Affine space

From Wikipedia, the free encyclopedia
  (Redirected from Affine line)
Jump to: navigation, search
Not to be confused with affinity space.
Line segments on a two-dimensional affine space

In mathematics, an affine space is a geometric structure that generalizes certain properties of parallel lines in Euclidean space. 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 vectors between two points of the space. Thus it makes sense to subtract two points of the space, giving a vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a vector to a point of an affine space, resulting in a new point displaced from the starting point by that vector.

The simplest example of an affine space is a linear subspace of a vector space that has been translated away from the origin. In finite dimensions, such an affine subspace corresponds to the solution set of an inhomogeneous linear system. The displacement vectors for that affine space live in the solution set of the corresponding homogeneous linear system, which is a linear subspace. Linear subspaces, in contrast, always contain the origin of the vector space.

Informal descriptions[edit]

The following characterization may be easier to understand than the usual formal 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 the French mathematician Marcel Berger, "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps"[1]). Imagine that Alice knows that a certain point is the actual origin, but Bob believes that another point — call it p — is the origin. Two vectors, a and b, are to be added. Bob draws an arrow from point p to point a and another arrow from point p to point b, and completes the parallelogram to find what Bob thinks is a + b, but Alice knows that he has actually computed

p + (ap) + (bp).

Similarly, Alice and Bob may evaluate any linear combination of a and b, 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 Bob travels to

λa + (1 − λ)b

then Alice can similarly travel to

p + λ(ap) + (1 − λ)(bp) = λa + (1 − λ)b.

Then, for all coefficients λ + (1 − λ) = 1, Alice and Bob describe the same point with the same linear combination, starting from different origins.

While Alice knows the "linear structure", both Alice and Bob 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.

Definition[edit]

An affine space[2] is a set A together with a vector space V over a field F and a faithful and transitive group action of V (with addition of vectors as group action) on A. (That is, an affine space is a principal homogeneous space for the action of V.)

Explicitly, an affine space is a point set A together with a map

l\colon V \times A \to A,\; (\mathbf{v}, a) \mapsto \mathbf{v} + a

with the following properties:.[3][4][5]

  1. Left identity
    \forall a \in A,\; \mathbf{0} + a = a
  2. Associativity
    \forall \mathbf{v}, \mathbf{w} \in V, \forall a \in A,\; \mathbf{v} + (\mathbf{w} + a) = (\mathbf{v} + \mathbf{w}) + a
  3. Uniqueness
    \forall a \in A,\; V \to A\colon \mathbf{v} \mapsto \mathbf{v} + a\quad is a bijection.

(Since the group V is abelian, there is no difference between its left and right actions, so it is also permissible to place vectors on the right.)

By choosing an origin, o, one can thus identify A with V, hence turn A into a vector space. Conversely, any vector space, V, is an affine space over itself.

Subtraction and Weyl's axioms[edit]

The uniqueness property ensures that subtraction of any two elements of A is well defined, producing a vector of V:

 a \,-\, b\; is the unique vector in V such that  \left(a \,-\, b\right) \,+\, b \;=\; a.

One can equivalently define an affine space as a point set A, together with a vector space V, and a subtraction map

\operatorname{\phi}:\; A \,\times\, A \;\to\; V,\; \left(a,\, b\right) \,\mapsto\, b \,-\, a \;\equiv\; \overrightarrow{ab}

with the following properties:[6]

  1.  \forall p \,\in\, A,\; \forall \mathbf{v}\,\in\, V there is a unique point  q \,\in\, A such that  q \,-\, p \;=\; \mathbf{v} and
  2.  \forall p,\, q,\, r \,\in\, A,\; (q \,-\, p) \,+\, (r \,-\, q) \;=\; r \,-\, p.

These two properties are called Weyl's axioms.

Affine combinations[edit]

For any choice of origin o, and two points a, b in A and scalar λ, there is a unique element of A, denoted by \lambda a + (1-\lambda)b such that

(\lambda a + (1 - \lambda)b) - o = \lambda (a - o) + (1 - \lambda)(b - o).

This element can be shown to be independent of the choice of origin o. Instead of arbitrary linear combinations, only such affine combinations of points have meaning.

Examples[edit]

  • When children find the answers to sums such as 4 + 3 or 4 − 2 by counting right or left on a number line, they are treating the number line as a one-dimensional affine space.
  • Any coset of a subspace V of a vector space is an affine space over that subspace.
  • If T is a matrix and b lies in its column space, the set of solutions of the equation Tx = b is an affine space over the subspace of solutions of Tx = 0.
  • 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 T : VW is a linear mapping and y lies in its image, the set of solutions xV to the equation Tx = y is a coset of the kernel of T , and is therefore an affine space over Ker T.

