Inmathematics, anaffine space is ageometricstructure that generalizes some of the properties ofEuclidean spaces in such a way that these are independent of the concepts ofdistance and measure ofangles, keeping only the properties related toparallelism andratio of lengths for parallelline segments. Affine space is the setting foraffine geometry.
As in Euclidean space, the fundamental objects in an affine space are calledpoints, which can be thought of as locations in the space without any size or shape: zero-dimensional. Through any pair of points an infinitestraight line can be drawn, a one-dimensional set of points; through any three points that are not collinear, a two-dimensionalplane can be drawn; and, in general, throughk + 1 points in general position, ak-dimensionalflat or affine subspace can be drawn. Affine space is characterized by a notion of pairs of parallel lines that lie within the same plane but never meet each-other (non-parallel lines within the same planeintersect in a point). Given any line, a line parallel to it can be drawn through any point in the space, and theequivalence class of parallel lines are said to share adirection.
Unlike for vectors in avector space, in an affine space there is no distinguished point that serves as anorigin. There is no predefined concept of adding or multiplying points together, or multiplying a point by a scalar number. However, for any affine space, an associated vector space can be constructed from the differences between start and end points, which are calledfree vectors,displacement vectors,translation vectors or simplytranslations.[1] Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point (of the same affine space) translated from the starting point by that vector. While points cannot be arbitrarily added together, it is meaningful to takeaffine combinations of points: weighted sums with numerical coefficients summing to 1, resulting in another point. These coefficients define abarycentric coordinate system for the flat through the points.
Anyvector space may be viewed as an affine space; this amounts to "forgetting" the special role played by thezero vector. In this case, elements of the vector space may be viewed either aspoints of the affine space or asdisplacement vectors ortranslations. When considered as a point, the zero vector is called theorigin. Adding a fixed vector to the elements of alinear subspace (vector subspace) of avector space produces anaffine subspace of the vector space. One commonly says that this affine subspace has been obtained by translating (away from the origin) the linear subspace by the translation vector (the vector added to all the elements of the linear subspace). In finite dimensions, such anaffine subspace is the solution set of aninhomogeneous linear system. The displacement vectors for that affine space are the solutions of the correspondinghomogeneous linear system, which is a linear subspace. Linear subspaces, in contrast, always contain the origin of the vector space.
Thedimension of an affine space is defined as thedimension of the vector space of its translations. An affine space of dimension one is anaffine line. An affine space of dimension 2 is anaffine plane. An affine subspace of dimensionn – 1 in an affine space or a vector space of dimensionn is anaffine hyperplane.
The followingcharacterization may be easier to understand than the usual formal definition: an affine space is what is left of avector space after one has forgotten which point is the origin (or, in the words of the French mathematicianMarcel Berger, "An affine space is nothing more than a vector space whose origin we try to forget about, by addingtranslations to the linear maps"[2]). Imagine that Alice knows that a certain point is the actual origin, but Bob believes that another point—call itp—is the origin. Two vectors,a andb, are to be added. Bob draws an arrow from pointp to pointa and another arrow from pointp to pointb, and completes the parallelogram to find what Bob thinks isa +b, but Alice knows that he has actually computed
Similarly,Alice and Bob may evaluate anylinear combination ofa andb, 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 coefficientsλ + (1 −λ) = 1, Alice and Bob describe the same point with the same linear combination, despite using different origins.
While only Alice knows the "linear structure", both Alice and Bob know the "affine structure"—i.e. the values ofaffine combinations, defined as linear combinations in which the sum of the coefficients is 1. A set with an affine structure is an affine space.
While affine space can be defined axiomatically (see§ Axioms below), analogously to the definition of Euclidean space implied byEuclid'sElements, for convenience most modern sources define affine spaces in terms of the well developed vector space theory.
Anaffine space is a setA together with avector space, and a transitive and freeaction of theadditive group of on the setA.[3] The elements of the affine spaceA are calledpoints. The vector space is said to beassociated to the affine space, and its elements are calledvectors,translations, or sometimesfree vectors.
