# Translation (geometry)

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.

A translation moves every point of a figure or a space by the same amount in a given direction.
The reflection of a red shape against an axis followed by a reflection of the resulting green shape against a second axis parallel to the first one results in a total motion which is a translation of the red shape to the position of the blue shape.

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.

A translation operator is an operator ${\displaystyle T_{\mathbf {\delta } }}$ such that ${\displaystyle T_{\mathbf {\delta } }f(\mathbf {v} )=f(\mathbf {v} +\mathbf {\delta } ).}$

If ${\displaystyle \mathbf {v} }$ is a fixed vector, then the translation ${\displaystyle T_{\mathbf {v} }}$ will work as ${\displaystyle T_{\mathbf {v} }(\mathbf {p} )=\mathbf {p} +\mathbf {v} }$.

If ${\displaystyle T}$ is a translation, then the image of a subset ${\displaystyle A}$ under the function ${\displaystyle T}$ is the translate of ${\displaystyle A}$ by ${\displaystyle T}$. The translate of ${\displaystyle A}$ by ${\displaystyle T_{\mathbf {v} }}$ is often written ${\displaystyle A+\mathbf {v} }$.

In a Euclidean space, any translation is an isometry. The set of all translations forms the translation group ${\displaystyle \mathbb {T} }$, which is isomorphic to the space itself, and a normal subgroup of Euclidean group ${\displaystyle E(n)}$. The quotient group of ${\displaystyle E(n)}$ by ${\displaystyle \mathbb {T} }$ is isomorphic to the orthogonal group ${\displaystyle O(n)}$:

${\displaystyle E(n)/\mathbb {T} \cong O(n)}$

## 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 ${\displaystyle \mathbf {v} =(v_{x},v_{y},v_{z})}$  using 4 homogeneous coordinates as ${\displaystyle \mathbf {v} =(v_{x},v_{y},v_{z},1)}$ .[2]

To translate an object by a vector ${\displaystyle \mathbf {v} }$ , each homogeneous vector ${\displaystyle \mathbf {p} }$  (written in homogeneous coordinates) can 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:

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

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]

If a body is moved from one position to another, and if the lines joining the initial and final points of each of the points of the body are a set of parallel straight lines of length , so that the orientation of the body in space is unaltered, the displacement is called a translation parallel to the direction of the lines, through a distance ℓ.

— E. T. Whittaker: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, p. 1

A translation is the operation changing the positions of all points ${\displaystyle (x,y,z)}$  of an object according to the formula

${\displaystyle (x,y,z)\to (x+\Delta x,y+\Delta y,z+\Delta z)}$

where ${\displaystyle (\Delta x,\ \Delta y,\ \Delta z)}$  is the same vector for each point of the object. The translation vector ${\displaystyle (\Delta x,\ \Delta y,\ \Delta z)}$  common to all points of the object describes a particular type of displacement of the object, usually called a linear displacement to distinguish it from displacements involving rotation, called angular displacements.

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