TheInfoList

OR: In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
s 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 segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...
s. In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead '' displacement vectors'', also called ''
translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transl ...
'' vectors or simply ''translations'', between two points of the space. Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. 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. Any
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
may be viewed as an affine space; this amounts to forgetting the special role played by the
zero vector In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An additive ident ...
. 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, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, ...
(vector subspace) of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
produces an ''affine subspace''. 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 Characterization or characterisation is the representation of persons (or other beings or creatures) in narrative and dramatic works. The term character development is sometimes used as a synonym. This representation may include direct methods ...
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 whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
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 Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transl ...
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 Alice and Bob are fictional characters commonly used as placeholders in discussions about cryptographic systems and protocols, and in other science and engineering literature where there are several participants in a thought experiment. The A ...
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 combination In mathematics, an affine combination of is a linear combination : \sum_^ = \alpha_ x_ + \alpha_ x_ + \cdots +\alpha_ x_, such that :\sum_^ =1. Here, can be elements ( vectors) of a vector space over a field , and the coefficients \alpha_ ...
s, defined as linear combinations in which the sum of the coefficients is 1. A set with an affine structure is an affine space.

# Definition

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 whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
$\overrightarrow$, and a transitive and free
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
of the additive group of $\overrightarrow$ on the set . The elements of the affine space are called ''points''. The vector space $\overrightarrow$ is said to be ''associated'' to the affine space, and its elements are called ''vectors'', ''translations'', or sometimes ''
free vector In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors a ...
s''. Explicitly, the definition above means that the action is a mapping, generally denoted as an addition, : $\begin A \times \overrightarrow &\to A \\ \left(a,v\right)\; &\mapsto a + v, \end$ that has the following properties. # Right identity: #: $\forall a \in A,\; a+0 = a$, where is the zero vector in $\overrightarrow$ # Associativity: #: $\forall v,w \in \overrightarrow, \forall a \in A,\; \left(a + v\right) + w = a + \left(v + w\right)$ (here the last is the addition in $\overrightarrow$) # Free and transitive action: #: For every $a \in A$, the mapping $\overrightarrow A \to A \colon v \mapsto a + v$ is a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
. 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
translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transl ...
s
• #:For all $v \in \overrightarrow A$, the mapping $A \to A \colon a \mapsto a + v$ is a bijection. Property 3 is often used in the following equivalent form (the 5th property). #
• Subtraction:
• #:For every in , there exists a unique $v\in\overrightarrow A$, denoted , such that $b = a + v$. Another way to express the definition is that an affine space is a
principal homogeneous space In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non ...
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 $\overrightarrow$. This vector, denoted $b - a$ or $\overrightarrow$, is defined to be the unique vector in $\overrightarrow$ such that : $a + \left(b - a\right) = b.$ 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: # $\forall a \in A,\; \forall v\in \overrightarrow$, there is a unique point $b \in A$ such that $b - a = v.$ # $\forall a,b,c \in A,\; \left(c - b\right) + \left(b - a\right) = c - a.$ In
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axiom ...
, the second of Weyl's axiom is commonly called the ''parallelogram rule''. Affine spaces can be equivalently defined as a point set , together with a vector space $\overrightarrow$, 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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, a ''linear manifold'') of an affine space is a subset of such that, given a point $a \in B$, the set of vectors $\overrightarrow = \$ is a
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, ...
of $\overrightarrow$. This property, which does not depend on the choice of , implies that is an affine space, which has $\overrightarrow$ as its associated vector space. The affine subspaces of are the subsets of of the form : $a + V = \,$ where is a point of , and a linear subspace of $\overrightarrow$. 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 $A \to A: a \mapsto a + v$ 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 $\overrightarrow$ and $\overrightarrow$, an ''affine map'' or ''affine homomorphism'' from to is a map : $f: A \to B$ such that : $\begin \overrightarrow: \overrightarrow &\to \overrightarrow\\ b - a &\mapsto f\left(b\right) - f\left(a\right) \end$ is a
well defined In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A func ...
linear map. By $f$ being well defined is meant that implies . This implies that, for a point $a \in A$ and a vector $v \in \overrightarrow$, one has : $f\left(a + v\right) = f\left(a\right) + \overrightarrow\left(v\right).$ Therefore, since for any given in , for a unique , is completely defined by its value on a single point and the associated linear map $\overrightarrow$.

## Endomorphisms

An ''affine transformation'' or ''endomorphism'' of an affine space $A$ is an affine map from that space to itself. One important family of examples is the translations: given a vector $\overrightarrow$, the translation map $T_: A\rightarrow A$ that sends $a\mapsto a + \overrightarrow$ for every $a$ in $A$ is an affine map. Another important family of examples are the linear maps centred at an origin: given a point $b$ and a linear map $M$, one may define an affine map $L_:A\rightarrow A$ by $L_(a) = b + M(a-b)$ for every $a$ in $A$. After making a choice of origin $b$, any affine map may be written uniquely as a combination of a translation and a linear map centred at $b$.

