Translation (geometry)

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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.

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 an Euclidean space, any translation is an isometry. The set of all translations form 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 ) / T ≅ O(n ).