Affine space

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In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point that serves as an origin. One-dimensional affine space is the affine line.

Physical space (in many pre-relativistic conceptions) is not only an affine space, but it also has a metric structure and in particular a conformal structure. In general, an affine space need have neither a preferred metric structure nor conformal structure.

Informal descriptions

The following characterization may be easier to understand than a precise definition: an affine space is what is left of a vector space after you've forgotten which point is the origin (or, in the words of mathematical physicist John Baez, "An affine space is a vector space that's forgotten its origin"). Imagine that Smith knows that a certain point is the true origin, and Jones believes that another point—call it p—is the origin. Two vectors, a and b, are to be added. Jones draws an arrow from p to a and another arrow from p to b, and completes the parallelogram to find what Jones thinks is a + b, but Smith knows that it is actually p + (a − p) + (b − p). Similarly, Jones and Smith may evaluate any linear combination of a and b, or of any finite set of vectors, and will generally get different answers. However—and note this well:

If the sum of the coefficients in a linear combination is 1, then Smith and Jones will agree on the answer!

The proof is a routine exercise. Here is the punch line: Smith knows the "linear structure", but both Smith and Jones know the "affine structure"—i.e. the values of affine combinations, defined as linear combinations in which the sum of the coefficients is 1. An underlying set with an affine structure is an affine space.

Another way of looking at this is that by forgetting about the zero vector, we're remembering the vector's tail and where it is situated. Denoting the position of the vector's tail by O, the process of adding vectors a and b involves taking the displacements of a and b relative to their tails and applying them sequentially starting from the tail, O. The resulting sum is O + (a − O) + (b − O) = a − O + b. The operation that emerges from this is the ternary affine operation a − b + c.

Precise definition

An affine space is a set with a faithful freely transitive vector space action, i.e. a torsor (or principal homogeneous space) for the vector space.

Alternatively an affine space is a set S, together with a vector space V, and a map

Failed to parse (Missing texvc executable; please see math/README to configure.): \Theta : S \times S \to V : (a, b) \mapsto \Theta(a, b).

The image Failed to parse (Missing texvc executable; please see math/README to configure.): \Theta(a, b)

is written as a - b and can be thought of as the vector from b to a.

The map has the properties that:

1. for every b in S the map

Failed to parse (Missing texvc executable; please see math/README to configure.): \Theta_b : S \to V : a \mapsto a - b\,

is a bijection, and

2. for every a, b and c in S we have

Failed to parse (Missing texvc executable; please see math/README to configure.): (a-b) + (b-c) = a-c.\,

Consequences

We can define addition of vectors and points as follows

Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi : S \times V \to S : (a, v) \mapsto a + v := \Theta_a^{-1}v.

By choosing an origin a we can thus identify S with V, hence change S into a vector space.

Conversely, any vector space V is an affine space for vector subtraction.

If O, a and b are points in S and Failed to parse (Missing texvc executable; please see math/README to configure.): \ell

is a real number, then 

Failed to parse (Missing texvc executable; please see math/README to configure.): \oplus_O : S^2 \to S : (a, b) \mapsto a \oplus_O b := O+\ell(a-O)+(1-\ell)(b-O)\,

is independent of O. Instead of arbitrary linear combinations, only such affine combinations of points have meaning.

Affine subspaces

An affine subspace of a vector space V is a subset closed under affine combinations of vectors in the space. For example, the set

Failed to parse (Missing texvc executable; please see math/README to configure.): S=\left \{\left. \sum^N_i \alpha_i \mathbf{v}_i \right\vert \sum^N_i\alpha_i=1\right\}

is an affine space, where {v_i}_i is a family of vectors in V. To see that this is indeed an affine space, observe that this set carries a transitive action of the vector subspace W of V

Failed to parse (Missing texvc executable; please see math/README to configure.): W=\left\{\left. \sum^N_i \beta_i\mathbf{v}_i \right\vert \sum^N_i \beta_i=0\right\}.

This vector subspace, and therefore also the affine subspace, is of dimension N–1. This affine subspace can be equivalently described as the coset of the W-action

Failed to parse (Missing texvc executable; please see math/README to configure.): S=\mathbf{p}+W,\,

where p is any element of S.

