
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an affine space is a
geometric structure
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
that generalizes some of the properties of
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
s in such a way that these are independent of the concepts of
distance and measure of
angle
In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
s, keeping only the properties related to
parallelism and
ratio
In mathematics, a ratio () shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
of lengths for parallel
line segment
In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...
s. 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-
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
al. 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 mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
, 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
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
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
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping'');
* linearity of a ''polynomial''.
An example of a li ...
(vector subspace) of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
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
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
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
In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
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
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
, 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
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
.
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 number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s, a ''linear manifold'') of an affine space is a
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of such that, given a point
, the set of vectors
is a
linear subspace
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping'');
* linearity of a ''polynomial''.
An example of a li ...
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
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
, 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 transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, '' affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More general ...
s 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.
*
Time
Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...
can be modelled as a one-dimensional affine space. Specific points in time (such as a date on the calendar) are points in the affine space, while durations (such as a number of days) are displacements.
* 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 space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
s
and
.
* Any
coset of a subspace of a vector space is an affine space over that subspace.
* In particular, a line in the plane that doesn't pass through the origin is an affine space that is not a vector space relative to the operations it inherits from
, although it can be given a canonical vector space structure by picking the point closest to the origin as the zero vector; likewise in higher dimensions and for any normed vector space
* 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
An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
, 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
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...
, 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
In mathematics, a tuple is a finite sequence or ''ordered list'' of numbers or, more generally, mathematical objects, which are called the ''elements'' of the tuple. An -tuple is a tuple of elements, where is a non-negative integer. There is o ...
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 is a
coordinate frame of an affine space, consisting 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
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
,
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
In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
. 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 barycentric coordinates
Barycentric coordinates are readily changed from one basis to another. Let
and
be affine bases of . For every in 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 . For each point of , 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
An affine transformation
is executed on a projective space
of
, by a 4 by 4 matrix with a special fourth column:
The transformation is affine instead of linear due to the inclusion of point
, the transformed output of which reveals the affine shift.
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 space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
s are affine spaces, and that these kinds of projections are fundamental in
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
.
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
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping'');
* linearity of a ''polynomial''.
An example of a li ...
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 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. 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
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation  ...
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
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, 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 function
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For di ...
s (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
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
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 ''i''th 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