Explicitly, the definition above means that the action is a mapping, generally denoted as an addition,
that has the following properties.[4][5][6]
The first two properties are simply defining properties of a (right) group action. The third property characterizes free and transitive actions, theonto character coming from transitivity, and then theinjective character follows from the action being free. There is a fourth property that follows from 1, 2 above:
Property 3 is often used in the following equivalent form (the 5th property).
Another way to express the definition is that an affine space is aprincipal homogeneous space for the action of theadditive 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.
The properties of the group action allows for the definition of subtraction for any given ordered pair(b,a) of points inA, producing a vector of. This vector, denoted or, is defined to be the unique vector in such that
Existence follows from the transitivity of the action, and uniqueness follows because the action is free.
This subtraction has the two following properties, calledWeyl's axioms:[7]
Theparallelogram property is satisfied in affine spaces, where it is expressed as: given four points the equalities and are equivalent. This results from the second Weyl's axiom, since
Affine spaces can be equivalently defined as a point setA, together with a vector space, and a subtraction satisfying Weyl's axioms. In this case, the addition of a vector to a point is defined from the first of Weyl's axioms.
Anaffine subspace (also called, in some contexts, alinear variety, aflat, or, over thereal numbers, alinear manifold)B of an affine spaceA is asubset ofA for which there exists a point such that the set of vectors is alinear subspace of. If is an affine subspace then the set is a linear subspace for all (that is, the choice of the point is irrelevant). An affine subspaceB is an affine space which has as its associated vector space.
The affine subspaces ofA are the subsets ofA of the form
wherea is a point ofA, andV a linear subspace of.
The linear subspace associated with an affine subspace is often called itsdirection, and two subspaces that share the same direction are said to beparallel.
This implies the following generalization ofPlayfair's axiom: Given a directionV, for any pointa ofA there is one and only one affine subspace of directionV, which passes througha, namely the subspacea +V.
Every translation maps any affine subspace to a parallel subspace.
The termparallel is also used for two affine subspaces such that the direction of one is included in the direction of the other.
Given two affine spacesA andB whose associated vector spaces are and, anaffine map oraffine homomorphism fromA toB is a map
such that
is awell defined linear map. By being well defined is meant thatb –a =d –c impliesf(b) –f(a) =f(d) –f(c).
This implies that, for a point and a vector, one has
Therefore, since for any givenb inA,b =a +v for a uniquev,f is completely defined by its value on a single point and the associated linear map.
Anaffine transformation orendomorphism of an affine space is an affine map from that space to itself. One importantfamily of examples is the translations: given a vector, the translation map that sends for every in is an affine map. Another important family of examples are the linear maps centred at an origin: given a point and a linear map, one may define an affine map byfor every in.
After making a choice of origin, any affine map may be written uniquely as a combination of a translation and a linear map centred at.
Every vector spaceV may be considered as an affine space over itself. This means that every element ofV may be considered either as a point or as a vector. This affine space is sometimes denoted(V,V) for emphasizing the double role of the elements ofV. When considered as a point, thezero vector is commonly denotedo (orO, when upper-case letters are used for points) and called theorigin.
IfA is another affine space over the same vector space (that is) the choice of any pointa inA defines a unique affine isomorphism, which is the identity ofV and mapsa too. In other words, the choice of an origina inA allows us to identifyA and(V,V)up to acanonical isomorphism. The counterpart of this property is that the affine spaceA may be identified with the vector spaceV in which "the place of the origin has been forgotten".
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 realinner product space of finite dimension, that is a vector space over the reals with apositive-definite quadratic formq(x). The inner product of two vectorsx andy is the value of thesymmetric bilinear form
The usualEuclidean distance between two pointsA andB is
In older definition of Euclidean spaces throughsynthetic geometry, vectors are defined asequivalence classes ofordered pairs of points underequipollence (the pairs(A,B) and(C,D) areequipollent if the pointsA,B,D,C (in this order) form aparallelogram). It is straightforward to verify that the vectors form a vector space, the square of theEuclidean distance is a quadratic form on the space of vectors, and the two definitions of Euclidean spaces are equivalent.