# 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 In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An additive ident ...
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 $V = \overrightarrow$) 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 Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...
a
canonical isomorphism In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word i ...
. 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 : $x \cdot y = \frac 12 \left(q\left(x + y\right) - q\left(x\right) - q\left(y\right)\right).$ The usual
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore o ...
between two points and is : $d\left(A, B\right) = \sqrt.$ In older definition of Euclidean spaces through synthetic geometry, vectors are defined as equivalence classes of
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
s of points under equipollence (the pairs and are ''equipollent'' if the points (in this order) form a
parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of eq ...
). It is straightforward to verify that the vectors form a vector space, the square of the
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore o ...
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 mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axiom ...
, 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 In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
. 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 $\lambda_1, \dots, \lambda_n$ be elements of the ground field. Suppose that $\lambda_1 + \dots + \lambda_n = 0$. For any two points and one has : $\lambda_1 \overrightarrow + \dots + \lambda_n \overrightarrow = \lambda_1 \overrightarrow + \dots + \lambda_n \overrightarrow.$ Thus, this sum is independent of the choice of the origin, and the resulting vector may be denoted : $\lambda_1 a_1 + \dots + \lambda_n a_n .$ When $n = 2, \lambda_1 = 1, \lambda_2 = -1$, one retrieves the definition of the subtraction of points. Now suppose instead that the field elements satisfy $\lambda_1 + \dots + \lambda_n = 1$. For some choice of an origin , denote by $g$ the unique point such that : $\lambda_1 \overrightarrow + \dots + \lambda_n \overrightarrow = \overrightarrow.$ One can show that $g$ is independent from the choice of . Therefore, if : $\lambda_1 + \dots + \lambda_n = 1,$ one may write : $g = \lambda_1 a_1 + \dots + \lambda_n a_n.$ The point $g$ is called the
barycenter In astronomy, the barycenter (or barycentre; ) is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit. A barycenter is a dynamical point, not a physical object. It is an important con ...
of the $a_i$ for the weights $\lambda_i$. One says also that $g$ is an
affine combination In mathematics, an affine combination of is a linear combination : \sum_^ = \alpha_ x_ + \alpha_ x_ + \cdots +\alpha_ x_, such that :\sum_^ =1. Here, can be elements ( vectors) of a vector space over a field , and the coefficients \alpha_ ...
of the $a_i$ with coefficients $\lambda_i$.

# Examples

* When children find the answers to sums such as or by counting right or left on a
number line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
, they are treating the number line as a one-dimensional affine space. * The space of energies is an affine space for $\mathbb$, 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 $\mathbb^3$ in non-relativistic settings and $\mathbb^$ 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, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
s $\text\left(3\right)$ and $\text\left(1,3\right)$. * Any
coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) h ...
of a subspace of a vector space is an affine space over that subspace. * If is a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
and lies in its
column space In linear algebra, the column space (also called the range or image) of a matrix ''A'' is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding mat ...
, 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 Connections may refer to: Television * '' Connections: An Investigation into Organized Crime in Canada'', a documentary television series * ''Connections'' (British documentary), a documentary television series and book by science historian Jam ...
(viewed from the vector bundle $E\xrightarrowM$, where $M$ is a smooth manifold) is an affine space for the vector space of $\text\left(E\right)$ valued 1-forms. The space of connections (viewed from the principal bundle $P\xrightarrowM$) is an affine space for the vector space of $\text\left(P\right)$-valued 1-forms, where $\text\left(P\right)$ is the associated adjoint bundle.

# Affine span and bases

For any 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 In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterized ...
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 system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is s ...
s 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 ordered list ( sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is de ...
$\left(\lambda_0, \dots, \lambda_n\right)$ of elements of such that : $\lambda_0 + \dots + \lambda_n = 1$ and : $x = \lambda_0 x_0 + \dots + \lambda_n x_n.$ The $\lambda_i$ are called the barycentric coordinates of over the affine basis $\$. If the are viewed as bodies that have weights (or masses) $\lambda_i$, the point is thus the
barycenter In astronomy, the barycenter (or barycentre; ) is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit. A barycenter is a dynamical point, not a physical object. It is an important con ...
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 $\lambda_0 + \dots + \lambda_n = 1$. 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.

