Affine transformation

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

In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, satisfying certain properties described below.

Geometrically, an affine transformation in Euclidean space is one that preserves

In general, an affine transformation is composed of linear transformations (rotation, scaling or shear) and a translation (or "shift"). Several linear transformations can be combined into a single one, so that the general formula given above is still applicable.

In the one-dimensional case, A and b are called, respectively, slope and intercept.

Contents

Representation

Ordinary vector algebra uses matrix multiplication to represent linear transformations, and vector addition to represent translations. Using an augmented matrix, it is possible to represent both using matrix multiplication. The technique requires that all vectors are augmented with a "1" at the end, and all matrices are augmented with an extra row of zeros at the bottom, an extra column—the translation vector—to the right, and a "1" in the lower right corner. If A is a matrix,

is equivalent to the following

This representation exhibits the set of all invertible affine transformations as the semidirect product of Kn and GL(n, k). This is a group under the operation of composition of functions, called the affine group.

Ordinary matrix-vector multiplication always maps the origin to the origin, and could therefore never represent a translation, in which the origin must necessarily be mapped to some other point. By appending a "1" to every vector, one essentially considers the space to be mapped as a subset of a space with an additional dimension. In that space, the original space occupies the subset in which the final index is 1. Thus the origin of the original space can be found at (0,0, ... 0, 1). A translation within the original space by means of a linear transformation of the higher-dimensional space is then possible (specifically, a shear transformation). This is an example of homogeneous coordinates.

The advantage of using homogeneous coordinates is that one can combine any number of affine transformations into one by multiplying the matrices. This property is used extensively by graphics software.

Properties

An affine transformation is invertible if and only if A is invertible. In the matrix representation, the inverse is:

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