Affine space

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 which 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 which 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 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 (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 space). 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 (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 —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 (see 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 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.
Right identity:
, where is the zero vector in
Associativity:
(here the last is the addition in )
Free and transitive action:
For every , the mapping 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 , the mapping is a bijection.
Property 3 is often used in the following equivalent form (the 5th property).
Subtraction:
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:
, 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 (also called, in some contexts, a linear variety, a flat, or, over the real numbers, a linear manifold) of an affine space is a subset of such that, given a point , the set of vectors is a linear subspace of . This property, which does not depend on the choice of , implies that 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 (or , when upper-case letters are used for points) and called the origin.
If is another affine space over the same vector space (that is ) 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".
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 . The inner product of two vectors and is the value of the symmetric bilinear form
The usual Euclidean distance between two points and is
In older definition of Euclidean spaces through synthetic geometry, vectors are defined as equivalence classes of ordered pairs of points under equipollence (the pairs and are equipollent if the points (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 be a collection of points in an affine space, and be elements of the ground field.
Suppose that . For any two points and one has
Thus, this sum is independent of the choice of the origin, and the resulting vector may be denoted
When , one retrieves the definition of the subtraction of points.
Now suppose instead that the field elements satisfy . For some choice of an origin , denote by the unique point such that
One can show that is independent from the choice of . Therefore, if
one may write
The point is called the barycenter of the for the weights . One says also that is an affine combination of the with coefficients .
Examples
When children find the answers to sums such as or by counting right or left on a number line, they are treating the number line as a one-dimensional affine space.
The space of energies is an affine space for , 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 in non-relativistic settings and in the relativistic setting. To distinguish them from the vector space these are sometimes called Euclidean spaces and .
Any coset of a subspace of a vector space is an affine space over that subspace.
If is a matrix and lies in its column space, the set of solutions of the equation is an affine space over the subspace of solutions of .
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 is a linear map and lies in its image, the set of solutions to the equation is a coset of the kernel of , and is therefore an affine space over .
The space of (linear) complementary subspaces of a vector subspace in a vector space is an affine space, over . That is, if 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 .
The space of connections (viewed from the vector bundle , where is a smooth manifold) is an affine space for the vector space of valued 1-forms. The space of connections (viewed from the principal bundle ) is an affine space for the vector space of -valued 1-forms, where is the associated adjoint bundle.
Affine span and bases
For any non-empty subset of an affine space , there is a smallest affine subspace that contains it, called the affine span of . It is the intersection of all affine subspaces containing , and its direction is the intersection of the directions of the affine subspaces that contain .
The affine span of is the set of all (finite) affine combinations of points of , and its direction is the linear span of the for and in . If one chooses a particular point , the direction of the affine span of is also the linear span of the for in .
One says also that the affine span of is generated by and that is a generating set of its affine span.
A set of points of an affine space is said to be or, simply, independent, if the affine span of any strict subset of is a strict subset of the affine span of . An or barycentric frame (see , 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 are the independent subsets of elements, or, equivalently, the generating subsets of elements. Equivalently, } is an affine basis of an affine space if and only if } 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 be an affine space of dimension over a field , and be an affine basis of . The properties of an affine basis imply that for every in there is a unique -tuple of elements of such that
and
The are called the barycentric coordinates of over the affine basis . If the are viewed as bodies that have weights (or masses) , the point is thus the barycenter of the , and this explains the origin of the term barycentric coordinates.
The barycentric coordinates define an affine isomorphism between the affine space and the affine subspace of defined by the equation .
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 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 with associated vector space , the origin belongs to , and the linear basis is a basis of (for simplicity of the notation, we consider only the case of finite dimension, the general case is similar).
For each point of , there is a unique sequence of elements of the ground field such that
or equivalently
The are called the affine coordinates of over the affine frame .
Example: In Euclidean geometry, Cartesian coordinates are affine coordinates relative to an orthonormal frame, that is an affine frame such that is an orthonormal basis.
Relationship between barycentric and affine coordinates
Barycentric coordinates and affine coordinates are strongly related, and may be considered as equivalent.
In fact, given a barycentric frame
one deduces immediately the affine frame
and, if
are the barycentric coordinates of a point over the barycentric frame, then the affine coordinates of the same point over the affine frame are
Conversely, if
is an affine frame, then
is a barycentric frame. If
are the affine coordinates of a point over the affine frame, then its barycentric coordinates over the barycentric frame are
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.
