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 (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.[1] 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.
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.
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.
\overrightarrow{A}
\overrightarrow{A}
\overrightarrow{A}
Explicitly, the definition above means that the action is a mapping, generally denoted as an addition,
\begin{align} A x \overrightarrow{A}&\toA\\ (a,v) &\mapstoa+v, \end{align}
\foralla\inA, a+0=a
\overrightarrow{A}
\forallv,w\in\overrightarrow{A},\foralla\inA, (a+v)+w=a+(v+w)
\overrightarrow{A}
For every
a\inA
\overrightarrowA\toA\colonv\mapstoa+v
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:
For all
v\in\overrightarrowA
A\toA\colona\mapstoa+v
Property 3 is often used in the following equivalent form (the 5th property).
For every in, there exists a unique
v\in\overrightarrowA
b=a+v
The properties of the group action allows for the definition of subtraction for any given ordered pair of points in, producing a vector of
\overrightarrow{A}
b-a
\overrightarrow{ab}
\overrightarrow{A}
a+(b-a)=b.
This subtraction has the two following properties, called Weyl's axioms:
\foralla\inA, \forallv\in\overrightarrow{A}
b\inA
b-a=v.
\foralla,b,c\inA, (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,
b-a=d-c
c-a=d-b
d-a=(d-b)+(b-a)=(d-c)+(c-a).
Affine spaces can be equivalently defined as a point set, together with a vector space
\overrightarrow{A}
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
a\inB
\overrightarrow{B}=\{b-a\midb\inB\}
\overrightarrow{A}
\overrightarrow{B}
The affine subspaces of are the subsets of of the form
a+V=\{a+w:w\inV\},
\overrightarrow{A}
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
A\toA:a\mapstoa+v
The term parallel is also used for two affine subspaces such that the direction of one is included in the direction of the other.
Given two affine spaces and whose associated vector spaces are
\overrightarrow{A}
\overrightarrow{B}
f:A\toB
\begin{align} \overrightarrow{f}:\overrightarrow{A}&\to\overrightarrow{B}\\ b-a&\mapstof(b)-f(a) \end{align}
f
This implies that, for a point
a\inA
v\in\overrightarrow{A}
f(a+v)=f(a)+\overrightarrow{f}(v).
Therefore, since for any given in, for a unique, is completely defined by its value on a single point and the associated linear map
\overrightarrow{f}
See main article: Affine transformation and Affine group.
An affine transformation or endomorphism of an affine space
A
\overrightarrow{v}
T\overrightarrow{v
a\mapstoa+\overrightarrow{v}
a
A
b
M
LM,b:A → A
a
A
After making a choice of origin
b
b
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
V=\overrightarrow{A}
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
x ⋅ y=
12 | |
(q(x |
+y)-q(x)-q(y)).
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 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.
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.
Let be a collection of points in an affine space, and
λ1,...,λn
Suppose that
λ1+...+λn=0
λ1\overrightarrow{oa1}+...+λn\overrightarrow{oan}=λ1\overrightarrow{o'a1}+...+λn\overrightarrow{o'an}.
λ1a1+...+λnan.
When
n=2,λ1=1,λ2=-1
Now suppose instead that the field elements satisfy
λ1+...+λn=1
g
λ1\overrightarrow{oa1}+...+λn\overrightarrow{oan}=\overrightarrow{og}.
g
λ1+...+λn=1,
g=λ1a1+...+λnan.
g
ai
λi
g
ai
λi
R
R3
R1,3
E(3)
E(1,3)
R2
E\xrightarrow{\pi}M
M
End(E)
P\xrightarrow{\pi}M
ad(P)
ad(P)
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.
There are two strongly related kinds of coordinate systems that may be defined on affine spaces.
See also: Barycentric coordinate system. Let be an affine space of dimension over a field, and
\{x0,...,xn\}
(λ0,...,λn)
λ0+...+λn=1
x=λ0x0+...+λnxn.
The
λi
\{x0,...,xn\}
λi
The barycentric coordinates define an affine isomorphism between the affine space and the affine subspace of defined by the equation
λ0+...+λn=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.
