In Euclidean geometry, a **translation** is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction.

In Euclidean geometry a transformation is a one-to-one correspondence between two sets of points or a mapping from one plane to another.^{[1]} A translation can be described as a rigid motion: the other rigid motions are rotations, reflections and glide reflections.

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.

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 a 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*):

# Matrix representation

A translation is an affine transformation with *no* fixed points. Matrix multiplications *always* have the origin as a fixed point. Nevertheless, there is a common workaround using homogeneous coordinates to represent a translation of a vector space with matrix multiplication: 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).^{[2]}

To translate an object by a vector **v**, each homogeneous vector **p** (written in homogeneous coordinates) can 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).

# Translations in physics

In physics, **translation** (Translational motion) is movement that changes the position of an object, as opposed to rotation. For example, according to Whittaker:^{[3]}

A translation is the operation changing the positions of all points (*x*, *y*, *z*) of an object according to the formula

When considering spacetime, a change of time coordinate is considered to be a translation. For example, the Galilean group and the PoincarĂ© group include translations with respect to time.