Affine transformation

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:

$x \mapsto A x+ b$

In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, which can be written as the matrix A with an extra column b. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Euclidean geometry, a translation is moving every point a constant distance in a specified direction

Physically, an affine transform is one that preserves

Collinearity between points, i. e. , three points which lie on a line continue to be collinear after the transformation
Ratios of distances along a line, i. e. , for distinct colinear points $p 1$ , $p 2$ , $p 3$ , the ratio $| p 2 - p 1 | / | p 3 - p 2 |$ is preserved

In general, an affine transform is composed of zero or more linear transformations (rotation, scaling or shear) and translation (shift). In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Geometry and Linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a Rigid body around a fixed In Euclidean geometry, uniform scaling or Isotropic scaling is a Linear transformation that enlarges or diminishes objects the Scale factor In Mathematics, a shear or transvection is a particular kind of Linear mapping. In Euclidean geometry, a translation is moving every point a constant distance in a specified direction Several linear transformations can be combined into a single matrix, thus the general formula given above is still applicable.

1 Representation of affine transformations
2 Properties of affine transformations
3 Affine transformations and linear transformations
4 Affine transformation of the plane
5 Example of an affine transformation
6 See also
7 External links

Representation of affine transformations

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. In Linear algebra, the augmented matrix of a matrix is obtained by combining two matrices In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix 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,

$\begin{bmatrix} \vec{y} \\ 1 \end{bmatrix} = \begin{bmatrix} A & \vec{b} \ \\ 0, \ldots, 0 & 1 \end{bmatrix} \begin{bmatrix} \vec{x} \\ 1 \end{bmatrix}$

is equivalent to the following

$\vec{y} = A \vec{x} + \vec{b}.$

Ordinary matrix-vector multiplication always maps the origin to the origin. Since the set of vectors with 1 in the last entry does not contain the origin, translations within this subset using linear transformations are possible. This is the homogeneous coordinates system. In Mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, allow Affine transformations

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

Properties of affine transformations

An affine transformation is invertible if and only if A is invertible. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to ↔ In the matrix representation, the inverse is:

$\begin{bmatrix} A^{-1} & -A^{-1}\vec{b} \ \\ 0,\ldots,0 & 1 \end{bmatrix}.$

The invertible affine transformations form the affine group, which has the general linear group of degree n as subgroup and is itself a subgroup of the general linear group of degree n + 1. In Mathematics, the affine group or general affine group of any Affine space over a field K is the group of all invertible In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation

The similarity transformations form the subgroup where A is a scalar times an orthogonal matrix. In Linear algebra, two n -by- n matrices A and B over the field K are called similar if there exists In Matrix theory, a real orthogonal matrix is a square matrix Q whose Transpose is its inverse: Q^T If and only if the determinant of A is 1 or –1 then the transformation preserves area; these also form a subgroup. Combining both conditions we have the isometries, the subgroup of both where A is an orthogonal matrix. For the Mechanical engineering and Architecture usage see Isometric projection.

Each of these groups has a subgroup of transformations which preserve orientation: those where the determinant of A is positive. See also Orientation (geometry. In Mathematics, an orientation on a real Vector space is a choice of which In the last case this is in 3D the group of rigid body motions (proper rotations and pure translations). In Physics, a rigid body is an idealization of a solid body of finite size in which Deformation is neglected In 3D Geometry, an improper rotation, also called rotoreflection or rotary reflection is depending on context a Linear transformation or

For any matrix A the following propositions are equivalent:

A – I is invertible
A does not have an eigenvalue equal to 1
for all b the transformation has exactly one fixed point
there is a b for which the transformation has exactly one fixed point
affine transformations with matrix A can be written as a linear transformation with some point as origin

If there is a fixed point we can take that as the origin, and the affine transformation reduces to a linear transformation. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In Mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function 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 is easier to get an 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. Describing such a transformation for an object tends to make more sense in terms of rotation about an axis through the center of that object, combined with a translation, rather than by just a rotation with respect to some distant point. For example "move 200 m north and rotate 90° anti-clockwise", rather than the equivalent "with respect to the point 141 m to the northwest, rotate 90° anti-clockwise".

Affine transformations in 2D without fixed point (so where A has eigenvalue 1) are:

pure translations
scaling in a given direction, with respect to a line in another direction (not necessarily perpendicular), combined with translation that is not purely in the direction of scaling; the scale factor is the other eigenvalue; taking "scaling" in a generalized sense it includes the cases that the scale factor is zero (projection) and negative; the latter includes reflection, and combined with translation it includes glide reflection. In Euclidean geometry, uniform scaling or Isotropic scaling is a Linear transformation that enlarges or diminishes objects the Scale factor A scale factor is a number which scales, or multiplies some quantity In Mathematics, a reflection (also spelled reflexion) is a map that transforms an object into its Mirror image. In Geometry, a glide reflection is a type of Isometry of the Euclidean plane: the combination of a reflection in a line and a translation
shear combined with translation that is not purely in the direction of the shear (there is no other eigenvalue than 1; it has algebraic multiplicity 2, but geometric multiplicity 1)

Affine transformations and linear transformations

In a geometric setting, affine transformations are precisely the functions that map straight lines to straight lines. In Mathematics, a shear or transvection is a particular kind of Linear mapping. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes

A linear transformation is a function that preserves all linear combinations; an affine transformation is a function that preserves all affine combinations. In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics An affine combination is a linear combination in which the sum of the coefficients is 1. In Mathematics, an affine combination of vectors x 1. x n is vector \sum_{i=1}^{n}{\alpha_{i} \cdot

An affine subspace of a vector space (sometimes called a linear manifold) is a coset of a linear subspace; i. In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH The concept of a linear subspace (or vector subspace) is important in Linear algebra and related fields of Mathematics. 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.

