# Translation (geometry)

In Euclidean geometry, a translation, or translation operator, is an affine transformation of Euclidean space which moves every point by a fixed distance in the same direction. It can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In other words, if v is a fixed vector, then the translation Tv will work as Tv(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 Tv is often written A + v.

Each translation is an isometry. The set of all translations form 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 ) / TO(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. Thus we write the 3-dimensional vector w = (wx, wy, wz) using 4 homogeneous coordinates as w = (wx, wy, wz, 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:

${\displaystyle T_{\mathbf {v} }={\begin{bmatrix}1&0&0&v_{x}\\0&1&0&v_{y}\\0&0&1&v_{z}\\0&0&0&1\end{bmatrix}}.\!}$

As shown below, the multiplication will give the expected result:

${\displaystyle T_{\mathbf {v} }\mathbf {p} ={\begin{bmatrix}1&0&0&v_{x}\\0&1&0&v_{y}\\0&0&1&v_{z}\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}p_{x}+v_{x}\\p_{y}+v_{y}\\p_{z}+v_{z}\\1\end{bmatrix}}=\mathbf {p} +\mathbf {v} .\!}$

The inverse of a translation matrix can be obtained by reversing the direction of the vector:

${\displaystyle T_{\mathbf {v} }^{-1}=T_{-\mathbf {v} }.\!}$

Similarly, the product of translation matrices is given by adding the vectors:

${\displaystyle T_{\mathbf {u} }T_{\mathbf {v} }=T_{\mathbf {u} +\mathbf {v} }.\!}$

Because addition of vectors is commutative, multiplication of translation matrices is therefore also commutative (unlike multiplication of arbitrary matrices).