Affine transformation


In Euclidean geometry, an affine transformation or affinity is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, an affine transformation is an automorphism of an affine space, that is, a function which maps an affine space onto itself while preserving both the dimension of any affine subspaces and the ratios of the lengths of parallel line segments. Consequently, sets of parallel affine subspaces remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.
If is the point set of an affine space, then every affine transformation on can be represented as the composition of a linear transformation on and a translation of. Unlike a purely linear transformation, an affine transformation need not preserve the origin of the affine space. Thus, every linear transformation is affine, but not every affine transformation is linear.
Examples of affine transformations include translation, scaling, homothety, similarity, reflection, rotation, hyperbolic rotation, shear mapping, and compositions of them in any combination and sequence.
Viewing an affine space as the complement of a hyperplane at infinity of a projective space, the affine transformations are the projective transformations of that projective space that leave the hyperplane at infinity invariant, restricted to the complement of that hyperplane.
A generalization of an affine transformation is an affine map between two affine spaces over the same field. Let and be two affine spaces with and the point sets and and the respective associated vector spaces over the field. A map is an affine map if there exists a linear map such that for all in.

Definition

Let be an affine space over a field, and be its associated vector space. An affine transformation is a bijection from onto itself that is an affine map; this means that a linear map from to is well defined by the equation here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that implies that
If the dimension of is at least two, a semiaffine transformation of is a bijection from onto itself satisfying:
  1. For every -dimensional affine subspace of, then is also a -dimensional affine subspace of.
  2. If and are parallel affine subspaces of, then and are parallel.
These two conditions are satisfied by affine transformations, and express what is precisely meant by the expression that " preserves parallelism".
These conditions are not independent as the second follows from the first. Furthermore, if the field has at least three elements, the first condition can be simplified to: is a collineation, that is, it maps lines to lines.

Structure

By the definition of an affine space, acts on, so that, for every pair in there is associated a point in. We can denote this action by . Here we use the convention that are two interchangeable notations for an element of. By fixing a point in one can define a function by. For any, this function is one-to-one, and so, has an inverse function given by. These functions can be used to turn into a vector space by defining:
This vector space has origin and formally needs to be distinguished from the affine space, but common practice is to denote it by the same symbol and mention that it is a vector space after an origin has been specified. This identification permits points to be viewed as vectors and vice versa.
For any linear transformation of, we can define the function by
Then is an affine transformation of which leaves the point fixed. It is a linear transformation of, viewed as a vector space with origin.
Let be any affine transformation of. Pick a point in and consider the translation of by the vector, denoted by. Translations are affine transformations and the composition of affine transformations is an affine transformation. For this choice of, there exists a unique linear transformation of such that
That is, an arbitrary affine transformation of is the composition of a linear transformation of and a translation of.
This representation of affine transformations is often taken as the definition of an affine transformation.

Representation

As shown above, an affine map is the composition of two functions: a translation and a linear map. Ordinary vector algebra uses matrix multiplication to represent linear maps, and vector addition to represent translations. Formally, in the finite-dimensional case, if the linear map is represented as a multiplication by an invertible matrix and the translation as the addition of a vector, an affine map acting on a vector can be represented as

Augmented matrix

Using an augmented matrix and an augmented vector, it is possible to represent both the translation and the linear map using a single matrix multiplication. The technique requires that all vectors be augmented with a "1" at the end, and all matrices be 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 is a matrix,
is equivalent to the following
The above-mentioned augmented matrix is called an affine transformation matrix. In the general case, when the last row vector is not restricted to be, the matrix becomes a projective transformation matrix.
This representation exhibits the set of all invertible affine transformations as the semidirect product of and. 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 the additional coordinate "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 additional coordinate is 1. Thus the origin of the original space can be found at. A translation within the original space by means of a linear transformation of the higher-dimensional space is then possible. The coordinates in the higher-dimensional space are an example of homogeneous coordinates. If the original space is Euclidean, the higher dimensional space is a real projective space.
The advantage of using homogeneous coordinates is that one can combine any number of affine transformations into one by multiplying the respective matrices. This property is used extensively in computer graphics, computer vision and robotics.

Example augmented matrix

Suppose you have three points that define a non-degenerate triangle in a plane, or four points that define a non-degenerate tetrahedron in 3-dimensional space, or generally points,..., that define a non-degenerate simplex in -dimensional space. Suppose you have corresponding destination points,...,, where these new points can lie in a space with any number of dimensions. The unique augmented matrix that achieves the affine transformation
for every is
using matrix inversion.

Properties

Properties preserved

An affine transformation preserves:
  1. collinearity between points: three or more points which lie on the same line continue to be collinear after the transformation.
  2. parallelism: two or more lines which are parallel, continue to be parallel after the transformation.
  3. convexity of sets: a convex set continues to be convex after the transformation. Moreover, the extreme points of the original set are mapped to the extreme points of the transformed set.
  4. ratios of lengths of parallel line segments: for distinct parallel segments defined by points and, respectively, the ratio of to is the same as that of to.
  5. barycenters of weighted collections of points.

    Groups

As an affine transformation is invertible, the square matrix appearing in its [|matrix representation] is invertible. The matrix representation of the inverse transformation is thus
The invertible affine transformations form the affine group, which has the general linear group of degree as subgroup and is itself a subgroup of the general linear group of degree.
The similarity transformations form the subgroup where is a scalar times an orthogonal matrix. For example, if the affine transformation acts on the plane and if the determinant of is 1 or −1 then the transformation is an equiareal mapping. Such transformations form a subgroup called the equi-affine group. A transformation that is both equi-affine and a similarity is an isometry of the plane taken with Euclidean distance.
Each of these groups has a subgroup of orientation-preserving or positive affine transformations: those where the determinant of is positive. In the last case this is in 3D the group of rigid transformations.
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 may give a clearer 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.

Affine maps

An affine map between two affine spaces is a map on the points that acts linearly on the vectors. In symbols, ' determines a linear transformation ' such that, for any pair of points :
or
We can interpret this definition in a few other ways, as follows.
If an origin is chosen, and denotes its image, then this means that for any vector :
If an origin is also chosen, this can be decomposed as an affine transformation that sends, namely
followed by the translation by a vector.
The conclusion is that, intuitively, consists of a translation and a linear map.