Translation (mathematics)
Introduction
In the field of mathematics, translation refers to a geometric transformation that moves every point of a figure or a space by the same distance in a given direction. It is one of the most basic forms of transformations in Euclidean space.
Definition
In a two-dimensional plane, a translation moves each point a constant distance in a specified direction. A translation can be described as a rigid motion: the 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.
Mathematical Description
In two-dimensional Euclidean space, translations are often identified with the displacement vector, which specifies the direction and distance of the one possible translation. When considered as transformations, which are a function from the plane to itself, translations are simply additive.
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).
Translation in Various Dimensions
One Dimension
In one dimension, a translation moves points a certain distance on a line. The direction of movement is either to the left or right along the line.
Two Dimensions
In two dimensions, translations move points a certain distance in a certain direction on a plane. The direction of movement can be anywhere on the plane.
Three Dimensions
In three dimensions, translations move points a certain distance in a certain direction in space. The direction of movement can be anywhere in the space.
Translation Matrices
A translation matrix can be used to represent a translation in linear algebra. In two dimensions, a translation matrix is a 3x3 matrix that is used to translate a point in a two-dimensional plane. In three dimensions, a translation matrix is a 4x4 matrix that is used to translate a point in a three-dimensional space.
Translation in Computer Graphics
In computer graphics, 2D geometric transformations are commonly used. These include not only translations, but also scaling, rotation, and shearing. In particular, translation is the simplest type of transformation, both in terms of the amount of computation and the concept.