Affine space
In
As in Euclidean space, the fundamental objects in an affine space are called
Unlike for vectors in a
Any
The dimension of an affine space is defined as the
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"[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
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 , and a transitive and free
Explicitly, the definition above means that the action is a mapping, generally denoted as an addition,
that has the following properties.[4][5][6]
- Right identity:
- , where 0 is the zero vector in
- Associativity:
- (here the last + is the addition in )
- 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
- 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 a, b in A, there exists a unique , denoted b – a, 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 (b, a) of points in A, 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:[7]
- , 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 A, 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) B of an affine space A is a subset of A such that, given a point , the set of vectors is a linear subspace of . This property, which does not depend on the choice of a, implies that B is an affine space, which has as its associated vector space.
The affine subspaces of A are the subsets of A of the form
where a is a point of A, and V a linear subspace of .
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 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 and , an
such that
is a
This implies that, for a point and a vector , one has
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 .
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
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 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
If A is another affine space over the same vector space (that is ) 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)
Relation to Euclidean spaces
Definition of Euclidean 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
The usual Euclidean distance between two points A and B 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 (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 be n elements of the ground field.
Suppose that . For any two points o and o' 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 o, denote by the unique point such that
One can show that is independent from the choice of o. 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 4 + 3 or 4 − 2 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 spaceis 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 V of a vector space is an affine space over that subspace.
- If T is a 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) short exact sequence of vector spaces, then the space of all splittingsof the exact sequence naturally carries the structure of an affine space over Hom(X, V).
- The space of connections (viewed from the vector bundle , where is a smooth manifold) is an affine space for the vector space of valued1-forms. The space of connections (viewed from the principal bundle) is an affine space for the vector space of -valued 1-forms, where is theassociated 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
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
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 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 of elements of k such that
and
The are called the barycentric coordinates of x over the affine basis . If the xi are viewed as bodies that have weights (or masses) , 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 .
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
For each point p of A, there is a unique sequence of elements of the ground field such that
or equivalently
The are called the affine coordinates of p over the affine frame (o, v1, ..., vn).
Example: In
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 (1, 0, 0), (0, 1, 0) and (0, 0, 1). The lines supporting the
Change of coordinates
Case of barycentric coordinates
Barycentric coordinates are readily changed from one basis to another. Let and be affine bases of A. For every x in A there is some tuple for which
Similarly, for every from the first basis, we now have in the second basis
for some tuple . Now we can rewrite our expression in the first basis as one in the second with
giving us coordinates in the second basis as the tuple .
Case of affine coordinates
Affine coordinates are also readily changed from one basis to another. Let , and , be affine frames of A. For each point p of A, there is a unique sequence of elements of the ground field such that
and similarly, for every from the first basis, we now have in the second basis
for tuple and tuples . Now we can rewrite our expression in the first basis as one in the second with
giving us coordinates in the second basis as the tuple .
Properties of affine homomorphisms
Matrix representation
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Image and fibers
Let
be an affine homomorphism, with
its associated linear map. The image of f is the affine subspace of F, 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 x of , the
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 E with associated vector space , let F be an affine subspace of direction , and D be a
This is an affine homomorphism whose associated linear map is defined by
for x and y in E.
The image of this projection is F, and its fibers are the subspaces of direction D.
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 E be an affine space, and D be a linear subspace of the associated vector space . 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
This quotient is an affine space, which has as associated vector space.
For every affine homomorphism , the image is isomorphic to the quotient of E by the kernel of the associated linear map. This is the
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.
Coxeter (1969, p. 192) 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 (Cameron 1991, chapter 2): (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 that 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. (Cameron 1991, chapter 3) 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
Further, transformations of projective space that preserve affine space (equivalently, that leave the
Affine algebraic geometry
In
The choice of a system of affine coordinates for an affine space of dimension n over a field k induces an affine isomorphism between and the affine
As the whole affine space is the set of the common zeros of the
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 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 is a k-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
Zariski topology
Affine spaces over
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
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
See also
- Affine hull – Smallest affine subspace that contains a subset
- Complex affine space – Affine space over the complex numbers
- Exotic affine space – Real affine space of even dimension that is not isomorphic to a complex affine space
- Space (mathematics) – Mathematical set with some added structure
- Barycentric coordinate system
Notes
- displacements include also rotations.
- ^ Berger 1987, p. 32
- ISBN 9780387909714
- ^ Berger 1987, p. 33
- ^ Snapper, Ernst; Troyer, Robert J. (1989), Metric Affine Geometry, p. 6
- ISBN 9780857297105
- ^ Nomizu & Sasaki 1994, p. 7
- ^ Hartshorne 1977, Ch. I, § 1.
References
- Berger, Marcel (1984), "Affine spaces", Problems in Geometry, Springer-Verlag, ISBN 978-0-387-90971-4
- ISBN 3-540-11658-3
- MR 1153019
- MR 0123930
- Dolgachev, I.V.; Shirokov, A.P. (2001) [1994], "Affine space", Encyclopedia of Mathematics, EMS Press
- Zbl 0367.14001.
- ISBN 978-0-521-44177-3
- ISBN 0-486-66108-3
- Reventós Tarrida, Agustí (2011), "Affine spaces", Affine Maps, Euclidean Motions and Quadrics, Springer, ISBN 978-0-85729-709-9