In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. 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. In a Euclidean space, any translation is an isometry.
As a function
As a function
See also: Displacement (geometry)
If {\displaystyle \mathbf {v} }{\displaystyle \mathbf {v} } is a fixed vector, known as the translation vector, and {\displaystyle \mathbf {p} }\mathbf {p} is the initial position of some object, then the translation function {\displaystyle T_{\mathbf {v} }}{\displaystyle T_{\mathbf {v} }} will work as {\displaystyle T_{\mathbf {v} }(\mathbf {p} )=\mathbf {p} +\mathbf {v} }{\displaystyle T_{\mathbf {v} }(\mathbf {p} )=\mathbf {p} +\mathbf {v} }.
If {\displaystyle T} T is a translation, then the image of a subset {\displaystyle A}A under the function {\displaystyle T} T is the translate of {\displaystyle A}A by {\displaystyle T}T. The translate of {\displaystyle A}A by {\displaystyle T_{\mathbf {v} }}{\displaystyle T_{\mathbf {v} }} is often written {\displaystyle A+\mathbf {v} }{\displaystyle A+\mathbf {v} }.
Horizontal and vertical translations
In geometry, a vertical translation (also known as vertical shift) is a translation of a geometric object in a direction parallel to the vertical axis of the Cartesian coordinate system.
See also
- Advection
- Parallel transport
- Rotation matrix
- Scaling (geometry)
- Transformation matrix
- Translational symmetry
External links
- Translation Transform at cut-the-knot
- Geometric Translation (Interactive Animation) at Math Is Fun
- Understanding 2D Translation and Understanding 3D Translation by Roger Germundsson, The Wolfram Demonstrations Project.
