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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, an affine space is a geometric
structure A structure is an arrangement and organization of interrelated elements in a material object or system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. ...
that generalizes some of the properties of
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
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 Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

line segment
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 vector In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
s'', also called ''
translation Translation is the communication of the meaning Meaning most commonly refers to: * Meaning (linguistics), meaning which is communicated through the use of language * Meaning (philosophy), definition, elements, and types of meaning discusse ...
'' 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
may be viewed as an affine space; this amounts to forgetting the special role played by the
zero vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. In this case, the 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
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. In finite dimensions, such an ''affine subspace'' is the solution set of an
inhomogeneous Homogeneity and heterogeneity are concepts often used in the sciences Science (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally s ...
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 In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of fi ...
. An affine subspace of dimension in an affine space or a vector space of dimension is an
affine hyperplane In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...
.


Informal description

The following
characterization Characterization or characterisation is the representation of persons (or other beings or creatures) in narrative A narrative, story or tale is any account of a series of related events or experiences, whether nonfiction Nonfiction (also spelle ...
may be easier to understand than the usual formal definition: an affine space is what is left of a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
after you've forgotten which point is the origin (or, in the words of the French mathematician
Marcel Berger Marcel Berger (14 April 1927 – 15 October 2016) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity ...
, "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 ''transla ...
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 In cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia ''-logy'' is a suffix in the English language, used with words originally adapted from Ancient Gr ...
may evaluate any
linear combination In mathematics, a linear combination 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'' and ''y'' would be ...
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 combinationIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
\overrightarrow, and a transitive and free
action ACTION is a bus operator in , Australia owned by the . History On 19 July 1926, the commenced operating public bus services between Eastlake (now ) in the south and in the north. The service was first known as Canberra City Omnibus Se ...
of the
additive group An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structure ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s''. Explicitly, the definition above means that the action is a mapping, generally denoted as an addition, : \begin A \times \overrightarrow &\to A \\ (a,v)\; &\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,\; (a + v) + w = a + (v + w) (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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

bijection
. The first two properties are simply defining properties of a (right) group action. The third property characterizes free and transitive actions, the onto character coming from transitivity, and then the injective character follows from the action being free. There is a fourth property that follows from 1, 2 above: #
  • Existence of one-to-one
    translation Translation is the communication of the meaning Meaning most commonly refers to: * Meaning (linguistics), meaning which is communicated through the use of language * Meaning (philosophy), definition, elements, and types of meaning discusse ...
    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. #
  • 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:''For the term "torsor" in algebraic geometry, see torsor (algebraic geometry).'' In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
    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 + (b - a) = 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 Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a Germany, German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he ...

    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,\; (c - b) + (b - a) = c - a. In
    Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
    , the second 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 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
    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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
    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 In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
    : 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(b) - f(a) \end is a well defined 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(a + v) = f(a) + \overrightarrow(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.


    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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
    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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
    a
    canonical isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
    . 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 Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called pa ...
    (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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
    of finite dimension, that is a vector space over the reals with a
    positive-definite quadratic formIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
    . The inner product of two vectors and is the value of the
    symmetric bilinear formA symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear function B tha ...
    : x \cdot y = \frac 12 (q(x + y) - q(x) - q(y)). The usual
    Euclidean distance In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
    between two points and is : d(A, B) = \sqrt. In older definition of Euclidean spaces through
    synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is conce ...
    , vectors are defined as
    equivalence class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
    es of
    ordered pair In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

    ordered pair
    s of points under equipollence (the pairs and are ''equipollent'' if the points (in this order) form a
    parallelogram In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...

    parallelogram
    ). It is straightforward to verify that the vectors form a vector space, the square of the
    Euclidean distance In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
    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 Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
    , 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 Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

    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 Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...
    s 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 fieldIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
    . 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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
    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 Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses ...

    barycenter
    of the a_i for the weights \lambda_i. One says also that g is an
    affine combinationIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
    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 300px, Both groups are equal to 5. Apples are frequently used to explain arithmetic in textbooks for children. Elementary mathematics consists of mathematics Mathematics (from Ancient Greek, Greek: ) include ...

    number line
    , they are treating the number line as a one-dimensional affine space. * Any
    coset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
    of a subspace of a vector space is an affine space over that subspace. * If is a
    matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
    and lies in its
    column space In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces a ...
    , 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 mapping 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 An exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Image (mathematics ...
    of vector spaces, then the space of all
    splittings
    splittings
    of the exact sequence naturally carries the structure of an affine space over .


