Test:Translation (geometry)

A translation moves every point of a figure or a space by the same amount in a given direction.

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

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.