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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.

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.

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):

# 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:

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

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

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

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]

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

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