One might like to define an affine subspace of an affine space as a set closed under affine combinations. However, affine combinations are only defined in vector spaces; one cannot add points of an affine space. Allowing a slightly more abstract definition, one may define an affine subspace of an affine space as a subset that is invariant under an affine transformation.

In affine geometry there is not only no notion of origin, but neither a notion of length nor of angle.

An affine transformation between two vector spaces is a combination of a linear transformation and a translation. For specifying one the origins are used, but the set of affine transformations does not depend on the origins.

Intrinsic definition of affine spaces

Affine spaces over general fields

Given a field Failed to parse (Missing texvc executable; please see math/README to configure.): F

other than the trivial field {0,1} and the field (0,1,2}, 

an affine space Failed to parse (Missing texvc executable; please see math/README to configure.): S

may be viewed as an algebra with a ternary operation Failed to parse (Missing texvc executable; please see math/README to configure.): [\ ]: S\times F\times S \to S
(it may be intuitively thought of as Failed to parse (Missing texvc executable; please see math/README to configure.): (1-r)A + rB

) such that

• (1) Failed to parse (Missing texvc executable; please see math/README to configure.): [A, 0, B] = A

• (2) Failed to parse (Missing texvc executable; please see math/README to configure.): [A, 1, B] = B

• (3) Failed to parse (Missing texvc executable; please see math/README to configure.): [A, rt(1-t),[B, s, C]] = [[A, rt(1-s), B], t, [A,rs(1-t),C]]

.

The case of the field {0,1} may be dealt with separately, and the resolution for the case of the field {0,1,2} may be captured by modifying identity (3) to

• (3') Failed to parse (Missing texvc executable; please see math/README to configure.): [[A,rt(1-s),B],t,[D,rs(1-t),C]] = [[A,x,D],rt(1-t),[B,s,C]]

where Failed to parse (Missing texvc executable; please see math/README to configure.): x

is a solution to Failed to parse (Missing texvc executable; please see math/README to configure.): xt(1 - rs(1-t)) = 1 - rt(1-t)

, and Failed to parse (Missing texvc executable; please see math/README to configure.): rt(1-t)

is assumed not to be 1.

The following properties may be derived

• (4) Failed to parse (Missing texvc executable; please see math/README to configure.): [A,t,A] = A

• (5) Failed to parse (Missing texvc executable; please see math/README to configure.): [A,m,B] = [B,1-m,A]

• (6) Failed to parse (Missing texvc executable; please see math/README to configure.): [A,m,[A,n,C]] = [A,mn,C]

• (7) Failed to parse (Missing texvc executable; please see math/README to configure.): [A,m,[B,n,C]] = [[A,m,B],n,[A,m,C]]

• (8) Failed to parse (Missing texvc executable; please see math/README to configure.): [[A,m,B],t,[A,n,B]] = [A,m(1-t)+nt,B]

• (9) Failed to parse (Missing texvc executable; please see math/README to configure.): [A,t,[B,s,C]] = [[A,t(1-s)/(1-t),B],t,[A,s,C]]

for t other than 1.

• (10) Failed to parse (Missing texvc executable; please see math/README to configure.): [A,m,[B,s,C]] = [C,1-ms,[B,(1-m)/(1-ms),A]]

if ms is not 1.

Property (4) is derived by taking r = 0 in axiom (3) and applying axiom (1).

For (5), the case m = 1 trivial. For m other than 1, we may set r = 1/(1-m), s = 0 and t = 1-m and apply axiom (3),

• Failed to parse (Missing texvc executable; please see math/README to configure.): [A,m,B] = [A,m,[B,0,C]] = [[A,1,B],1-m,[A,0,C]] = [B,1-m,A]

For (6), the case m = 1 is also trivial. In other cases, we may take r=n/(1-m), s = 1, t = m in axiom (3), which leads directly to the result.

For (7), the cases n = 0 or 1 are trivial. In other cases, we may write r = m/(n(1-n)) and s = t = n, in axiom (3) to directly arrive at the result.

For (8), the cases t = 0 or 1 are trivial. Otherwise, if m(1-t)+nt = 0, one use property (5) to rewrite this as Failed to parse (Missing texvc executable; please see math/README to configure.): [[B,1-n,A],1-t,[B,1-m,A]] = [B,(1-n)(1-(1-t))+(1-m)(1-t),A]

and prove property (8) for this, instead. Otherwise, we may take r = ((1-t)m+tn)/(t(1-t)), s = nt/((1-t)m+nt) and derive the result directly from axiom (3).

