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. In a geometric setting, these 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.

An affine subspace of a vector space 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. 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.

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,. ., v_n

are linearly dependent if scalars

a₁, a₂,. ., a_n

exist, not all 0, such that

a₁v₁ +. .. + a_nv_n = 0.

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

a₁ +. .. + a_n = 0.

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

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).

Affine transformation of the plane

To visualise the general affine transformation of the Euclidean plane, take labelled parallelograms ABCD and A′B′C′D′. Whatever the choices of points, there is an affine transformation T of the plane taking A to A′, and each vertex similarly. Supposing we exclude the degenerate case where ABCD has zero area, there is a unique such affine transformation T. Drawing out a whole grid of parallelograms based on ABCD, the image T(P) of any point P is determined by noting that T(A) = A′, T applied to the vector AB is A′B′, T applied to the vector AC is A′C′, and T respects scalar multiples of vectors based at A. Geometrically T transforms the grid based on ABCD to that based in A′B′C′D′.

Affine transformations don't respect lengths or angles; they multiply area by a constant factor

area of A′B′C′D′/area of ABCD.

A given T may either be direct (respect orientation), or indirect (reverse orientation), and this may be determined by its effect on signed areas (as defined, for example, by the cross product of vectors).

Example of an affine transformation

The following equation expresses an affine transformation in GF(2) (with "+" representing XOR):

{a'} = [M]{a} + {v}

where [M] is the matrix

<math>\begin{bmatrix}

1&0&0&0&1&1&1&1 \\ 1&1&0&0&0&1&1&1 \\ 1&1&1&0&0&0&1&1 \\ 1&1&1&1&0&0&0&1 \\ 1&1&1&1&1&0&0&0 \\ 0&1&1&1&1&1&0&0 \\ 0&0&1&1&1&1&1&0 \\ 0&0&0&1&1&1&1&1\end{bmatrix}<math>

and {v} is the vector

<math>\begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \\ 0 \\ 1 \\ 1 \\ 0 \end{bmatrix}.<math>

For instance, the affine transformation of the element {a} = x⁷ + x⁶ + x³ + x = {11001010} in big-endian binary notation = {CA} in big-endian hexadecimal notation, is calculated as follows:

<math>a_0' = a_0 \oplus a_4 \oplus a_5 \oplus a_6 \oplus a_7 \oplus 1 = 0 \oplus 0 \oplus 0 \oplus 1 \oplus 1 \oplus 1 = 1<math>

<math>a_1' = a_0 \oplus a_1 \oplus a_5 \oplus a_6 \oplus a_7 \oplus 1 = 0 \oplus 1 \oplus 0 \oplus 1 \oplus 1 \oplus 1 = 0<math>

<math>a_2' = a_0 \oplus a_1 \oplus a_2 \oplus a_6 \oplus a_7 \oplus 0 = 0 \oplus 1 \oplus 0 \oplus 1 \oplus 1 \oplus 0 = 1<math>

<math>a_3' = a_0 \oplus a_1 \oplus a_2 \oplus a_3 \oplus a_7 \oplus 0 = 0 \oplus 1 \oplus 0 \oplus 1 \oplus 1 \oplus 0 = 1<math>

<math>a_4' = a_0 \oplus a_1 \oplus a_2 \oplus a_3 \oplus a_4 \oplus 0 = 0 \oplus 1 \oplus 0 \oplus 1 \oplus 0 \oplus 0 = 0<math>

<math>a_5' = a_1 \oplus a_2 \oplus a_3 \oplus a_4 \oplus a_5 \oplus 1 = 1 \oplus 0 \oplus 1 \oplus 0 \oplus 0 \oplus 1 = 1<math>

<math>a_6' = a_2 \oplus a_3 \oplus a_4 \oplus a_5 \oplus a_6 \oplus 1 = 0 \oplus 1 \oplus 0 \oplus 0 \oplus 1 \oplus 1 = 1<math>

<math>a_7' = a_3 \oplus a_4 \oplus a_5 \oplus a_6 \oplus a_7 \oplus 0 = 1 \oplus 0 \oplus 0 \oplus 1 \oplus 1 \oplus 0 = 1.<math>

Thus, {a'} = x⁷ + x⁶ + x⁵ + x³ + x² + 1 = {11101101} = {ED}

Affine transformation

Affine transformation of the plane

Example of an affine transformation

See also

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