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affine transformation : ウィキペディア英語版
affine transformation

In geometry, an affine transformation, affine map〔Berger, Marcel (1987), p. 38.〕 or an affinity (from the Latin, ''affinis'', "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines 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.
Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination and sequence.
If X and Y are affine spaces, then every affine transformation f : X \to Y is of the form x \mapsto Mx + b, where M is a linear transformation on X and b is a vector in Y. Unlike a purely linear transformation, an affine map need not preserve the zero point in a linear space. Thus, every linear transformation is affine, but not every affine transformation is linear.
For many purposes an affine space can be thought of as Euclidean space, though the concept of affine space is far more general (i.e., all Euclidean spaces are affine, but there are affine spaces that are non-Euclidean). In affine coordinates, which include Cartesian coordinates in Euclidean spaces, each output coordinate of an affine map is a linear function (in the sense of calculus) of all input coordinates. Another way to deal with affine transformations systematically is to select a point as the origin; then, any affine transformation is equivalent to a linear transformation (of position vectors) followed by a translation.
==Mathematical definition==

An affine map〔 f:\mathcal \to \mathcal between two affine spaces is a map on the points that acts linearly on the vectors (that is, the vectors between points of the space). In symbols, ''f'' determines a linear transformation ''\varphi'' such that, for any pair of points P, Q \in \mathcal:
:\overrightarrow = \varphi(\overrightarrow)
or
:f(Q)-f(P) = \varphi(Q-P).
We can interpret this definition in a few other ways, as follows.
If an origin O \in \mathcal is chosen, and B denotes its image f(O) \in \mathcal, then this means that for any vector \vec:
:f: (O+\vec) \mapsto (B+\varphi(\vec)) .
If an origin O' \in \mathcal is also chosen, this can be decomposed as an affine transformation g : \mathcal \to \mathcal that sends O \mapsto O', namely
:g: (O+\vec) \mapsto (O'+\varphi(\vec)) ,
followed by the translation by a vector \vec = \overrightarrow.
The conclusion is that, intuitively, f consists of a translation and a linear map.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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