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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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#### Transcription

- [Voiceover] Triangle ABC undergoes a translation, and we're using the notation capital T for it, and then we see what the translation has to be. We're gonna move, it's kind of small, I hope you can see it on your video screen. We're gonna move positive eight. Every point here is gonna move positive eight in the x direction. Its x coordinate is going to increase by eight, or the corresponding point in the image, its x coordinate, is going to increase by eight, and the corresponding point in the image's y coordinate is going to decrease by one, so let's do that. And I'll focus on the vertices, whoops, let me drag that to the trash, I didn't mean to do that. I'm going to focus on the vertices well, that's just the easiest thing for my brain to worth with. And actually, this is what the tool expects as well. So the point B, is going to move eight to the right, or its corresponding point in the image is going to have an x coordinate eight larger. So right now, the x coordinate is negative four, if you added eight to that, it would be positive four, and its y coordinate is going to be one lower. Right now, point B's y coordinate is eight, one lower than that is seven. So, in the image, the corresponding point of the image would going to be right over there. And you see we moved eight to the right, and one down. Let's do that with point C. It's at x equals negative seven, if you move eight to the right, if you increase your x coordinate by eight, you're gonna move to x equals one, and then if you change your y coordinate by negative one, you're gonna move down one, then you're gonna get to that point right over there. Now, let's do it with point A. So point A's x coordinate is negative one. If you add eight to it, it's going to be positive seven, and its current y coordinate is two. If you take one away from it, you're gonna get to a y coordinate of one. And so there you have it. Let's see, how do I connect these two? Oh, there you go, and we can check our answer. And we got it right. We have performed the translation.

## 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.