Affine space


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 points in general position, a -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. 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 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 of a vector space produces an affine subspace of the vector space. One commonly says that this affine subspace has been obtained by translating 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 in an affine space or a vector space of dimension 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. Imagine that Alice knows that a certain point is the actual origin, but Bob believes that another point—call it —is the origin. Two vectors, and, are to be added. Bob draws an arrow from point to point and another arrow from point to point, and completes the parallelogram to find what Bob thinks is, but Alice knows that he has actually computed
Similarly, Alice and Bob may evaluate any linear combination of and, 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
then Bob can similarly travel to
Under this condition, for all coefficients, 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, 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 together with a vector space, and a transitive and free action of the additive group of on the set. The elements of the affine space are called points. The vector space 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,
that has the following properties.
  1. Right identity:
  2. :, where is the zero vector in
  3. Associativity:
  4. :
  5. Free and transitive action:
  6. : For every, the mapping is a bijection.
The first two properties are simply defining properties of a 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:
  2. Existence of one-to-one translations
  3. : For all, the mapping is a bijection.
Property 3 is often used in the following equivalent form.
  2. Subtraction:
  3. : For every in, there exists a unique, denoted, such that.
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 of points in, producing a vector of. This vector, denoted or, is defined to be the unique vector in such that
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:
  1. , there is a unique point such that
The parallelogram property is satisfied in affine spaces, where it is expressed as: given four points the equalities and are equivalent. This results from the second Weyl's axiom, since
Affine spaces can be equivalently defined as a point set, together with a vector space, 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 of an affine space is a subset of for which there exists a point such that the set of vectors is a linear subspace of. If is an affine subspace then the set is a linear subspace for all . An affine subspace is an affine space which has as its associated vector space.
The affine subspaces of are the subsets of of the form
where is a point of, and a linear subspace of.
The linear subspace associated with an affine subspace is often called its , 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, for any point of there is one and only one affine subspace of direction, which passes through, namely the subspace.
Every translation 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 and whose associated vector spaces are and, an affine map or affine homomorphism from to is a map
such that
is a well defined linear map. By being well defined is meant that implies .
This implies that, for a point and a vector, one has
Therefore, since for any given in, for a unique, is completely defined by its value on a single point and the associated linear map.

Endomorphisms

An affine transformation or endomorphism of an affine space is an affine map from that space to itself. One important family of examples is the translations: given a vector, the translation map that sends for every in is an affine map. Another important family of examples are the linear maps centred at an origin: given a point and a linear map, one may define an affine map by
for every in.
After making a choice of origin, any affine map may be written uniquely as a combination of a translation and a linear map centred at.

Vector spaces as affine spaces

Every vector space may be considered as an affine space over itself. This means that every element of may be considered either as a point or as a vector. This affine space is sometimes denoted for emphasizing the double role of the elements of. When considered as a point, the zero vector is commonly denoted and called the origin.
If is another affine space over the same vector space the choice of any point in defines a unique affine isomorphism, which is the identity of and maps to. In other words, the choice of an origin in allows us to identify and up to a canonical isomorphism. The counterpart of this property is that the affine space may be identified with the vector space in which "the place of the origin has been forgotten".