![Shield cfg By [R]affine[R]i~ Sxe Bomb Cfg](http://web.archive.org./web/20110824141921im_/http://i.ytimg.com/vi/oLJ3qVvEMLg/0.jpg "Shield cfg By [R]affine[R]i~ Sxe Bomb Cfg")
Hurricane Irene could hit U.S. as a monster
CBS News
Why Palestinians and Palestinian Statehood May Cease to Exist if Perry Becomes President
WorldNews.com
Gaddafi son's appearance in Tripoli sparks doubts
m&c;
Gaddafi, the ghost
Independent online (SA)
England riots: 1,400 suspects have appeared in court
BBC News
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: :
In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, satisfying certain properties described below.
Geometrically, an affine transformation in Euclidean space is one that preserves # The collinearity relation between points; i.e., the points which lie on a line continue to be collinear after the transformation # Ratios of distances along a line; i.e., for distinct collinear points the ratio is preserved
In general, an affine transformation is composed of linear transformations (rotation, scaling or shear) and a translation (or "shift"). Several linear transformations can be combined into a single one, so that the general formula given above is still applicable.
In the one-dimensional case, A and b are called, respectively, slope and intercept.
:
is equivalent to the following
:
The above mentioned augmented matrix is called affine transformation matrix, or projective transformation matrix (as it can also be used to perform Projective transformations).
This representation exhibits the set of all invertible affine transformations as the semidirect product of Kn and GL(n, k). 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 (0,0, ... 0, 1). A translation within the original space by means of a linear transformation of the higher-dimensional space is then possible (specifically, a shear transformation). 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.
:
The invertible affine transformations (of an affine space onto itself) 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. 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.
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).
For any 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. 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:
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).
:
where [M] is the matrix
:
and {v} is the vector
:
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:
: : : : : : : :
Thus, {a′} = x7 + x6 + x5 + x3 + x2 + 1 = {11101101} = {ED}.
:
Transforming the three corner points of the original triangle (in red) gives three new points which form the new triangle (in blue). This transformation skews and translates the original triangle.
Category:Affine geometry Category:Linear algebra Category:Functions and mappings
This text is licensed under the Creative Commons CC-BY-SA License. This text was originally published on Wikipedia and was developed by the Wikipedia community.
