- Translation (geometry)
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 the coordinate system. A translation operator is an operatorsuch that
If v is a fixed vector, then the translation "T"v will work as "T"v(p) = p + v.
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 subgroupof Euclidean group"E"("n" ). The quotient groupof "E"("n" ) by "T" is isomorphic to the orthogonal group"O"("n" )::"E"("n" ) "/ T" ≅ "O"("n" ).
Since a translation is an
affine transformationbut not a linear transformation, homogeneous coordinatesare 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:
As shown below, the multiplication will give the expected result::
The inverse of a translation matrix can be obtained by reversing the direction of the vector::
Similarly, the product of translation matrices is given by adding the vectors:: Because addition of vectors is
commutative, multiplication of translation matrices is therefore also commutative (unlike multiplication of arbitrary matrices).
* [http://www.cut-the-knot.org/Curriculum/Geometry/Translation.shtml Translation Transform] at
* [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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