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Las tranformaciones afines invertibles (de un espacio afín en sí mismo) forman el llamado grupo afín que como se ha mencionado tiene al [[grupo lineal]] de orden ''n'' como subgrupo. El propio grupo afín de orden ''n'' es a su vez subgrupo del grupo lineal de orden ''n''+1.
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==Properties==
The [[Similar matrix|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 [[isometry|isometries]], the subgroup of both where ''A'' is an orthogonal matrix.
Each of these groups has a subgroup of transformations which preserve [[Orientation (mathematics)|orientation]]: those where the determinant of ''A'' is positive. In the last case this is in 3D the group of [[rigid body]] motions ([[improper rotation|proper rotation]]s and pure translations).
For any matrix ''A'' the following propositions are equivalent:
*''A'' − ''I'' is invertible
*''A'' does ''not'' have an [[Eigenvalue, eigenvector, and eigenspace|eigenvalue]] equal to 1
*for all ''b'' the transformation has exactly one [[Fixed point (mathematics)|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. As an 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 (geometry)|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 (linear algebra)|projection]]) and negative; the latter includes [[Reflection (mathematics)|reflection]], and combined with translation it includes [[glide reflection]].
*[[Shear (mathematics)|shear]] combined with translation that is not purely in the direction of the shear (there is no other eigenvalue than 1; it has algebraic [[Eigenvalue, eigenvector, and eigenspace#Multiplicity|multiplicity]] 2, but geometric multiplicity 1)
==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''. [If ''A'', ''E'', ''F'' are collinear 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).
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== Referencias ==
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