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# Translation (geometry)

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 geometric transformation that moves every point of a figure or a space by the same amount 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.

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 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 ).

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• Translation example | Transformations | Geometry | Khan Academy
• Math Shorts Episode 3 - Translation
• Formal translation tool example | Transformations | Geometry | Khan Academy

#### Transcription

- Let's do an example on the performing translations exercise. Use the translate tool to find the image of triangle W I N for a translation of six units, positive six units, in the X direction and negative three units in the Y direction. Alright, so we wanna go positive six units in the X direction and negative three units in the Y direction, alright. So, I click on the translate tool. Click on the translate tool and I wanna go, so, I wanna go positive six units in the X direction. So, I can pick any point and go six to the right and everything else is gonna come with it. So, one, two, three, four, five, six. So, I did that part, I translated positive six units in the X direction and negative three units in the Y direction, so everything needs to go down by three. One, two, three and notice, I focused on point N and this is it's image now, or the image of point N, this whole triangle is the image of this entire triangle, the triangle W I N after the transformation, but you see that every point shifted six to the right, six to the right and three down. This point over here, six to the right would take you, let's see, it's at one and a half right now. It's X coordinate is one and a half. It's new X coordinate is seven and a half, so it's X coordinate increased by six and it's old Y coordinate, or the original Y coordinate was six and now, in the image, the corresponding Y coordinate is three. So, it has, we have shifted it down three. So, we see that that's happened to every point here and we're done.

## 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:

${\displaystyle T_{\mathbf {v} }={\begin{bmatrix}1&0&0&v_{x}\\0&1&0&v_{y}\\0&0&1&v_{z}\\0&0&0&1\end{bmatrix}}}$

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:

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

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

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

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.