- Translation (geometry)
In

Euclidean geometry , a**translation**is moving every point a constant distance in a specified direction. It is one of the rigid motions (other rigid motions include rotation and reflection). A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of thecoordinate system . A**translation operator**is anoperator $T\_mathbf\{delta\}$ such that $T\_mathbf\{delta\}\; f(mathbf\{v\})\; =\; f(mathbf\{v\}+mathbf\{delta\}).$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 an

Euclidean space , any translation is anisometry . The set of all translations forms the translation group "T", which is isomorphic to the space itself, and anormal subgroup ofEuclidean group "E"("n" ). Thequotient group of "E"("n" ) by "T" is isomorphic to theorthogonal group "O"("n" )::"E"("n" ) "/ T" ≅ "O"("n" ).**Matrix representation**Since a translation is an

affine transformation but not alinear transformation ,homogeneous coordinates are normally used to represent the translation operator by a matrix and thus to make it linear. Thus we 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).To translate an object by a vector

**v**, each homogeneous vector**p**(written in homogeneous coordinates) would need to be multiplied by this**translation matrix**:: $T\_\{mathbf\{v\; =\; egin\{bmatrix\}1\; 0\; 0\; v\_x\; \backslash 0\; 1\; 0\; v\_y\; \backslash 0\; 0\; 1\; v\_z\; \backslash 0\; 0\; 0\; 1\; end\{bmatrix\}.\; !$

As shown below, the multiplication will give the expected result:: $T\_\{mathbf\{v\; mathbf\{p\}\; =egin\{bmatrix\}1\; 0\; 0\; v\_x\; \backslash 0\; 1\; 0\; v\_y\; \backslash 0\; 0\; 1\; v\_z\; \backslash 0\; 0\; 0\; 1end\{bmatrix\}egin\{bmatrix\}p\_x\; \backslash \; p\_y\; \backslash \; p\_z\; \backslash \; 1end\{bmatrix\}=egin\{bmatrix\}p\_x\; +\; v\_x\; \backslash \; p\_y\; +\; v\_y\; \backslash \; p\_z\; +\; v\_z\; \backslash \; 1end\{bmatrix\}=\; mathbf\{p\}\; +\; mathbf\{v\}\; .\; !$

The inverse of a translation matrix can be obtained by reversing the direction of the vector:: $T^\{-1\}\_\{mathbf\{v\; =\; T\_\{-mathbf\{v\; .\; !$

Similarly, the product of translation matrices is given by adding the vectors:: $T\_\{mathbf\{uT\_\{mathbf\{v\; =\; T\_\{mathbf\{u\}+mathbf\{v\; .\; !$Because addition of vectors is

commutative , multiplication of translation matrices is therefore also commutative (unlike multiplication of arbitrary matrices).**See also***

Translation (physics)

*Translational symmetry

*Transformation matrix **External links*** [

*http://www.cut-the-knot.org/Curriculum/Geometry/Translation.shtml Translation Transform*] atcut-the-knot

* [*http://www.mathsisfun.com/geometry/translation.html Geometric Translation (Interactive Animation)*] at Math Is Fun

* [*http://demonstrations.wolfram.com/Understanding2DTranslation/ Understanding 2D Translation*] and [*http://demonstrations.wolfram.com/Understanding3DTranslation/ Understanding 3D Translation*] by Roger Germundsson,The Wolfram Demonstrations Project .

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