Example of the triangle
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 , and . 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 .
Change of coordinates
Case of affine coordinates
Case of barycentric coordinates
Properties of affine homomorphisms
Matrix representation
Image and fibers
Let
be an affine homomorphism, with
its associated linear map. The image of is the affine subspace of , which has 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 does, and if we denote by its kernel, then for any point of , the inverse image of is an affine subspace of whose direction is . This affine subspace is called the fiber of .
Projection
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 with associated vector space , let be an affine subspace of direction , and be a complementary subspace of in (this means that every vector of may be decomposed in a unique way as the sum of an element of and an element of ). For every point of , its projection to parallel to is the unique point in such that
This is an affine homomorphism whose associated linear map is defined by
for and in .
The image of this projection is , and its fibers are the subspaces of direction .
Quotient space
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 be an affine space, and be a linear subspace of the associated vector space . The quotient of by is the quotient of by the equivalence relation such that and are equivalent if
This quotient is an affine space, which has as associated vector space.
For every affine homomorphism , the image is isomorphic to the quotient of by the kernel of the associated linear map. This is the first isomorphism theorem for affine spaces.
Axioms
Affine spaces are usually studied by analytic geometry using coordinates, or equivalently vector spaces. They can also be studied as synthetic geometry by writing down axioms, though this approach is much less common. There are several different systems of axioms for affine space.
axiomatizes the special case of affine geometry over the reals as ordered geometry together with an affine form of Desargues's theorem and an axiom stating that in a plane there is at most one line through a given point not meeting a given line.
Affine planes satisfy the following axioms :
(in which two lines are called parallel if they are equal or
disjoint):
Any two distinct points lie on a unique line.
Given a point and line there is a unique line which contains the point and is parallel to the line
There exist three non-collinear points.
As well as affine planes over fields (or division rings), there are also many non-Desarguesian planes satisfying these axioms. gives axioms for higher-dimensional affine spaces.
Purely axiomatic affine geometry is more general than affine spaces and is treated in a separate article.
Relation to projective spaces
Affine spaces are contained in projective spaces. For example, an affine plane can be obtained from any projective plane by removing one line and all the points on it, and conversely any affine plane can be used to construct a projective plane as a closure by adding a line at infinity whose points correspond to equivalence classes of parallel lines. Similar constructions hold in higher dimensions.
Further, transformations of projective space that preserve affine space (equivalently, that leave the hyperplane at infinity invariant as a set) yield transformations of affine space. Conversely, any affine linear transformation extends uniquely to a projective linear transformation, so the affine group is a subgroup of the projective group. For instance, Möbius transformations (transformations of the complex projective line, or Riemann sphere) are affine (transformations of the complex plane) if and only if they fix the point at infinity.
Affine algebraic geometry
In algebraic geometry, an affine variety (or, more generally, an affine algebraic set) is defined as the subset of an affine space that is the set of the common zeros of a set of so-called polynomial functions over the affine space. For defining a polynomial function over the affine space, one has to choose an affine frame. Then, a polynomial function is a function such that the image of any point is the value of some multivariate polynomial function of the coordinates of the point. As a change of affine coordinates may be expressed by linear functions (more precisely affine functions) of the coordinates, this definition is independent of a particular choice of coordinates.
The choice of a system of affine coordinates for an affine space of dimension over a field induces an affine isomorphism between and the affine coordinate space . This explains why, for simplification, many textbooks write , and introduce affine algebraic varieties as the common zeros of polynomial functions over .
As the whole affine space is the set of the common zeros of the zero polynomial, affine spaces are affine algebraic varieties.
Ring of polynomial functions
By the definition above, the choice of an affine frame of an affine space allows one to identify the polynomial functions on with polynomials in variables, the ith variable representing the function that maps a point to its th coordinate. It follows that the set of polynomial functions over is a -algebra, denoted , which is isomorphic to the polynomial ring .
When one changes coordinates, the isomorphism between and changes accordingly, and this induces an automorphism of , which maps each indeterminate to a polynomial of degree one. It follows that the total degree defines a filtration of , 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.
Zariski topology
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 to the maximal ideal . 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.
Cohomology
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, for all coherent sheaves F, and integers . 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.
See also
Barycentric coordinate system
Notes
References
Affine geometry
Linear algebra
Space (mathematics)