For (9), take r = 1/(1-t) and apply axiom (3).

For (10), the case s = 0 follows from (5). Otherwise, one may take r = 1/(s(1-ms)) and t = ms and apply axiom (3), and then property (5).

With these properties in hand, we may show that a vector space may be defined by first selecting a point O to designate as the zero vector and then defining the operations

• rA = Failed to parse (Missing texvc executable; please see math/README to configure.): [O,r,A]

• B+C = Failed to parse (Missing texvc executable; please see math/README to configure.): [B/(1-t),t,C/t]

for any t other than 0 or 1

The second operation may then be proven to be independent of t, ultimately using (9).

The properties of a vector space may be derived and one may prove that with these definitions that Failed to parse (Missing texvc executable; please see math/README to configure.): [A,r,B]

reduces to (1-r)A+rB.

From (6), we get Failed to parse (Missing texvc executable; please see math/README to configure.): (rs)A = [O,rs,A] = [O,r,[O,s,A]] = r(sA) .

From (1), we have Failed to parse (Missing texvc executable; please see math/README to configure.): 0A = [O,0,A] = O .

From (2), we have Failed to parse (Missing texvc executable; please see math/README to configure.): 1A = [O,1,A] = A .

From (7), we have Failed to parse (Missing texvc executable; please see math/README to configure.): m[B,n,C] = [mB,n,mC] .

Using these results, we may employ (9) to show that Failed to parse (Missing texvc executable; please see math/README to configure.): [B/(1-t),t,C/t] = 1/(t(1-s)) [t(1-s)B/(1-t),t,(1-s)C] = 1/(t(1-s)) (t [B,s,(1-s)C/s]) = 1/(1-s) [B,s,(1-s)C/s] = [B/(1-s),s,C/s]

for s and t other than 0 and 1.

Commutativity of addition then follows from (5) with Failed to parse (Missing texvc executable; please see math/README to configure.): A+B=[A/t,t,B/(1-t)] = [B/(1-t),1-t,A/t] = B+A .

Multiplication of the O vector yields O since Failed to parse (Missing texvc executable; please see math/README to configure.): rO = [O,r,O] = O , by (4).

The additive identity property then follows with Failed to parse (Missing texvc executable; please see math/README to configure.): O+A=[O/(1-t),t,A/t] = [O,t,A/t] = t(1/t)A = 1A = A .

Distributivity over vector addition follows with Failed to parse (Missing texvc executable; please see math/README to configure.): r(A+B)=r[A/(1-t),t,B/t] = [rA/(1-t),t,rB/t] = rA+rB .

Distributivity over scalar addition follows from (8) with Failed to parse (Missing texvc executable; please see math/README to configure.): rA+sA = [[O,r/(1-t),A],t,[O,s/t,A]] = [O,r+s,A] = (r+s)A .

Associativity follows from (10), with Failed to parse (Missing texvc executable; please see math/README to configure.): A+(B+C) = [A/(1-t),t,[B/(t(1-s)),s,C/(ts)]] = [C/st,1-st,[B/(t(1-s),(1-t)/(1-s),A/(1-t)]] = [C/st,1-st,(A+B)/(1-st)] = C+(A+B) = (A+B)+C . This requires s and t to be chosen such that neither is 0 nor 1 and such that st is not 1.

Finally, the identification of this operator is established with Failed to parse (Missing texvc executable; please see math/README to configure.): (1-r)A+rB = [(1-r)A/(1-r),r,rB/r] = [A,r,B] . The cases r = 0 and r = 1 are handled separately using (0) and (1).

