affine mapping - meaning and definition. What is affine mapping
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What (who) is affine mapping - definition

GEOMETRIC TRANSFORMATION THAT PRESERVES LINES BUT NOT ANGLES NOR THE ORIGIN
Affine map; Affine function; Affine transformations; Affine transform; Affine mapping; Affinity (mathematics); Affine Transform; Affine transformation matrix; Affine-linear function
  • Effect of applying various 2D affine transformation matrices on a unit square. Note that the reflection matrices are special cases of the scaling matrix.
  • Affine transformations on the 2D plane can be performed by linear transformations in three dimensions. Translation is done by shearing along over the z axis, and rotation is performed around the z axis.
  • A central dilation. The triangles A1B1Z, A1C1Z, and B1C1Z get mapped to A2B2Z, A2C2Z, and B2C2Z, respectively.
  • Barnsley's fern]]) that exhibits affine [[self-similarity]]. Each of the leaves of the fern is related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the dark blue leaf and any of the light blue leaves by a combination of reflection, rotation, scaling, and translation.
  • A simple affine transformation on the real plane

Affine transformation         
In Euclidean geometry, an affine transformation, or an affinity (from the Latin, affinis, "connected with"), is a geometric transformation that preserves lines and parallelism (but not necessarily distances and angles).
affine transformation         
<mathematics> A linear transformation followed by a translation. Given a matrix M and a vector v, A(x) = Mx + v is a typical affine transformation. (1995-04-10)
Quantum affine algebra         
Affine quantum group; Quantum affine algebras
In mathematics, a quantum affine algebra (or affine quantum group) is a Hopf algebra that is a q-deformation of the universal enveloping algebra of an affine Lie algebra. They were introduced independently by and as a special case of their general construction of a quantum group from a Cartan matrix.

Wikipedia

Affine transformation

In Euclidean geometry, an affine transformation or affinity (from the Latin, affinis, "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.

More generally, an affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space onto itself while preserving both the dimension of any affine subspaces (meaning that it sends points to points, lines to lines, planes to planes, and so on) and the ratios of the lengths of parallel line segments. Consequently, sets of parallel affine subspaces remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

If X is the point set of an affine space, then every affine transformation on X can be represented as the composition of a linear transformation on X and a translation of X. Unlike a purely linear transformation, an affine transformation need not preserve the origin of the affine space. Thus, every linear transformation is affine, but not every affine transformation is linear.

Examples of affine transformations include translation, scaling, homothety, similarity, reflection, rotation, shear mapping, and compositions of them in any combination and sequence.

Viewing an affine space as the complement of a hyperplane at infinity of a projective space, the affine transformations are the projective transformations of that projective space that leave the hyperplane at infinity invariant, restricted to the complement of that hyperplane.

A generalization of an affine transformation is an affine map (or affine homomorphism or affine mapping) between two (potentially different) affine spaces over the same field k. Let (X, V, k) and (Z, W, k) be two affine spaces with X and Z the point sets and V and W the respective associated vector spaces over the field k. A map f: XZ is an affine map if there exists a linear map mf : VW such that mf (xy) = f (x) − f (y) for all x, y in X.