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: 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: : while an element 1 is added at the bottom of column vectors: : ( Homogeneous Coordinates ). An affine transformation is invertible Iff 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. The Similarity Transformations form the subgroup where A is a scalar times an Orthogonal Matrix . Iff 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. Each of these groups has a subgroup of transformations which preserve motions ( Proper Rotation s and pure translations). For matrix A the following propositions are equivalent:
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". Affine transformations in 2D without fixed point (so where A has eigenvalue 1) are:
In a geometric setting, affine transformations are precisely the functions that map straight lines to straight lines. A linear transformation is a function that preserves all Linear Combination s; 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. 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. For example, in 3D lines and planes through the origin, the origin, and the whole space are the 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 v are linearly dependent if scalars a exist, not all 0, such that a Similarly they are affinely dependent if the same is true and also a Such a vector (''a''1,. .., ''a''''n'') is an affine dependence among the vectors ''v''1, ''v''2,. .., ''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 ''K''''n'' and GL(''n'', ''k''). AFFINE TRANSFORMATION OF THE PLANE To visualise the general affine transformation of the Euclidean Plane , take labelled Parallelogram s ''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''. ''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). EXAMPLE OF AN AFFINE TRANSFORMATION The following equation expresses an affine transformation in GF (2) (with "+" representing XOR ): : where {Link without Title} is the Matrix : and {''v''} is the Vector : For instance, the affine transformation of the element {a} = ''x''7 + ''x''6 + ''x''3 + ''x'' = {11001010} in Big-endian Binary notation = {CA} in big-endian Hexadecimal notation, is calculated as follows: : : : : : : : : Thus, {''a''′} = ''x''7 + ''x''6 + ''x''5 + ''x''3 + ''x''2 + 1 = {11101101} = {ED} SEE ALSO
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