翻訳と辞書 Words near each other ・ Affonso Beato ・ Affonso Celso Pastore ・ Affonso Eduardo Reidy ・ Affine manifold (disambiguation) ・ Affine monoid ・ Affine plane ・ Affine plane (incidence geometry) ・ Affine pricing ・ Affine q-Krawtchouk polynomials ・ Affine representation ・ Affine root system ・ Affine shape adaptation ・ Affine space ・ Affine sphere ・ Affine term structure model ・ Affine transformation ・ Affine variety ・ Affine vector field ・ Affine-regular polygon ・ Affing ・ Affing House ・ Affinger Bach ・ Affinghausen ・ Affinia ・ Affinia Group ・ Affinia Hotel Collection ・ Affiniam ・ Affinine ・ Affinion Group ・ Affinisine Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク 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 $f\; :\; X\; \backslash to\; Y$ is of the form $x\; \backslash mapsto\; Mx\; +\; b$, 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〔 $f:\backslash mathcal\; \backslash to\; \backslash mathcal$ 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 ''$\backslash varphi$'' such that, for any pair of points $P,\; Q\; \backslash in\; \backslash mathcal$: :$\backslash overrightarrow\; =\; \backslash varphi(\backslash overrightarrow)$ or :$f(Q)-f(P)\; =\; \backslash varphi(Q-P)$. We can interpret this definition in a few other ways, as follows. If an origin $O\; \backslash in\; \backslash mathcal$ is chosen, and $B$ denotes its image $f(O)\; \backslash in\; \backslash mathcal$, then this means that for any vector $\backslash vec$: :$f:\; (O+\backslash vec)\; \backslash mapsto\; (B+\backslash varphi(\backslash vec))\; .$ If an origin $O\text{'}\; \backslash in\; \backslash mathcal$ is also chosen, this can be decomposed as an affine transformation $g\; :\; \backslash mathcal\; \backslash to\; \backslash mathcal$ that sends $O\; \backslash mapsto\; O\text{'}$, namely :$g:\; (O+\backslash vec)\; \backslash mapsto\; (O\text{'}+\backslash varphi(\backslash vec))\; ,$ followed by the translation by a vector $\backslash vec\; =\; \backslash overrightarrow$. 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）』 ■ウィキペディアで「Affine transformation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.