    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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
    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 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
    of the associated vector space.


    Coordinates

    There are two strongly related kinds of
    coordinate system In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

    coordinate system
    s that may be defined on affine spaces.


    Barycentric coordinates

    Let be an affine space of dimension over a
    field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
    , and \ be an affine basis of . The properties of an affine basis imply that for every in there is a unique -
    tuple In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
    (\lambda_0, \dots, \lambda_n) 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 Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses ...

    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 \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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
    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 Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
    ,
    Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

    Cartesian coordinates
    are affine coordinates relative to an orthonormal frame, that is an affine frame such that is an
    orthonormal basis In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
    .


    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 : (x_0, \dots, x_n), one deduces immediately the affine frame : (x_0, \overrightarrow, \dots, \overrightarrow) = \left(x_0, x_1 - x_0, \dots, x_n - x_0\right), and, if : \left(\lambda_0, \lambda_1, \dots, \lambda_n\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(\lambda_1, \dots, \lambda_n\right). Conversely, if : \left(o, v_1, \dots, v_n\right) is an affine frame, then : \left(o, o + v_1, \dots, o + v_n\right) is a barycentric frame. If : \left(\lambda_1, \dots, \lambda_n\right) are the affine coordinates of a point over the affine frame, then its barycentric coordinates over the barycentric frame are : \left(1 - \lambda_1 - \dots - \lambda_n, \lambda_1, \dots, \lambda_n\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 affinity 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 In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The ...

    triangle
    form an affine basis of the
    Euclidean plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
    . The barycentric coordinates allows easy characterization of the elements of the triangle that do not involve angles or distance: 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 a zero coordinate and two nonnegative coordinates. The interior of the triangle are the points whose all coordinates are positive. The
    medians bas-relief Relief is a sculptural technique where the sculpted elements remain attached to a solid background of the same material. The term ''relief Relief is a sculptural technique where the sculpted elements remain attached to a sol ...
    are the points that have two equal coordinates, and the
    centroid In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

    centroid
    is the point of coordinates .


    Change of coordinates


    Case of affine coordinates


    Case of barycentric coordinates


    Properties of affine homomorphisms


    Matrix representation


    Image and fibers

    Let : 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(\overrightarrow) as associated vector space. As an affine space does not have a
    zero element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
    , an affine homomorphism does not have a
    kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
    . However, for any point of , the
    inverse image In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

    inverse image
    of is an affine subspace of , of direction \overrightarrow^(\overrightarrow). 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 classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
    s are affine spaces, and that this kind of projections is fundamental in
    Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
    . 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(x) - x \in D. This is an affine homomorphism whose associated linear map \overrightarrow is defined by : \overrightarrow(x - y) = p(x) - p(y), 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
    of the associated vector space \overrightarrow. The quotient of by is the
    quotient In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne' ...
    of by the
    equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
    : 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 In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for ...

    first isomorphism theorem
    for affine spaces.


    Affine transformation


    Axioms

    Affine space is usually studied as
    analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches ...
    using coordinates, or equivalently vector spaces. It can also be studied as
    synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is conce ...
    by writing down axioms, though this approach is much less common. There are several different systems of axioms for affine space. axiomatizes
    affine geometry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
    (over the reals) as
    ordered geometryOrdered geometry is a form of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned wit ...
    together with an affine form of
    Desargues's theorem for the truth of the second. In projective geometry, Desargues's theorem, named after Girard Desargues, states: :Two triangle A triangle is a polygon In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' ...
    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 ringIn algebra, a division ring, also called a skew field, is a ring (mathematics), ring in which division (mathematics), division is possible. Specifically, it is a zero ring, nonzero ring in which every nonzero element has a multiplicative inverse, th ...
    s), there are also many
    non-Desarguesian planeIn mathematics, a non-Desarguesian plane is a projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect in a si ...
    s satisfying these axioms. gives axioms for higher-dimensional affine spaces.


