In Euclidean geometry, a translation is moving every point a constant distance in a specified direction. It is one of the rigid motions (other rigid motions include rotation and reflection). 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 such that
If v is a fixed vector, then the translation T_{v} will work as T_{v}(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 T_{v} 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 ):
Since a translation is an affine transformation but not a linear transformation, homogeneous coordinates are normally used to represent the translation operator by a matrix and thus to make it linear. Thus we write the 3dimensional vector w = (w_{x}, w_{y}, w_{z}) using 4 homogeneous coordinates as w = (w_{x}, w_{y}, w_{z}, 1).
To translate an object by a vector v, each homogeneous vector p (written in homogeneous coordinates) would need to 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).
