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.

In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors, also called translation vectors or simply translations, between two points of the space.^[1] Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector.

Any vector space may be considered as an affine space, and this amounts to forgetting the special role played by the zero vector. In this case, the 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 of a vector space produces an affine subspace. One commonly says that this affine subspace has been obtained by translating (away from the origin) the linear subspace by the translation vector. 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 you've 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"^[2]). 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

An affine space is a set A together with a vector space ${\displaystyle {\overrightarrow {A}}}$, and a transitive and free action of the additive group of ${\displaystyle {\overrightarrow {A}}}$ on the set A.^[3] The elements of the affine space A are called points. The vector space ${\displaystyle {\overrightarrow {A}}}$ 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,

In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors, also called translation vectors or simply translations, between two points of the space.^[1] Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector.

Any vector space may be considered as an affine space, and this amounts to forgetting the special role played by the zero vector. In this case, the 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 of a vector space produces an affine subspace. One commonly says that this affine subspace has been obtained by translating (away from the origin) the linear subspace by the translation vector. 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.

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 you've 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"^[2]). 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

An affine space is a set A together with a vector space ${\displaystyle {\overrightarrow {A}}}$, and a transitive and free action of the additive group of ${\displaystyle {\overrightarrow {A}}}$ on the set A.^[3] The elements of the affine space A are called points. The vector space ${\displaystyle {\overrightarrow {A}}}$ 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,

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

If Alice travels to

then Bob can similarly travel to

p + λ(a − p) + (1 − λ)

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 combi

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.

An affine space is a set A together with a vector space ${\displaystyle {\overrightarrow {A}}}$, and a transitive and free action of the additive group of ${\displaystyle {\overrightarrow {A}}}$ on the set A.^[3] The elements of the affine space A are called points. The vector space ${\displaystyle {\overrightarrow {A}}}$ 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,

1. Right identity:

   ${\displaystyle \forall a\in A,\;a+0=a}$, where 0 is the zero vector in ${\displaystyle {\overrightarrow {A}}}$

2. Associativity:

4. Existence of one-to-one translations

For all ${\displaystyle v\in {\overrightarrow {A}}}$, the mapping ${\displaystyle A\to A\colon a\mapsto a+v}$ is a bijection.

Property 3 is often used in the following equivalent form.

5. Subtraction:

For every a, b in A, there exists a unique ${\displaystyle v\in {\overrightarrow {A}}}$, denoted b – a, such that ${\displaystyle b=a+v}$.

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 ${\displaystyle {\overrightarrow {A}}}$. This vector, denoted ${\displaystyle b-a}$

Subtraction:

For every a, b in A, there exists a unique ${\displaystyle v\in {\overrightarrow {A}}}$, denoted b – a, such that ${\displaystyle b=a+v}$.

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 ${\displaystyle {\overrightarrow {A}}}$. This vector, denoted ${\displaystyle b-a}$

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

${\displaystyle {\overrightarrow {A}}}$