Affine subspaces[edit]

An affine subspace (sometimes called a linear manifold, linear variety, or a flat) of a vector space V is a subset closed under affine combinations of vectors in the space. For example, the set

A=\Bigl\{\sum^N_i \alpha_i \mathbf{v}_i \Big| \sum^N_i\alpha_i=1\Bigr\}

is an affine space, where \scriptstyle \{ \mathbf{v}_i \}_{i\in I} is a family of vectors in V; this space is the affine span of these points. 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=\Bigl\{\sum^N_i \beta_i\mathbf{v}_i \Big| \sum^N_i \beta_i=0\Bigr\}.

This affine subspace can be equivalently described as the coset of the W-action

S=\mathbf{p}+W,\,

where p is any element of A, or equivalently as any level set of the quotient map VV/W. A choice of p gives a base point of A and an identification of W with A, but there is no natural choice, nor a natural identification of W with A.

A linear transformation is a function that preserves all linear combinations; an affine transformation is a function that preserves all affine combinations. A linear subspace is an affine subspace containing the origin, or, equivalently, a subspace that is closed under linear combinations.

For example, in \scriptstyle {\mathbb R^3}, the origin, lines and planes through the origin and the whole space are linear subspaces, while points, lines and planes in general as well as the whole space are the affine subspaces.

Affine combinations and affine dependence[edit]

Main article: Affine combination

An affine combination is a linear combination in which the sum of the coefficients is 1. Just as members of a set of vectors are linearly independent if none is a linear combination of the others, so also they are affinely independent if none is an affine combination of the others. The set of linear combinations of a set of vectors is their "linear span" and is always a linear subspace; the set of all affine combinations is their "affine span" and is always an affine subspace. For example, the affine span of a set of two points is the line that contains both; the affine span of a set of three non-collinear points is the plane that contains all three.

Vectors

v1, v2, … , vn

are linearly dependent if there exist scalars a1, a2, … , an, not all zero, for which

a1v1 + a2v2 + ⋯ + anvn = 0

 

 

 

 

(1)

Similarly they are affinely dependent if in addition the sum of coefficients is zero:

 \sum_{i=1}^n a_i = 0

a condition that ensures that the combination (1) makes sense as a displacement vector.

Geometric objects as points and vectors[edit]

In an affine space, geometric objects have two different (although related) descriptions on languages of points (elements of A) and vectors (elements of V ). A vector description can specify an object only up to translations.

Geometry Points Vectors
A point One point P none (zero vector space)
A line (1-subspace) Can be specified with two distinct points A non-zero vector up to multiplication to (non-zero) scalars
A line segment Two (independent) points:
P, Q
One vector PQ, or
two dependent (mutually opposite) vectors PQ and QP
A plane (2-subspace) Can be specified with three points not lying on one line A linear 2-subspace,
can be specified with two linearly-independent vectors
A triangle Three (independent) points:
PQR
Three dependent vectors related as
PR = PQ + QR, or
PQ + QR + RP = 0, or
just two independent vectors
A parallelogram Four points: ▱PQRS
of which any three determine the fourth
Two independent vectors:
PQ = SR
PS = QR

Axioms[edit]

Affine space is usually studied as analytic geometry using coordinates, or equivalently vector spaces. It can also be studied as synthetic 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 affine geometry (over the reals) as ordered geometry together with an affine form of Desargues'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 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 rings), there are also many non-Desarguesian planes satisfying these axioms. (Cameron 1991, chapter 3) gives axioms for higher-dimensional affine spaces.

Relation to projective spaces[edit]

An affine space is a subspace of projective space, which is in turn a quotient of a vector space.

Affine spaces are subspaces of projective spaces: an affine plane can be obtained from any projective plane 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 whose points correspond to equivalence classes of parallel lines.

Further, transformations of projective space that preserve affine space (equivalently, that leave the hyperplane at infinity invariant as a set) yield transformations of affine space. Conversely, any affine linear transformation extends uniquely to a projective linear transformations, so the affine group is a subgroup of the projective group. For instance, Möbius transformations (transformations of the complex projective line, or Riemann sphere) are affine (transformations of the complex plane) if and only if they fix the point at infinity.

However, one cannot take the projectivization of an affine space, so projective spaces are not naturally quotients of affine spaces: one can only take the projectivization of a vector space, since the projective space is lines through a given point, and there is no distinguished point in an affine space. If one chooses a base point (as zero), then an affine space becomes a vector space, which one may then projectivize, but this requires a choice.

See also[edit]

Notes[edit]

  1. ^ Berger 1987, p. 32
  2. ^ Berger, Marcel (1984), "Affine spaces", Problems in Geometry, p. 11, ISBN 9780387909714 
  3. ^ Berger 1987, p. 33
  4. ^ Snapper, Ernst; Troyer, Robert J. (1989), Metric Affine Geometry, p. 6 
  5. ^ Tarrida, Agusti R. (2011), "Affine spaces", Affine Maps, Euclidean Motions and Quadrics, pp. 1–2, ISBN 9780857297105 
  6. ^ Nomizu & Sasaki 1994, p. 7

References[edit]