affine geometry

Affine Spaces

Let V be a left (right) vector spaceMathworldPlanetmath over a division ring D. An affine spacePlanetmathPlanetmath associated with V is a set A, together with a function f:A×AV, with the following conditions:

  1. 1.

    for every PA, the function f(P,-):AV is onto.

  2. 2.

    f is non-degenerate in the sense that f(P,Q)=0 implies P=Q.

  3. 3.


An affine space is an affine space associated with some vector space. Elements of A are called points of A. The dimensionMathworldPlanetmathPlanetmathPlanetmath of an affine space is just the dimension of its associated vector space. The function f is called a direction. We sometimes write PQ for f(P,Q).

For example, V itself is an affine space associated with V, with f given by f(v,w)=w-v.

Below are some properties of f:

  1. 1.

    f(P,P)=0, because f(P,P)+f(P,P)=f(P,P) by condition 3, hence the result.

  2. 2.

    f(P,-) is one-to-one, and therefore a bijectionMathworldPlanetmath, for if f(P,Q)=f(P,R), then f(P,Q)+f(Q,R)=f(P,R)=f(P,Q), so f(Q,R)=0, or Q=R, by condition 2.

  3. 3.

    f(P,Q)=-f(Q,P), because f(P,Q)+f(Q,P)=f(P,P)=0.

  4. 4.

    f(-,Q) is also a bijection as a result.

Vector Spaces Induced by an Affine Space

An affine space is often called a “vector space without the origin”. In other words, singling out a point in an affine space gives us a vector space, in the following sense: fix a point P in an affine space A, define a vector space AP as follows:

  1. 1.

    vectors of AP are points of A,

  2. 2.

    define +P:AP×APAP by Q+PR:=S, where S is the point determined by f(P,Q)+f(P,R)=f(P,S). Because of property 2 above, S is uniquely determined.

  3. 3.

    define P:D×APAP by dPQ:=T, where T is the point determined by df(P,Q)=f(P,T). Again, T is unique.

Both +P and P are well-defined, because f(P,-) is a bijection. When there is no confusion, we may drop the subscript P. It is also easy to verify that AP, together with + and , is indeed a left vector space over D, with P as the origin, written 0P. Furthermore, AP is isomorphicPlanetmathPlanetmath to V. Hence, APAQ for any two points P,QA; there is nothing special about P, and any point of A can be used as the origin of a vector space.

Affine Subspaces

Continue to assume that V is a left vector space over a division ring D, and (A,f) an affine space associated with V. An affine subspace of A is the collectionMathworldPlanetmath B of points of A that is mapped to a vector subspace S of V by the induced function f(P,-) for some point PA. In other words, B is the inverse image of S under the function f(P,-):


If f is restricted to B×B, then B is an affine space associated with W, since f(Q,R)=f(P,R)-f(P,Q)W, given that Q,RB.

For example, if V is considered as an affine space associated with V with the map f(v,w)=w-v, then an affine subspace B of V is just a coset of a subspaceMathworldPlanetmathPlanetmath of V. In other words, B=S+v, where S is a subspace of V and vV is a vector. It is evident that B is uniquely determined by S, and v up to translationMathworldPlanetmathPlanetmath by a vector in S. In other words, any two cosets of S are affinely isomorphic.

In an affine space A, an affine point, affine line, or affine planeMathworldPlanetmath is a 0,1, or 2 dimensional affine subspace. Thus, an affine point is just the inverse image of the origin 0V. The codimension of an affine subspace is the codimension of the associated vector subspace. An affine hyperplane is an affine subspace with codimension 1. When there is no confusion, we may drop the word “affine” in affine point, affine line, etc… Affine subspaces are also called flats.

Affine Geometry

Affine geometry is, generally speaking, the study the geometric properties of affine subspaces. In particular, it is the study of the incidence structure on affine subspaces. Operationally, we may define an affine geometry 𝒜(V) of a vector space V to be the poset of all affine subspaces of V, orderd by set theoretic inclusion. Points in 𝒜(V) are commonly written without the set theoretic brackets, so that v𝒜(V) means {v}𝒜(V).

Next, we can define an incidence relation I on 𝒜(V) so that (A,B)I iff AB or BA. Together with I, 𝒜(V) becomes an incidence geometry. Two flats A and B are said to be parallelMathworldPlanetmathPlanetmathPlanetmath if they have the same associated subspace. As a result, two parallel flats are never incidentMathworldPlanetmath unless they are equal. Also, given a point v𝒜(V) not incident with A, we can always find a flat B incident with v and parallel to A. If A=S+w with wv, simply take B=S+v. This makes 𝒜(V) an affine incidence geometry.

