# A.3 Affine transforms

An affine transform is a generalization of a rigid transform, in which the rotational component is replaced by a general matrix . This means that an affine transform implements a generalized basis transformation combined with an offset of the origin (Figure A.7). As with for rigid transforms, the columns of still describe the transformed basis vectors , , and , but these are generally no longer orthonormal. Figure A.7: A position vector and a general matrix describing the affine position and basis transform of frame B with respect to frame A.

Expressed in terms of homogeneous coordinates, the affine transform takes the form (A.18)

with (A.19)

As with rigid transforms, when an affine transform is applied to a vector instead of a point, only the matrix is applied and the translation component is ignored.

Affine transforms are typically used to effect transformations that require stretching and shearing of a coordinate frame. By the polar decomposition theorem, can be factored into a regular rotation plus a symmetric shearing/scaling matrix : (A.20)

Affine transforms can also be used to perform reflections, in which is orthogonal (so that ) but with .