InEuclidean geometry, the common phrase "affine property" refers to a property that can be proved in affine spaces, that is, it can be proved without using the quadratic form and its associated inner product. In other words, an affine property is a property that does not involve lengths and angles. Typical examples areparallelism, and the definition of atangent. A non-example is the definition of anormal.
Equivalently, an affine property is a property that is invariant underaffine transformations of the Euclidean space.
Leta1, ...,an be a collection ofn points in an affine space, and ben elements of theground field.
Suppose that. For any two pointso ando' one has
Thus, this sum is independent of the choice of the origin, and the resulting vector may be denoted
When, one retrieves the definition of the subtraction of points.
Now suppose instead that thefield elements satisfy. For some choice of an origino, denote by the unique point such that
One can show that is independent from the choice ofo. Therefore, if
one may write
The point is called thebarycenter of the for the weights. One says also that is anaffine combination of the withcoefficients.
For any non-empty subsetX of an affine spaceA, there is a smallest affine subspace that contains it, called theaffine span ofX. It is the intersection of all affine subspaces containingX, and its direction is the intersection of the directions of the affine subspaces that containX.
The affine span ofX is the set of all (finite) affine combinations of points ofX, and its direction is thelinear span of thex −y forx andy inX. If one chooses a particular pointx0, the direction of the affine span ofX is also the linear span of thex –x0 forx inX.
One says also that the affine span ofX isgenerated byX and thatX is agenerating set of its affine span.
A setX of points of an affine space is said to beaffinely independent or, simply,independent, if the affine span of anystrict subset ofX is a strict subset of the affine span ofX. Anaffine basis orbarycentric frame (see§ Barycentric coordinates, below) of an affine space is a generating set that is also independent (that is aminimal generating set).
Recall that thedimension of an affine space is the dimension of its associated vector space. The bases of an affine space of finite dimensionn are the independent subsets ofn + 1 elements, or, equivalently, the generating subsets ofn + 1 elements. Equivalently,{x0, ...,xn} is an affine basis of an affine spaceif and only if{x1 −x0, ...,xn −x0} is alinear basis of the associated vector space.
There are two strongly related kinds ofcoordinate systems that may be defined on affine spaces.
LetA be an affine space of dimensionn over afieldk, and be an affine basis ofA. The properties of an affine basis imply that for everyx inA there is a unique(n + 1)-tuple of elements ofk such that
and
The are called thebarycentric coordinates ofx over the affine basis. If thexi are viewed as bodies that have weights (or masses), the pointx is thus thebarycenter of thexi, and this explains the origin of the termbarycentric coordinates.
The barycentric coordinates define an affine isomorphism between the affine spaceA and the affine subspace ofkn + 1 defined by the equation.
For affine spaces of infinite dimension, the same definition applies, using only finite sums. This means that for each point, only a finite number of coordinates are non-zero.
Anaffine frame is acoordinate frame of an affine space, consisting of a point, called theorigin, and alinear basis of the associated vector space. More precisely, for an affine spaceA with associated vector space, the origino belongs toA, and the linear basis is a basis(v1, ...,vn) of (for simplicity of the notation, we consider only the case of finite dimension, the general case is similar).
For each pointp ofA, there is a unique sequence of elements of the ground field such that
or equivalently
The are called theaffine coordinates ofp over the affine frame(o,v1, ...,vn).Anaffine coordinate system is acoordinate system on an affine space where each coordinate is anaffine map to thenumber line. In other words, it is aninjective affine map from an affine space A to thecoordinate spaceKn, whereK is thefield ofscalars, for example, thereal numbers R.
A system ofn coordinates onn-dimensional space is defined by a (n+1)-tuple(O, R1, … Rn) ofpoints not belonging to anyaffine subspace of a lesser dimension. Any given coordinaten-tuple gives the point by the formula:
Note thatRj − O aredifferencevectors with the origin in O and ends in Rj .
An affine space cannot have a coordinate system withn less than itsdimension, butn may indeed be greater, which means that the coordinate map is not necessary surjective.Examples ofn-coordinate system in an (n−1)-dimensional space arebarycentric coordinates and affine "homogeneous" coordinates (1, x1, … , xn−1). In the latter case thex0 coordinate is equal to 1 on all space, but this "reserved" coordinate allows formatrix representation ofaffine maps similar to one used forprojective maps.