In addition, we define AB to be the smallest flat in 𝒜(V) that contains both A and B. By Zorn’s lemma, AB exists. Since AB is also unique, is well-defined. This turns 𝒜(V) into an upper semilatticePlanetmathPlanetmath. If S1 is the associated subspace of A and S2 is the associated subspace of B, then span(S1S2) is the associated subspace of AB. The definition of can be extended to an arbitrary set of flats, so that 𝒮 is the smallest flat that contains all flats in 𝒮𝒜. In fact, it is not hard to see that 𝒜(V) is complete semilattice.

However, since AB may be empty, 𝒜(V) is not a latticeMathworldPlanetmath in general via the “meet” (=) operationMathworldPlanetmath. But when AB, AB𝒜(V). So is a partially defined operator on 𝒜(V). If one adjoins the empty setMathworldPlanetmath to 𝒜(V), then 𝒜(V) becomes a lattice. is called the null subspace and its dimension is defined to be -1. One can show that 𝒜(V) is a geometric lattice.

Although 0𝒜(V), it is not special, since all points are treated equally; there is no notion of an origin in 𝒜(V). The notion of a metric is also absent, since the underlying vector space is not assumed to have an inner product. In fact, perpendicularity is not defined in 𝒜(V). In contrast, both notions are important in Euclidean geometryMathworldPlanetmath, where an inner product has been defined, so that 0 is the unique vector with 0 length.

Affine versus Projective

Affine geometry and projective geometryMathworldPlanetmath are intimately related. Given an affine geometry 𝒜(V) one can construct projective geometries. One easy way is to identify flats that are parallel to each other. Because the parallel relationMathworldPlanetmath is an equivalence relationMathworldPlanetmath, we can partitionMathworldPlanetmathPlanetmath 𝒜(V) into equivalence classesMathworldPlanetmath. Since each equivalence class is represented by exactly one subspace S of V, so 𝒜(V)/ can be identified with PG(V). Of course, PG(V) can also be viewed as a sub-poset of 𝒜(V) (simply by taking all the subspaces of V in 𝒜(V)). More generally, if we fix any point v𝒜(V), and take all flats that are incident with v, the resulting subset PGv(V) forms a modular complemented geometric lattice that is isomorphic to PG(V). In fact, PGv(V) has the structureMathworldPlanetmath of a projective geometry.

Another way to construct a projective geometry from an affine one is to adjoin extra elements to 𝒜(V). Remember that 𝒜(V) itself is not a lattice, but simply adjoining to 𝒜(V) won’t give us a projective geometry either, because the resulting lattice is not modular (take two parallel lines 1,2 and a point P lying on 1; then P(21)=P, while (P2)1=1). We start by taking a vector space U such that V is a subspace of U of codimension 1 (This can be done by linear algebra). Our objective is to show that 𝒜(V) is embeddable in PG(U).

Let uU be a non-zero vector and look at the affine hyperplane V+u. Each affine subspace of the form S+v in V has the form S+v+u in V+u, where S is a subspace of V and vV. Let 𝒜u(V) the collection of all affine subspaces S+v+u (affine subspaces of U that are incident with V+u). There is an obvious one-to-one order preserving correspondence between 𝒜(V) and 𝒜u(V).

Next, every affine subspace S+v+u is the intersectionMathworldPlanetmath of V+u and a subspace W of U such that S+v+uW and dim(W)=S+1. Just take W=span(S,v+u). Clearly S+v+uWV+u. In addition, v+uSV, or else uV gives us a contradictionMathworldPlanetmathPlanetmath. So dim(W)=S+1. Finally, if xV+uW, then x=y+u=s+k(v+u), where yV, sS, and kD. So xuku(modV). This implies (1-k)uV. But uV, k=1. Therefore x=s+u+vS+v+u. This means that WV+u=S+v+u.

The above paragraph shows there is a one-to-one order preserving map from 𝒜(V) to PG(U). If we delete all subspaces of V from PG(U), and call

PG(U)/PG(V)={WPG(U)W is not a subspace of V},

then we actually get an order-preserving bijection between 𝒜(V) and PG(U)/PG(V).

Title affine geometry
Canonical name AffineGeometry
Date of creation 2013-03-22 15:58:20
Last modified on 2013-03-22 15:58:20
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 21
Author CWoo (3771)
Entry type Definition
Classification msc 51A45
Classification msc 51A15
Synonym associated subspace
Related topic AffineCombination
Related topic AffineTransformation
Defines affine space
Defines flat
Defines associated linear subspace
Defines affine subspace
Defines dimension of an affine space
Defines affine transformation
Defines affine hyperplane
Defines affine line
Defines affine point