Affine transformation

Affine Transformation

An affine transformation or affine map is a type of linear transformation in geometry that includes translation, scaling, rotation, and shearing. It is a function between affine spaces which preserves points, straight lines and planes. Also, it multiplies all areas by the same positive number, and it preserves the ratio of the lengths of two segments lying on a straight line.

Definition

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

x ↦ Ax + b

In the finite-dimensional case, each output vector is a linear combination of the input vectors, with coefficients given by functions of the point at which the map is evaluated.

Properties

Affine transformations are invertible and the inverse of an affine transformation is again an affine transformation. As a result, the set of all affine transformations has the structure of a group. This group is called the affine group.

Applications

Affine transformations are widely used in computer graphics and image processing for tasks such as 2D and 3D rendering, image scaling, image rotation, and image translation.

See also

• Homography
• Projective transformation
• Transformation matrix