# Translation (geometry)

(Redirected from Translation (mathematics))
A translation moves every point of a figure or a space by the same amount in a given direction.
A reflection against an axis followed by a reflection against a second axis parallel to the first one results in a total motion which is a translation.

In Euclidean geometry, a translation is a function that moves every point a constant distance in a specified direction. (Also in Euclidean geometry a transformation is a one to one correspondence between two sets of points or a mapping from one plane to another.(master math Geometry, Debra Anne, Ross)[1] 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 $T_\mathbf{\delta}$ such that $T_\mathbf{\delta} f(\mathbf{v}) = f(\mathbf{v}+\mathbf{\delta}).$

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).[2]

To translate an object by a vector v, each homogeneous vector p (written in homogeneous coordinates) can be multiplied by this translation matrix:

$T_{\mathbf{v}} = \begin{bmatrix} 1 & 0 & 0 & v_x \\ 0 & 1 & 0 & v_y \\ 0 & 0 & 1 & v_z \\ 0 & 0 & 0 & 1 \end{bmatrix}$

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

$T_{\mathbf{v}} \mathbf{p} = \begin{bmatrix} 1 & 0 & 0 & v_x \\ 0 & 1 & 0 & v_y\\ 0 & 0 & 1 & v_z\\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} p_x \\ p_y \\ p_z \\ 1 \end{bmatrix} = \begin{bmatrix} p_x + v_x \\ p_y + v_y \\ p_z + v_z \\ 1 \end{bmatrix} = \mathbf{p} + \mathbf{v}$

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

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

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

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

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

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

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

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