Translation (geometry)
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[[Image:TraslazioneOK.png|right|thumb|A translation moves every point of a figure or a space by the same amount in a given direction.]]
[[Image:Simx2=traslOK.png|right|thumb|A [[reflection (mathematics)|reflection]] against an axis followed by a reflection against a second axis parallel to the first one results in a total motion which is a translation.]]
In [[Euclidean geometry]], a '''translation''' is moving every point a constant distance in a specified direction. It is one of the [[Euclidean group|rigid motion]]s (other rigid motions include rotation and reflection). A translation can also be interpreted as the addition of a constant [[vector space|vector]] to every point, or as shifting the [[Origin (mathematics)|origin]] of the [[coordinate system]]. A '''translation operator''' is an [[operator]] <math>T_\mathbf{\delta}</math> such that <math>T_\mathbf{\delta} f(\mathbf{v}) = f(\mathbf{v}+\mathbf{\delta}).</math>
If '''v''' is a fixed vector, then the translation ''T''<sub>'''v'''</sub> will work as ''T''<sub>'''v'''</sub>('''p''') = '''p''' + '''v'''.
If ''T'' is a translation, then the [[image (mathematics)|image]] of a subset ''A'' under the [[function (mathematics)|function]] ''T'' is the '''translate''' of ''A'' by ''T''. The translate of ''A'' by ''T''<sub>'''v'''</sub> is often written ''A'' + '''v'''.
In an [[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 ==<!-- This section is linked from [[Affine transformation]] -->
Since a translation is an [[affine transformation]] but not a [[linear transformation]], [[homogeneous coordinates]] are normally used to represent the translation operator by a [[matrix (math)|matrix]] and thus to make it linear. Thus we write the 3-dimensional vector '''w''' = (''w''<sub>''x''</sub>, ''w''<sub>''y''</sub>, ''w''<sub>''z''</sub>) using 4 homogeneous coordinates as '''w''' = (''w''<sub>''x''</sub>, ''w''<sub>''y''</sub>, ''w''<sub>''z''</sub>, 1).
To translate an object by a [[vector (spatial)|vector]] '''v''', each homogeneous vector '''p''' (written in homogeneous coordinates) would need to be multiplied by this '''translation matrix''':
: <math> T_{\mathbf{v}} =
\begin{bmatrix}
1 & 0 & 0 & v_x \\
0 & 1 & 0 & v_y \\
0 & 0 & 1 & v_z \\
0 & 0 & 0 & 1
\end{bmatrix}
. \! </math>
As shown below, the multiplication will give the expected result:
: <math> T_{\mathbf{v}} \mathbf{p} =
\begin{bmatrix}
1 & 0 & 0 & v_x \\
0 & 1 & 0 & v_y \\
0 & 0 & 1 & v_z \\
0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
p_x \\ p_y \\ p_z \\ 1
\end{bmatrix}
=
\begin{bmatrix}
p_x + v_x \\ p_y + v_y \\ p_z + v_z \\ 1
\end{bmatrix}
= \mathbf{p} + \mathbf{v} . \! </math>
The inverse of a translation matrix can be obtained by reversing the direction of the vector:
: <math> T^{-1}_{\mathbf{v}} = T_{-\mathbf{v}} . \! </math>
Similarly, the product of translation matrices is given by adding the vectors:
: <math> T_{\mathbf{u}}T_{\mathbf{v}} = T_{\mathbf{u}+\mathbf{v}} . \! </math>
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] at [[cut-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]].
[[Category:Functions and mappings]]
[[Category:Euclidean symmetries]]
[[Category:Geometry]]
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