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# Translation group

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 Title: Translation group Author: World Heritage Encyclopedia Language: English Subject: Plane symmetry Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Translation group

In Euclidean geometry, a translation is a function that moves every point a constant distance in a specified direction. A translation can be described as a rigid motion: other rigid motions include rotations and 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 $T_\mathbf\left\{\delta\right\}$ such that $T_\mathbf\left\{\delta\right\} f\left(\mathbf\left\{v\right\}\right) = f\left(\mathbf\left\{v\right\}+\mathbf\left\{\delta\right\}\right).$

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.

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 ):

E(n ) / TO(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 = (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) can be multiplied by this translation matrix:

$T_\left\{\mathbf\left\{v\right\}\right\} =$

\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:

$T_\left\{\mathbf\left\{v\right\}\right\} \mathbf\left\{p\right\} =$

\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:

$T^\left\{-1\right\}_\left\{\mathbf\left\{v\right\}\right\} = T_\left\{-\mathbf\left\{v\right\}\right\} . \!$

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

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

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:

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 (x, y, z) of an object according to the formula

$\left(x,y,z\right) \to \left(x+\Delta x,y+\Delta y, z+\Delta z\right)$

where $\left(\Delta x,\ \Delta y,\ \Delta z\right)$ is the same vector for each point of the object. The translation vector $\left(\Delta x,\ \Delta y,\ \Delta z\right)$ 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.

A translation of space (or time) should not be confused with a translation of an object. Such translations have no fixed points.