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; taking "scaling" in a generalized sense it includes the cases that the scale factor is zero (projection) or negative; the latter includes reflection, and combined with translation it includes glide reflection, rotation combined with a homothety and a translation, shear mapping combined with a homothety and a translation, or squeeze mapping combined with a homothety and a translation. 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 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 do not 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). Examples Over the real numbers The functions with and in and , are precisely the affine transformations of the real line. In plane geometry In , the transformation shown at left is accomplished using the map given by: 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. In fact, all triangles are related to one another by affine transformations. This is also true for all parallelograms, but not for all quadrilaterals. See also Anamorphosis – artistic applications of affine transformations Affine geometry 3D projection Homography Flat (geometry) Bent function Notes References External links Geometric Operations: Affine Transform, R. Fisher, S. Perkins,