Translation (geometry) 204682 215278111 2008-05-27T14:15:31Z Pleasantville 3058640 rvv to last version by Kubigula {{unreferenced|date=December 2007}} [[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]] [[de:Parallelverschiebung]] [[es:Traslación (geometría)]] [[eo:Movo (geometrio)]] [[fr:Translation (géométrie)]] [[gl:Translación (movemento)]] [[it:Traslazione (geometria)]] [[lb:Translatioun (Mathematik)]] [[nl:Translatie (meetkunde)]] [[oc:Translacion (geometria)]] [[pl:Translacja (matematyka)]] [[pt:Translação]] [[ru:Параллельный перенос]] [[sr:Транслација (геометрија)]] [[fi:Translaatio (matematiikka)]] [[th:การเลื่อนขนาน]] [[zh:平移]]