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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_\mathbf such that T_\mathbf f(\mathbf) = f(\mathbf+\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^_} . \!
Similarly, the product of translation matrices is given by adding the vectors:
: T_} = T_} . \!
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) \to (x+\Delta x,y+\Delta y, z+\Delta z)
where (\Delta x,\ \Delta y,\ \Delta z) is the same vector for each point of the object. The translation vector (\Delta x,\ \Delta y,\ \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



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