The most important case of affine coordinates inEuclidean spaces is the real-valuedCartesian coordinate system, which areorthogonal affine coordinate systems, while others are referred to asoblique affine coordinate systems.In other words,Cartesian coordinates are affine coordinates relative to anorthonormal frame, that is an affine frame(o,v1, ...,vn) such that(v1, ...,vn) is anorthonormal basis. However, general affine coordinate axes are not necessarily orthogonal straight lines.
Barycentric coordinates and affine coordinates are strongly related, and may be considered as equivalent.
In fact, given a barycentric frame
one deduces immediately the affine frame
and, if
are the barycentric coordinates of a point over the barycentric frame, then the affine coordinates of the same point over the affine frame are
Conversely, if
is an affine frame, then
is a barycentric frame. If
are the affine coordinates of a point over the affine frame, then its barycentric coordinates over the barycentric frame are
Therefore, barycentric and affine coordinates are almost equivalent. In most applications, affine coordinates are preferred, as involving less coordinates that are independent. However, in the situations where the important points of the studied problem are affinely independent, barycentric coordinates may lead to simpler computation, as in the following example.
The vertices of a non-flattriangle form an affine basis of theEuclidean plane. The barycentric coordinates allows easy characterization of the elements of the triangle that do not involve angles or distances:
The vertices are the points of barycentric coordinates(1, 0, 0),(0, 1, 0) and(0, 0, 1). The lines supporting theedges are the points that have a zero coordinate. The edges themselves are the points that have one zero coordinate and two nonnegative coordinates. Theinterior of the triangle are the points whose coordinates are all positive. Themedians are the points that have two equal coordinates, and thecentroid is the point of coordinates(1/3,1/3,1/3).
Barycentric coordinates are readily changed from one basis to another. Let and be affine bases ofA. For everyx inA there is some tuple for which
Similarly, for every from the first basis, we now have in the second basis
for some tuple. Now we can rewrite our expression in the first basis as one in the second with
giving us coordinates in the second basis as the tuple.
Affine coordinates are also readily changed from one basis to another. Let, and, be affine frames ofA. For each pointp ofA, there is a unique sequence of elements of the ground field such that
and similarly, for every from the first basis, we now have in the second basis
for tuple and tuples. Now we can rewrite our expression in the first basis as one in the second with
giving us coordinates in the second basis as the tuple.
An affine transformation is executed on a projective space of, by a 4 by 4 matrix with a special[8] fourth column:
The transformation is affine instead of linear due to the inclusion of point, the transformed output of which reveals the affine shift.
Let
be an affine homomorphism, with
its associated linear map. Theimage off is the affine subspace ofF, which has as associated vector space. As an affine space does not have azero element, an affine homomorphism does not have akernel. However, the linear map does, and if we denote by its kernel, then for any pointx of, theinverse image ofx is an affine subspace ofE whose direction is. This affine subspace is called thefiber ofx.
An important example is the projection parallel to some direction onto an affine subspace. The importance of this example lies in the fact thatEuclidean spaces are affine spaces, and that these kinds of projections are fundamental inEuclidean geometry.
More precisely, given an affine spaceE with associated vector space, letF be an affine subspace of direction, andD be acomplementary subspace of in (this means that every vector of may be decomposed in a unique way as the sum of an element of and an element ofD). For every pointx ofE, itsprojection toF parallel toD is the unique pointp(x) inF such that
This is an affine homomorphism whose associated linear map is defined by
forx andy inE.
The image of this projection isF, and its fibers are the subspaces of directionD.
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.
LetE be an affine space, andD be alinear subspace of the associated vector space. ThequotientE/D ofE byD is thequotient ofE by theequivalence relation such thatx andy are equivalent if
This quotient is an affine space, which has as associated vector space.
For every affine homomorphism, the image is isomorphic to the quotient ofE by the kernel of the associated linear map. This is thefirst isomorphism theorem for affine spaces.