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
\overrightarrow{A}
\overrightarrow{A}
For each point of, there is a unique sequence
λ1,...,λn
p=o+λ1v1+...+λnvn,
\overrightarrow{op}=λ1v1+...+λnvn.
The
λi
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.
Barycentric coordinates and affine coordinates are strongly related, and may be considered as equivalent.
In fact, given a barycentric frame
(x0,...,xn),
(x0,\overrightarrow{x0x1},...,\overrightarrow{x0xn})=\left(x0,x1-x0,...,xn-x0\right),
\left(λ0,λ1,...,λn\right)
\left(λ1,...,λn\right).
Conversely, if
\left(o,v1,...,vn\right)
\left(o,o+v1,...,o+vn\right)
\left(λ1,...,λn\right)
\left(1-λ1-...-λn,λ1,...,λn\right).
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.
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 .
Barycentric coordinates are readily changed from one basis to another. Let
\{x0,...,xn\}
\{x'0,...,x'n\}
\{λ0,...,λn\}
x=λ0x0+...+λnxn.
xi\in\{x0,...,xn\}
xi=λi,0x'0+...+λi,jx'j+...+λi,nx'n
\{λi,0,...,λi,n\}
x=
n | |
\sum | |
i=0 |
λixi=
n | |
\sum | |
i=0 |
λi
n | |
\sum | |
j=0 |
λi,jx'j=
n | |
\sum | |
j=0 |
l(
n | |
\sum | |
i=0 |
λiλi,jr)x'j,
giving us coordinates in the second basis as the tuple .
Affine coordinates are also readily changed from one basis to another. Let
o
\{v1,...,vn\}
o'
\{v'1,...,v'n\}
λ1,...,λn
p=o+λ1v1+...+λnvn,
vi\in\{v1,...,vn\}
o=o'+λo,1v'1+...+λo,jv'j+...+λo,nv'n
vi=λi,1v'1+...+λi,jv'j+...+λi,nv'n
\{λo,1,...,λo,n\}
\{λi,1,...,λi,n\}
\begin{align} p&=o+
n | |
\sum | |
i=1 |
λivi =l(o'+
n | |
\sum | |
j=1 |
λo,jv'jr)+
n | |
\sum | |
i=1 |
λi
n | |
\sum | |
j=1 |
λi,jv'j\\ &=o'+
n | |
\sum | |
j=1 |
l(λo,j+
n | |
\sum | |
i=1 |
λiλi,jr)v'j, \end{align}
giving us coordinates in the second basis as the tuple .
Let
f\colonE\toF
\overrightarrow{f}\colon\overrightarrow{E}\to\overrightarrow{F}
f(E)=\{f(a)\mida\inE\}
\overrightarrow{f}(\overrightarrow{E})
\overrightarrow{f}
K=\{v\in\overrightarrow{E}\mid\overrightarrow{f}(v)=0\}
f(E)
f-1(x)
K
See also: Projection (mathematics). 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
\overrightarrow{E}
\overrightarrow{F}
\overrightarrow{F}
\overrightarrow{E}
\overrightarrow{E}
\overrightarrow{F}
p(x)-x\inD.
This is an affine homomorphism whose associated linear map
\overrightarrow{p}
\overrightarrow{p}(x-y)=p(x)-p(y),
The image of this projection is, and its fibers are the subspaces of direction .
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
\overrightarrow{E}
x-y\inD.
\overrightarrow{E}/D
For every affine homomorphism
E\toF
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 ordisjoint):
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.
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.
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
n | |
A | |
k |
n | |
A | |
k |
n | |
A | |
k |
=kn
As the whole affine space is the set of the common zeros of the zero polynomial, affine spaces are affine algebraic varieties.
By the definition above, the choice of an affine frame of an affine space
n | |
A | |
k |
n | |
A | |
k |
n | |
A | |
k |
n\right] | |
k\left[A | |
k |
k\left[X1,...,Xn\right]
When one changes coordinates, the isomorphism between
n\right] | |
k\left[A | |
k |
k[X1,...,Xn]
k\left[X1,...,Xn\right]
n\right] | |
k\left[A | |
k |
See also: 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
\left(a1,...,an\right)
\left\langleX1-a1,...,Xn-an\right\rangle
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.
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,
n,F\right) | |
H | |
k |
=0
i>0