Affine spaces over the 3-element field

The one loose end in the proof of associativity is for the 3-element field, {0,1,2}. The only definition for addition available is Failed to parse (Missing texvc executable; please see math/README to configure.): A+B = 2[A,2,B] . To establish associativity, one needs to show that Failed to parse (Missing texvc executable; please see math/README to configure.): [[A,2,B],2,2C] = [2A,2,[B,2,C]] . A systematic exploration of all the combinations of axiom (3) shows that for the 3-element field, the following identities will hold:

• Failed to parse (Missing texvc executable; please see math/README to configure.): [A,2,A] = A

• Failed to parse (Missing texvc executable; please see math/README to configure.): [A,2,[A,2,B]] = B

• Failed to parse (Missing texvc executable; please see math/README to configure.): [A,2,B] = [B,2,A]

• Failed to parse (Missing texvc executable; please see math/README to configure.): [A,2,[B,2,C]] = [[A,2,B],2,[A,2,C]]

Writing the operation Failed to parse (Missing texvc executable; please see math/README to configure.): [A,2,B]

more compactly as AB, the required properties may be more succinctly stated as

• AA = A, A(AB) = B, AB = BA, A(BC) = (AC)(BC)

from which it is desired to prove that (AB)(CD) = (AC)(BD). Indeed, it is not too difficult to show that the free algebra, defined by these relations, generated by 3 elements {A,B,C} is just {A,B,C,AB,AC,BC,A(BC),B(AC),C(AB)}, which is the 2-dimensional affine space over {0,1,2}. However, the free algebra on 4 elements may not even be finite.

Instead, one may take as the defining postulates

• AA = A, A(AB) = B, AB = BA, (AB)(CD) = (AC)(BD).

From the last property, one proves that A(BC) = (AA)(BC) = (AB)(AC).

Affine spaces over the 2-element field and "affine groups"

The other loose end is in the definition of addition, which breaks down for the field {0,1}. Since a vector space over {0,1} does not have any non-trivial multiplication by scalars it may be equivalently characterized as an abelian group with A+A = 1A+1A = (1+1)A = 0A = 0. A suitable choice for an operation is the ternary operator ABC = A+B+C, for which one may pose the following properties

• (G1) AAB = B
• (G2) AB(CDE) = (ABC)DE
• (G3) ABC = CBA
• (G4) ABA = B.

Arbitrarily designating an element E as the identity, one may then define group operations by

• AB = AEB, A^{-1} = EAE.

Under (G1) and (G2), these operations define a group, with

• AA^{-1} = AE(EAE) = (AEE)AE = (EEA)AE = AAE = E
• A^{-1}A = (EAE)EA = EA(EEA) = EAA = E
• AE = AEE = EEA = E
• EA = EEA = A
• A(BC) = AE(BEC) = (AEB)EC = (AB)C.

Thus, (G1) and (G2) define what may be considered as the "affine" generalization of a group. With respect to these definitions, it can then be proven that

• AB^{-1}C = AE((EBE)EC) = AE(EB(EEC)) = AE(EBC) = (AEE)BC = (EEA)BC = ABC.

Under (G3), one also has commutativity

• AB = AEB = BEA = BA,

thus defining Abelian groups. Finally, under (G4), one has

• AA = AEA = E

Another characterization

David Kay's description of three dimensional affine space is as follows.^[1]

An affine space is any system of points, lines, and planes which satisfy the following axioms:

AS1. Two distinct points determine a unique line.

AS2. Three noncollinear points determine a unique plane.

AS3. If two points lie in a plane, then the line determined by those points lies in that plane.

AS4. If two planes meet, their intersection is a line.

AS5. There exist at least four noncoplanar points and at least one plane. Each plane contains at least three noncollinear points.

AS6. Given any two noncoplanar lines, there exists a unique plane through the first line which is parallel to the second line

Affine algebras

In universal algebra, an algebra A is called affine^[2] if there exists an abelian group operation + such that δ(x,y,z) = x - y + z is a term function of A, and every basic operation of A is a homomorphism with respect to δ.

References

• Ernst Snapper and Robert J. Troyer, Metric Affine Geometry, Dover Publications; Reprint edition (October 1989)

1. ^ Coxeter, H. S. M. (1961), Introduction to geometry, Wiley, pp. 192
2. ^ R. W. Quackenbush, Quasi-affine algebras, Algebra Universalis 20 (1985), no. 3, pp. 318–327. doi:10.1007/BF01195141

• Affine Space on PlanetMath

See also

• affine geometry
• affine combination
• affine transformation
• affine group
• interval measurement, an affine observable in statisticsca:Espai afí

cs:Afinní prostor de:Affiner Raum es:Espacio afín fr:Espace affine ko:아핀공간 it:Spazio affine ja:アフィン空間 pl:Przestrzeń afiniczna pt:Espaço afim ru:Аффинное пространство uk:Афінний простір