## 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 $\overrightarrow$, the origin belongs to , and the linear basis is a basis of $\overrightarrow$ (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 $\lambda_1, \dots, \lambda_n$ of elements of the ground field such that : $p = o + \lambda_1 v_1 + \dots + \lambda_n v_n,$ or equivalently : $\overrightarrow = \lambda_1 v_1 + \dots + \lambda_n v_n.$ The $\lambda_i$ are called the affine coordinates of over the affine frame . Example: In
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axiom ...
,
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
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 : $\left(x_0, \dots, x_n\right),$ one deduces immediately the affine frame : $\left(x_0, \overrightarrow, \dots, \overrightarrow\right) = \left\left(x_0, x_1 - x_0, \dots, x_n - x_0\right\right),$ and, if : $\left\left(\lambda_0, \lambda_1, \dots, \lambda_n\right\right)$ are the barycentric coordinates of a point over the barycentric frame, then the affine coordinates of the same point over the affine frame are : $\left\left(\lambda_1, \dots, \lambda_n\right\right).$ Conversely, if : $\left\left(o, v_1, \dots, v_n\right\right)$ is an affine frame, then : $\left\left(o, o + v_1, \dots, o + v_n\right\right)$ is a barycentric frame. If : $\left\left(\lambda_1, \dots, \lambda_n\right\right)$ are the affine coordinates of a point over the affine frame, then its barycentric coordinates over the barycentric frame are : $\left\left(1 - \lambda_1 - \dots - \lambda_n, \lambda_1, \dots, \lambda_n\right\right).$ 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.

### Example of the triangle

The vertices of a non-flat
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
form an affine basis of the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
. 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 The Medes (Old Persian: ; Akkadian: , ; Ancient Greek: ; Latin: ) were an ancient Iranian people who spoke the Median language and who inhabited an area known as Media between western and northern Iran. Around the 11th century BC, t ...
are the points that have two equal coordinates, and the
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any o ...
is the point of coordinates .

# Properties of affine homomorphisms

## Image and fibers

Let : $f \colon E \to F$ be an affine homomorphism, with : $\overrightarrow \colon \overrightarrow \to \overrightarrow$ as associated linear map. The image of is the affine subspace of , which has $\overrightarrow\left(\overrightarrow\right)$ as associated vector space. As an affine space does not have a zero element, an affine homomorphism does not have a kernel. However, for any point of , the
inverse image In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through ...
is an affine subspace of , of direction $\overrightarrow^\left(x\right)$. 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, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
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 mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axiom ...
. More precisely, given an affine space with associated vector space $\overrightarrow$, let be an affine subspace of direction $\overrightarrow$, and be a complementary subspace of $\overrightarrow$ in $\overrightarrow$ (this means that every vector of $\overrightarrow$ may be decomposed in a unique way as the sum of an element of $\overrightarrow$ and an element of ). For every point of , its projection to parallel to is the unique point in such that : $p\left(x\right) - x \in D.$ This is an affine homomorphism whose associated linear map $\overrightarrow$ is defined by : $\overrightarrow\left(x - y\right) = p\left(x\right) - p\left(y\right),$ 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, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, ...
of the associated vector space $\overrightarrow$. The quotient of by is the
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of by the
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
such that and are equivalent if : $x - y \in D.$ This quotient is an affine space, which has $\overrightarrow/D$ as associated vector space. For every affine homomorphism $E \to F$, 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 In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engine ...
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 In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle. As the notion of ''parallel lines'' is one of the main properties that is ind ...
over the reals as
ordered geometry Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry, omitting the basic notion of measurement. Ordered geometry is a fundamental geometry forming a common framework for affi ...
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 space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
s. For example, an affine plane can be obtained from any
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...
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, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of the projective group. For instance,
Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad ...
s (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 geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. A ...
.

# Affine algebraic geometry

In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, an
affine variety In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime idea ...
(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 dist ...
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 $\mathbb_k^n$ of dimension over a field induces an affine
isomorphism In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word i ...
between $\mathbb_k^n$ and the affine
coordinate space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
. This explains why, for simplification, many textbooks write $\mathbb_k^n = k^n$, 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 $\mathbb_k^n$ allows one to identify the polynomial functions on $\mathbb_k^n$ 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 $\mathbb_k^n$ is a -algebra, denoted
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variable ...
filtration Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter m ...
of
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
s.

## Zariski topology

Affine spaces over topological fields, such as the real or the complex numbers, have a natural
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. 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 ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
s (that is the
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
) of its ring of polynomial functions. When affine coordinates have been chosen, this function maps the point of coordinates $\left\left(a_1, \dots, a_n\right\right)$ to the maximal ideal $\left\langle X_1 - a_1, \dots, X_n - a_n\right\rangle$. This function is a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
(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
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s,
charts A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent tab ...
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 In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
of affine space is trivial. More precisely, $H^i\left\left(\mathbb_k^n,\mathbf\right\right) = 0$ for all coherent sheaves F, and integers $i > 0$. 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.