# 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 *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**.

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*)*/ 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. 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).

## See also

## External link

- Translation Transform at cut-the-knot
- Geometric Translation (Interactive Animation) at Math Is Fun