Translation (geometry)
In Euclidean geometry, a translation is a function that moves every point a constant distance in a specified direction. A translation can be described as a rigid motion: other rigid motions include rotations and 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 such that
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 ) / T ≅ O(n ).
Read more about Translation (geometry): Matrix Representation, Translations in Physics
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