For example, in R³, the origin, lines and planes through the origin and the whole space are linear subspaces, while points, lines and planes in general as well as the whole space are the affine subspaces.

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. In Linear algebra, a family of vectors is linearly independent if none of them can be written as a Linear combination of finitely many other vectors 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 there exists a vector a

a = [a₁, a₂, … ,a_n]

such that both:

∃ i∊[1…n]: a_i ≠ 0

and

[v₁^T, v₂^T, … , v_n^T] × a^T = 0

are true.

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

∑_i a_i; i ∊ [1…n] = 0

Vector a 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. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element That group is called the affine group, and is the semidirect product of Kⁿ and GL(n, k). In Mathematics, the affine group or general affine group of any Affine space over a field K is the group of all invertible In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can

Affine transformation of the plane

To visualise the general affine transformation of the Euclidean plane, take labelled parallelograms ABCD and A′B′C′D′. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides 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. Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. 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 line segment AB is A′B′, T applied to the line segment AC is A′C′, and T respects scalar multiples of vectors based at A. [If A, E, F are colinear then the ratio length(AF)/length(AE) is equal to length(A′F′)/length(A′E′). ] 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). In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which

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

$\begin{bmatrix}1&0&0&0&1&1&1&1 \\1&1&0&0&0&1&1&1 \\1&1&1&0&0&0&1&1 \\1&1&1&1&0&0&0&1 \\1&1&1&1&1&0&0&0 \\0&1&1&1&1&1&0&0 \\0&0&1&1&1&1&1&0 \\0&0&0&1&1&1&1&1\end{bmatrix}$

and {v} is the vector

$\begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \\ 0 \\ 1 \\ 1 \\ 0 \end{bmatrix}.$

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:

$a_0' = a_0 \oplus a_4 \oplus a_5 \oplus a_6 \oplus a_7 \oplus 1 = 0 \oplus 0 \oplus 0 \oplus 1 \oplus 1 \oplus 1 = 1$

$a_1' = a_0 \oplus a_1 \oplus a_5 \oplus a_6 \oplus a_7 \oplus 1 = 0 \oplus 1 \oplus 0 \oplus 1 \oplus 1 \oplus 1 = 0$

$a_2' = a_0 \oplus a_1 \oplus a_2 \oplus a_6 \oplus a_7 \oplus 0 = 0 \oplus 1 \oplus 0 \oplus 1 \oplus 1 \oplus 0 = 1$

$a_3' = a_0 \oplus a_1 \oplus a_2 \oplus a_3 \oplus a_7 \oplus 0 = 0 \oplus 1 \oplus 0 \oplus 1 \oplus 1 \oplus 0 = 1$

$a_4' = a_0 \oplus a_1 \oplus a_2 \oplus a_3 \oplus a_4 \oplus 0 = 0 \oplus 1 \oplus 0 \oplus 1 \oplus 0 \oplus 0 = 0$

$a_5' = a_1 \oplus a_2 \oplus a_3 \oplus a_4 \oplus a_5 \oplus 1 = 1 \oplus 0 \oplus 1 \oplus 0 \oplus 0 \oplus 1 = 1$

$a_6' = a_2 \oplus a_3 \oplus a_4 \oplus a_5 \oplus a_6 \oplus 1 = 0 \oplus 1 \oplus 0 \oplus 0 \oplus 1 \oplus 1 = 1$

$a_7' = a_3 \oplus a_4 \oplus a_5 \oplus a_6 \oplus a_7 \oplus 0 = 1 \oplus 0 \oplus 0 \oplus 1 \oplus 1 \oplus 0 = 1.$

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

External links

Geometric Operations: Affine Transform, R. In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added The binary numeral system, or base-2 number system, is a Numeral system that represents numeric values using two symbols usually 0 and 1. In Mathematics and Computer science, hexadecimal (also base -, hexa, or hex) is a Numeral system with a In Linear algebra, Linear transformations can be represented by matrices. In Euclidean geometry, a translation is moving every point a constant distance in a specified direction Affine geometry is a form of Geometry featuring the unique parallel line property (see the parallel postulate) but where the notion of angle is undefined and lengths In Mathematics, a homothety (or homothecy or dilation) is a transformation of space which takes each line into a parallel line (in essence a In Linear algebra, two n -by- n matrices A and B over the field K are called similar if there exists The word linear comes from the Latin word linearis, which means created by lines. 3D projection is any method of mapping three-dimensional points to a two-dimensional plane In Geometry, a flat is a subset of ''n''-dimensional space that is congruent to a Euclidean space of lower Dimension. Fisher, S. Perkins, A. Walker and E. Wolfart.
Eric W. Weisstein, Affine Transform at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
Affine Transform by Bernard Vuilleumier, The Wolfram Demonstrations Project.
Affine Transform on PlanetMath

Dictionary

affine transformation

-noun

(geometry, linear algebra) a linear transformation between vector spaces followed by a translation

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