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 correspondence between two sets of points or a mapping from one plane to another. )A translation can be described as a rigid motion. A translation can also be interpreted as the addition of a constant vector to every point, a translation operator is an operator T δ such that T δ f = f. If v is a vector, then the translation Tv will work as Tv. 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. The quotient group of E by T is isomorphic to the orthogonal group O, E / T ≅ O, a translation is an affine transformation with no fixed points. Matrix multiplications always have the origin as a fixed point, similarly, the product of translation matrices is given by adding the vectors, T u T v = T u + v. Because addition of vectors is commutative, multiplication of matrices is therefore also commutative. In physics, translation is movement that changes the position of an object, for example, according to Whittaker, A translation is the operation changing the positions of all points of an object according to the formula → where is the same vector for each point of the object. 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
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.