Affine space

Affine space

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Affine space

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
Precise definition
Consequences
Affine subspaces
Intrinsic definition of affine spaces

Affine spaces over general fields
Affine spaces over the 3-element field
Affine spaces over the 2-element field and "affine groups"

Another characterization
Affine algebras
See also
References

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 can most easily be defined in terms of a vector space, as here, but can also be defined intrinsically, without reference to an auxiliary vector space.

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

\Theta : S \times S \to V.

The image \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

\Theta_b : S \to V : a \mapsto a - b\,

is a bijection, and

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

(a-b) + (b-c) = a-c.\,

Consequences

We can define addition of vectors and points as follows

\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 \ell is a real number, then

\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

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

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

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 F other than the trivial field {0,1} and the field {0,1,2}, an affine space S may be viewed as an algebra with a ternary operation [\ ]: S\times F\times S \to S (it may be intuitively thought of as (1-r)A + rB) such that

(1) [A, 0, B] = A
(2) [A, 1, B] = B
(3) [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') [[A,rt(1-s),B],t,[D,rs(1-t),C]] = [[A,x,D],rt(1-t),[B,s,C]]

where x is a solution to xt(1 - rs(1-t)) = 1 - rt(1-t), and rt(1-t) is assumed not to be 1.

The following properties may be derived

(4) [A,t,A] = A
(5) [A,m,B] = [B,1-m,A]
(6) [A,m,[A,n,C]] = [A,mn,C]
(7) [A,m,[B,n,C]] = [[A,m,B],n,[A,m,C]]
(8) [[A,m,B],t,[A,n,B]] = [A,m(1-t)+nt,B]
(9) [A,t,[B,s,C]] = [[A,t(1-s)/(1-t),B],t,[A,s,C]] for t other than 1.
(10) [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),

[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 [[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 = [O,r,A]
B+C = [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 [A,r,B] reduces to (1-r)A+rB.

From (6), we get (rs)A = [O,rs,A] = [O,r,[O,s,A]] = r(sA).

From (1), we have 0A = [O,0,A] = O.

From (2), we have 1A = [O,1,A] = A.

From (7), we have m[B,n,C] = [mB,n,mC].

Using these results, we may employ (9) to show that [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 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 rO = [O,r,O] = O, by (4).

The additive identity property then follows with O+A=[O/(1-t),t,A/t] = [O,t,A/t] = t(1/t)A = 1A = A.

Distributivity over vector addition follows with 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 rA+sA = [[O,r/(1-t),A],t,[O,s/t,A]] = [O,r+s,A] = (r+s)A.

Associativity follows from (10), with 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 (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 A+B = 2[A,2,B]. To establish associativity, one needs to show that [[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:

[A,2,A] = A
[A,2,[A,2,B]] = B
[A,2,B] = [B,2,A]
[A,2,[B,2,C]] = [[A,2,B],2,[A,2,C]]

Writing the operation [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 is a term function of A, and every basic operation of A is a homomorphism with respect to ?.