NationMaster - Encyclopedia: Affine transformation

In geometry, an affine transformation or affine map (from the Latin, affinis, "connected with") between two vector spaces (strictly speaking, two affine spaces) consists of a linear transformation followed by a translation: Table of Geometry, from the 1728 Cyclopaedia. ... 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 transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ...

$x mapsto A x+ b$

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. An affine transformation corresponds to multiplication of a matrix and a vector, and composition of affine transformations corresponds to ordinary matrix multiplication, if an extra row is added at the bottom of the matrix containing only zeros except a 1 at the right:

$begin{bmatrix} A & b 0..0 & 1 end{bmatrix}$

while an element 1 is added at the bottom of column vectors:

$begin{bmatrix} x 1 end{bmatrix}$

(homogeneous coordinates). In mathematics, homogeneous coordinates, introduced by August Ferdinand MÃ¶bius, allow affine transformations to be easily represented by a matrix. ...

An affine transformation is invertible if and only if A is invertible. 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 idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication. ... It has been suggested that this article or section be merged with Logical biconditional. ... In mathematics, the affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself. ... In mathematics, the general linear group of degree n is the set of nÃ—n invertible matrices, together with the operation of ordinary matrix multiplication. ...

The similarity transformations form the subgroup where A is a scalar times an orthogonal matrix. 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. Several equivalence relations in mathematics are called similarity. ... In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ...

Each of these groups has a subgroup of transformations which preserve orientation: those where the determinant of A is positive. In the last case this is in 3D the group of rigid body motions (proper rotations and pure translations). In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ... In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. ... In geometry, an improper rotation is the combination of an ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with a coordinate inversion (a vector x goes to −x). ...

For 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. 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". In this transformation of the Mona Lisa, the blue vector has been rotated, but the red one has not. ... In mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function. ...

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.
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)

In a geometric setting, affine transformations are precisely the functions that map straight lines to straight lines. In Euclidean geometry, uniform scaling is a linear transformation that enlarges or diminishes objects; the scale factor is the same in all directions; it is also called a homothety. ... A scale factor is a number which scales some quantity. ... In linear algebra, a projection is a linear transformation P such that P2 = P, i. ... IT IS KNOWN AS MARK a lunitice insain int gw brain ... Example of a glide reflection In geometry, a glide reflection is a type of isometry of the Euclidean plane. ... In physics and mechanics, shear refers to a deformation that causes parallel surfaces to slide past one another (as opposed to compression and tension, which cause parallel surfaces to move towards or away from one another). ... In this transformation of the Mona Lisa, the blue vector has been rotated, but the red one has not. ...

A linear transformation is a function that preserves all linear combinations; an affine transformation is a function that preserves all affine combinations. An affine combination is a linear combination in which the sum of the coefficients is 1. In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In mathematics, an affine combination of vectors x1, ..., xn is a linear combination in which the sum of the coefficients is 1, thus: . Here the vectors are supposed to lie in given vector space V over a field K; and the coefficients are scalars in K. This concept is important...

An affine subspace of a vector space (sometimes called a linear manifold) is a coset of a linear subspace; i.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. In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G... The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ...

For example, in R³, 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. 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 In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ... A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ...

v₁, v₂,. ., v_n

are linearly dependent if scalars

a₁, a₂,. ., a_n

exist, not all 0, such that

a₁v₁ +. .. + a_nv_n = 0.

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

a₁ + ... + a_n = 0.

Such a vector (a₁,. .., a_n) 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. That group is called the affine group, and is the semidirect product of Kⁿ and GL(n, k). This picture illustrates how the hours in a clock form a group. ... In mathematics, the affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself. ... In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ...

Affine transformation of the plane

To visualise the general affine transformation of the Euclidean plane, take labelled parallelograms ABCD and A′B′C′D′. 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. 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′. In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... A parallelogram. ... Area is a physical quantity expressing the size of a part of a surface. ...

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 vectors in a three-dimensional Euclidean space. ...

Example of an affine transformation

The following equation expresses an affine transformation in GF(2) (with "+" representing XOR): In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ... Exclusive disjunction (usual symbol xor) is a logical operator that results in true if one of the operands (not both) is true. ...

${,a',} = M{,a,} + {,v,}.$

where [M] is the matrix In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...

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

and {v} is the vector In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...

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

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: In computing, endianness is the ordering used to represent some kind of data as a sequence of smaller units. ... The binary numeral system (base 2 numerals) represents numeric values using two symbols, typically 0 and 1. ... In mathematics and computer science, base-16, hexadecimal, or simply hex, is a numeral system with a radix or base of 16, usually written using the symbols 0â€“9 and Aâ€“F or aâ€“f. ...

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

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

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

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

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

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

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

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

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

Example of an affine transformation

See also