Translation (geometry)


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 the coordinate system. A translation operator is an operator 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 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

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 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 \0 & 1 & 0 & v_y \0 & 0 & 1 & v_z \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 \0 & 1 & 0 & v_y \0 & 0 & 1 & v_z \0 & 0 & 0 & 1end{bmatrix}egin{bmatrix}p_x \ p_y \ p_z \ 1end{bmatrix}=egin{bmatrix}p_x + v_x \ p_y + v_y \ p_z + v_z \ 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] 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.


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