Affine transformation

In geometry, an affine transformation or affine map (from the Latin, affinis, "connected with") between two vector spaces consists of a linear transformation followed by a translation:

Related Topics:
Geometry - Vector space - Linear transformation - Translation

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:x mapsto A x+ b.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, which can be written as the matrix A with an extra column b. An affine transformation corresponds to multiplication of a matrix and a vector, and composition of affine transformations corresponds to ordinary matrix multiplication if an extra row is added at the bottom of the matrix, containing only zeros, except a 1 at the right:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

egin{bmatrix} A & b \ 0..0 & 1 end{bmatrix}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

while an element 1 is added at the bottom of column vectors:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

egin{bmatrix} x \ 1 end{bmatrix}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

(homogeneous coordinates).

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

An affine transformation is invertible iff A is invertible. The invertible affine transformation form the affine group, which has the general linear group of degree n as subgroup and itself is a subgroup of the general linear group of degree n+1.

Related Topics:
Iff - Affine group - General linear group

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The similarity transformations form the subgroup where A is a scalar times an orthogonal matrix. Iff the determinant of A is 1 or -1 then the transformation preserves area; these also form a subgroup. Combining both conditions we have the isometries, the subgroup of both where A is an orthogonal matrix.

Related Topics:
Similarity transformations - Orthogonal matrix - Isometries

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Each of these groups has a subgroup of transformations which preserve orientation: those where the determinant of A is positive. In the last case this is in 3D the group of rigid body motions (proper rotations and pure translations).

Related Topics:
Orientation - Rigid body - Proper rotation

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

For matrix A the following propositions are equivalent:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• A - I is invertible
• A has not eigenvalue 1
• for all b the transformation has exactly one fixed point
• there is a b for which the transformation has exactly one fixed point
• affine transformations with matrix A can be written as a linear transformation with some point as origin

If there is a fixed point we can take that as the origin, and the affine transformation reduces to a linear transformation. This may make it easier to classify and understand the transformation. For example, describing a transformation as a rotation by a certain angle with respect to a certain axis is easier to get an idea of the overall behavior of the transformation than describing it as a combination of a translation and a rotation. However, this depends on application and context. Describing such a transformation for an object tends to make more sense in terms of rotation about an axis through the center of that object, combined with a translation, rather than by just a rotation with respect to some distant point. For example "move 200 m north and rotate 90° anti-clockwise", rather than the equivalent "with respect to the point 141 m to the northwest, rotate 90° anti-clockwise".

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Affine transformations in 2D without fixed point (so where A has eigenvalue 1) are:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• pure translations
• scaling in a given direction, with respect to a line in another direction (not necessarily perpendicular), combined with translation that is not purely in the direction of scaling; the scale factor is the other eigenvalue; taking "scaling" in a generalized sense it includes the cases that the scale factor is zero (projection) and negative; the latter includes reflection, and combined with translation it includes glide reflection.
• shear combined with translation that is not purely in the direction of the shear (there is no other eigenvalue than 1; it has algebraic multiplicity 2, but geometric multiplicity 1)

In a geometric setting, affine transformations are precisely the functions that map straight lines to straight lines.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

A linear transformation is a function that preserves all linear combinations; an affine transformation is a function that preserves all affine combinations. An affine combination is a linear combination in which the sum of the coefficients is 1.

Related Topics:
Linear combination - Affine combination

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

An affine subspace of a vector space (sometimes called a linear manifold) is a coset of a linear subspace; i.e., it is the result of adding a constant vector to every element of the linear subspace.

Related Topics:
Coset - Linear subspace

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

A linear subspace of a vector space is a subset that is closed under linear combinations; an affine subspace is one that is closed under affine combinations.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

For example, in 3D lines and planes through the origin, the origin, and the whole space are the linear subspaces, while points, lines and planes in general as well as the whole space are the affine subspaces.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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 scalars

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:a1, a2,. ., an

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

exist, not all 0, such that

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:a1v1 +. .. + anvn = 0.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Similarly they are affinely dependent if the same is true and also

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:a1 +. .. + an = 0.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Such a vector (a1,. .., an) is an affine dependence among the vectors v1, v2,. .., vn.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The set of all invertible affine transformations forms a group under the operation of composition of functions. That group is called the affine group, and is the semidirect product of Kn and GL(n, k).

Related Topics:
Group - Affine group - Semidirect product

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

 
Affine transformation: Affine transformation of the plane ►