Affine spaces are usually studied byanalytic geometry using coordinates, or equivalently vector spaces. They can also be studied assynthetic geometry by writing down axioms, though this approach is much less common. There are several different systems of axioms for affine space.
Coxeter (1969, p. 192) axiomatizes the special case ofaffine geometry over the reals asordered geometry together with an affine form ofDesargues's theorem and an axiom stating that in a plane there is at most one line through a given point not meeting a given line.
Affine planes satisfy the following axioms (Cameron 1991, chapter 2):(in which two lines are called parallel if they are equal ordisjoint):
As well as affine planes over fields (ordivision rings), there are also manynon-Desarguesian planes satisfying these axioms.Cameron (1991, chapter 3) gives axioms for higher-dimensional affine spaces.
Purely axiomatic affine geometry is more general than affine spaces and is treated in the articleAffine geometry.
Affine spaces are contained inprojective spaces. For example, an affine plane can be obtained from anyprojective plane by removing one line and all the points on it, and conversely any affine plane can be used to construct a projective plane as aclosure by adding aline at infinity whose points correspond to equivalence classes ofparallel lines. Similar constructions hold in higher dimensions.
Further, transformations of projective space that preserve affine space (equivalently, that leave thehyperplane at infinityinvariant as a set) yield transformations of affine space. Conversely, any affine linear transformation extends uniquely to aprojective linear transformation, so theaffine group is asubgroup of theprojective group. For instance,Möbius transformations (transformations of thecomplex projective line, orRiemann sphere) are affine (transformations of thecomplex plane) if and only if they fix thepoint at infinity.
Inalgebraic geometry, anaffine variety (or, more generally, anaffine algebraic set) is defined as the subset of an affine space that is the set of the common zeros of a set of so-calledpolynomial functions over the affine space. For defining apolynomial function over the affine space, one has to choose anaffine frame. Then, a polynomial function is a function such that the image of any point is the value of some multivariatepolynomial function of the coordinates of the point. As a change of affine coordinates may be expressed bylinear functions (more precisely affine functions) of the coordinates, this definition is independent of a particular choice of coordinates.
The choice of a system of affine coordinates for an affine space of dimensionn over afieldk induces an affineisomorphism between and the affinecoordinate spacekn. This explains why, for simplification, many textbooks write, and introduce affinealgebraic varieties as the common zeros of polynomial functions overkn.[9]
As the whole affine space is the set of the common zeros of thezero polynomial, affine spaces are affine algebraic varieties.
By the definition above, the choice of an affine frame of an affine space allows one to identify the polynomial functions on with polynomials inn variables, theith variable representing the function that maps a point to itsith coordinate. It follows that the set of polynomial functions over is ak-algebra, denoted, which is isomorphic to thepolynomial ring.
When one changes coordinates, the isomorphism between and changes accordingly, and this induces an automorphism of, which maps each indeterminate to a polynomial of degree one. It follows that thetotal degree defines afiltration of, which is independent from the choice of coordinates. The total degree defines also agraduation, but it depends on the choice of coordinates, as a change of affine coordinates may map indeterminates on non-homogeneous polynomials.
Affine spaces overtopological fields, such as the real or the complex numbers, have a naturaltopology. 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 whoseclosed sets areaffine 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 iscoarser than the natural topology.
There is a natural injective function from an affine space into the set ofprime ideals (that is thespectrum) of its ring of polynomial functions. When affine coordinates have been chosen, this function maps the point of coordinates to themaximal ideal. This function is ahomeomorphism (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 analgebraically 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 isHilbert's Nullstellensatz).
This is the starting idea ofscheme theory ofGrothendieck, 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 allowsgluing together algebraic varieties in a similar way as, formanifolds,charts are glued together for building a manifold.
Like all affine varieties, local data on an affine space can always be patched together globally: thecohomology of affine space is trivial. More precisely, for allcoherent sheavesF, andintegers. This property is also enjoyed by all otheraffine varieties (seeSerre's theorem on affineness). But also all of theétale cohomology groups on affine space are trivial. In particular, everyline bundle is trivial. More generally, theQuillen–Suslin theorem implies thatevery algebraicvector bundle over an affine space is trivial.
