ V to the equation Tx = y is a coset of the kernel of T , and is therefore an affine space over Ker T . The space of (linear) complementary subspaces of a vector subspace V in a vector space W is an affine space, over Hom(W/V, V). That is, if 0 → V → W → X → 0 is a short exact sequence of vector spaces, then the space of all splittings of the exact sequence naturally carries the structure of an affine space over Hom(X, V). The space of connections (viewed from the vector bundle E → π M {\displaystyle E\xrightarrow {\pi } M} , where M {\displaystyle M} is a smooth manifold) is an affine space for the vector space of End ( E ) {\displaystyle {\text{End}}(E)} valued 1-forms. The space of connections (viewed from the principal bundle P → π M {\displaystyle P\xrightarrow {\pi } M} ) is an affine space for the vector space of ad ( P ) {\displaystyle {\text{ad}}(P)} -valued 1-forms, where ad ( P ) {\displaystyle {\text{ad}}(P)} is the associated adjoint bundle.
Affine span and bases For any non-empty subset X of an affine space A, there is a smallest affine subspace that contains it, called the affine span of X. It is the intersection of all affine subspaces containing X, and its direction is the intersection of the directions of the affine subspaces that contain X. The affine span of X is the set of all (finite) affine combinations of points of X, and its direction is the linear span of the x − y for x and y in X. If one chooses a particular point x0, the direction of the affine span of X is also the linear span of the x – x0 for x in X. One says also that the affine span of X is generated by X and that X is a generating set of its affine span. A set X of points of an affine space is said to be affinely independent or, simply, independent, if the affine span of any strict subset of X is a strict subset of the affine span of X. An affine basis or barycentric frame (see § Barycentric coordinates, below) of an affine space is a generating set that is also independent (that is a minimal generating set). Recall that the dimension of an affine space is the dimension of its associated vector space. The bases of an affine space of finite dimension n are the independent subsets of n + 1 elements, or, equivalently, the generating subsets of n + 1 elements. Equivalently, {x0, ..., xn} is an affine basis of an affine space if and only if {x1 − x0, ..., xn − x0} is a linear basis of the associated vector space.
Coordinates There are two strongly related kinds of coordinate systems that may be defined on affine spaces.
Barycentric coordinates
Let A be an affine space of dimension n over a field k, and { x 0 , … , x n } {\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 ( λ 0 , … , λ n ) {\displaystyle (\lambda _{0},\dots ,\lambda _{n})} of elements of k such that
λ 0 + ⋯ + λ n = 1 {\displaystyle \lambda _{0}+\dots +\lambda _{n}=1}
and
x = λ 0 x 0 + ⋯ + λ n x n . {\displaystyle x=\lambda _{0}x_{0}+\dots +\lambda _{n}x_{n}.}
The λ i {\displaystyle \lambda _{i}} are called the barycentric coordinates of x over the affine basis { x 0 , … , x n } {\displaystyle \{x_{0},\dots ,x_{n}\}} . If the xi are viewed as bodies that have weights (or masses) λ i {\displaystyle \lambda _{i}} , the point x is thus the barycenter of the xi, 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 kn + 1 defined by the equation λ 0 + ⋯ + λ n = 1 {\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.
Affine coordinates An affine frame is a coordinate frame of an affine space, consisting 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 A → {\displaystyle {\overrightarrow {A}}} , the origin o belongs to A, and the linear basis is a basis (v1, ..., vn) of A → {\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 λ 1 , … , λ n {\displaystyle \lambda _{1},\dots ,\lambda _{n}} of elements of the ground field such that
p = o + λ 1 v 1