    Relation to projective spaces

    Affine spaces are subspaces of
    projective space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
    s: an affine plane can be obtained from any
    projective plane In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
    by removing a 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 In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

    line at infinity
    whose points correspond to equivalence classes of
    parallel lines In geometry, parallel lines are line (geometry), lines in a plane (geometry), plane which do not meet; that is, two straight lines in a plane that do not intersecting lines, intersect at any point are said to be parallel. Colloquially, curves tha ...

    parallel lines
    . Further, transformations of projective space that preserve affine space (equivalently, that leave the
    hyperplane at infinityIn geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...
    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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
    is a
    subgroup In group theory, a branch of mathematics, given a group (mathematics), 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 ...
    of the projective group. For instance,
    Möbius transformation In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
    s (transformations of the complex projective line, or
    Riemann sphere In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

    Riemann sphere
    ) are affine (transformations of the complex plane) if and only if they fix the
    point at infinity 150px, The real line with the point at infinity; it is called the real projective line. In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic ...
    .


    Affine algebraic geometry

    In
    algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

    algebraic geometry
    , an
    affine variety Affine (pronounced /əˈfaɪn/) relates to connections or affinities. It may refer to: *Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher *Affine combinationIn mathemati ...
    (or, more generally, an
    affine algebraic set Affine (pronounced /əˈfaɪn/) relates to connections or affinities. It may refer to: *Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine combination, a certai ...
    ) 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 In mathematics, a polynomial is an expression (mathematics), expression consisting of variable (mathematics), variables (also called indeterminate (variable), indeterminates) and coefficients, that involves only the operations of addition, subtra ...
    of the coordinates of the point. As a change of affine coordinates may be expressed by
    linear function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

    linear function
    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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
    induces an affine
    isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

    isomorphism
    between \mathbb_k^n and the affine coordinate space . 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
    , 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 k\left mathbb_k^n\right/math>, which is isomorphic to the
    polynomial ring In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
    k\left _1, \dots, X_n\right/math>. When one changes coordinates, the isomorphism between k\left mathbb_k^n\right/math> and k _1, \dots, X_n/math> changes accordingly, and this induces an automorphism of k\left _1, \dots, X_n\right/math>, which maps each indeterminate to a polynomial of degree one. It follows that the
    total degree In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
    defines a
    filtration Filtration is a physical separation process A separation process is a method that converts a mixture or solution of chemical substances into two or more distinct product mixtures. At least one of results of the separation is enriched in one or m ...
    of k\left mathbb A_k^n\right/math>, which is independent from the choice of coordinates. The total degree defines also a
    graduation Graduation is the award of academic degree An academic degree is a qualification awarded to students upon successful completion of a course of study in higher education, usually at a college or university. These institutions commonly offer d ...
    , but it depends on the choice of coordinates, as a change of affine coordinates may map indeterminates on non-
    homogeneous polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
    s.


    Zariski topology

    Affine spaces over
    topological fieldIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
    s, such as the real or the complex numbers, have a natural
    topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
    . 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 Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
    are
    affine algebraic set Affine (pronounced /əˈfaɪn/) relates to connections or affinities. It may refer to: *Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine combination, a certai ...
    s (that is sets of the common zeros of polynomials 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 a ring, spectrum) of its ring of polynomial functions. When affine coordinates have been chosen, this function maps the point of coordinates \left(a_1, \dots, a_n\right) to the maximal ideal \left\langle X_1 - a_1, \dots, X_n - a_n\right\rangle. This function is a homeomorphism (for the Zariski topology of the affine space and of the spectrum of the ring of polynomial functions) of the affine space onto the image of the function. The case of an algebraically closed field, 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, chart (topology), charts are glued together for building a manifold.


    Cohomology

    Like all affine varieties, local data on an affine space can always be patched together globally: the cohomology of affine space is trivial. More precisely, H^i\left(\mathbb_k^n,\mathbf\right) = 0 for all coherent sheaves F, and integers i > 0. This property is also enjoyed by all other affine variety, affine varieties. But also all of the etale cohomology groups on affine space are trivial. In particular, every line bundle is trivial. More generally, the Quillen–Suslin theorem implies that ''every'' algebraic vector bundle over an affine space is trivial.


    See also

    * * * *


    Notes


    References

    * * * * * * * * * {{DEFAULTSORT:Affine Space Affine geometry Linear algebra