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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Translation (mathematics) ： ウィキペディア英語版 Translation (geometry) In Euclidean geometry, a translation is a function that moves every point a constant distance in a specified direction. (Also in Euclidean geometry a transformation is a one to one correspondence between two sets of points or a mapping from one plane to another.) A translation can be described as a rigid motion: other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. A translation operator is an operator $T\_\backslash mathbf$ such that $T\_\backslash mathbf\; f(\backslash mathbf)\; =\; f(\backslash mathbf+\backslash mathbf).$ If v is a fixed vector, then the translation ''T''v will work as ''T''v(p) = p + v. If ''T'' is a translation, then the image of a subset ''A'' under the function ''T'' is the translate of ''A'' by ''T''. The translate of ''A'' by ''T''v is often written ''A'' + v. In a Euclidean space, any translation is an isometry. The set of all translations forms the translation group ''T'', which is isomorphic to the space itself, and a normal subgroup of Euclidean group ''E''(''n'' ). The quotient group of ''E''(''n'' ) by ''T'' is isomorphic to the orthogonal group ''O''(''n'' ): :''E''(''n'' ) ''/ T'' ≅ ''O''(''n'' ). ==Matrix representation== A translation is an affine transformation with ''no'' fixed points. Matrix multiplications ''always'' have the origin as a fixed point. Nevertheless, there is a common workaround using homogeneous coordinates to represent a translation of a vector space with matrix multiplication: Write the 3-dimensional vector w = (''w''''x'', ''w''''y'', ''w''''z'') using 4 homogeneous coordinates as w = (''w''''x'', ''w''''y'', ''w''''z'', 1).〔Richard Paul, 1981, (Robot manipulators: mathematics, programming, and control : the computer control of robot manipulators ), MIT Press, Cambridge, MA〕 To translate an object by a vector v, each homogeneous vector p (written in homogeneous coordinates) can be multiplied by this translation matrix: : $T\_$ 1 & 0 & 0 & v_x \\ 0 & 1 & 0 & v_y \\ 0 & 0 & 1 & v_z \\ 0 & 0 & 0 & 1 \end As shown below, the multiplication will give the expected result: : $T\_\; =$ \begin 1 & 0 & 0 & v_x \\ 0 & 1 & 0 & v_y\\ 0 & 0 & 1 & v_z\\ 0 & 0 & 0 & 1 \end \begin p_x \\ p_y \\ p_z \\ 1 \end = \begin p_x + v_x \\ p_y + v_y \\ p_z + v_z \\ 1 \end = \mathbf + \mathbf The inverse of a translation matrix can be obtained by reversing the direction of the vector: : $T^\_\}\; .\; \backslash !$ Similarly, the product of translation matrices is given by adding the vectors: : $T\_\}\; =\; T\_\}\; .\; \backslash !$ Because addition of vectors is commutative, multiplication of translation matrices is therefore also commutative (unlike multiplication of arbitrary matrices). ==Translations in physics== In physics, translation (Translational motion) is movement that changes the position of an object, as opposed to rotation. For example, according to Whittaker: A translation is the operation changing the positions of all points ''(x, y, z)'' of an object according to the formula :$(x,y,z)\; \backslash to\; (x+\backslash Delta\; x,y+\backslash Delta\; y,\; z+\backslash Delta\; z)$ where $(\backslash Delta\; x,\backslash \; \backslash Delta\; y,\backslash \; \backslash Delta\; z)$ is the same vector for each point of the object. The translation vector $(\backslash Delta\; x,\backslash \; \backslash Delta\; y,\backslash \; \backslash Delta\; z)$ common to all points of the object describes a particular type of displacement of the object, usually called a ''linear'' displacement to distinguish it from displacements involving rotation, called ''angular'' displacements. When considering spacetime, a change of time coordinate is considered to be a translation. For example, the Galilean group and the Poincaré group include translations with respect to time. 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「Translation (geometry)」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.