Editing Translation (geometry) (section)

==As a function==
{{see also|Displacement (geometry)}}
If <math>\mathbf{v} </math> is a fixed vector, known as the ''translation vector'', and <math>\mathbf{p}</math> is the initial position of some object, then the translation function <math>T_{\mathbf{v}} </math> will work as <math> T_{\mathbf{v}}(\mathbf{p})=\mathbf{p}+\mathbf{v}</math>.

If <math> T</math> is a translation, then the [[image (mathematics)|image]] of a subset <math> A </math> under the [[function (mathematics)|function]] <math> T</math> is the '''translate''' of <math> A </math> by <math> T </math>. The translate of <math>A </math> by <math>T_{\mathbf{v}} </math> is often written as <math>A+\mathbf{v} </math>.

===Application in classical physics===
In [[classical physics]], translational motion is movement that changes the [[Position (geometry)|position]] of an object, as opposed to [[rotation]]. For example, according to Whittaker:<ref name=Whittaker>{{cite book |title=A Treatise on the Analytical Dynamics of Particles and Rigid Bodies |author=Edmund Taylor Whittaker|author-link=E. T. Whittaker |isbn=0-521-35883-3 |publisher=Cambridge University Press |year=1988 |url=https://books.google.com/books?id=epH1hCB7N2MC&q=rigid+bodies+translation&pg=PA4 |edition=Reprint of fourth edition of 1936 with foreword by William McCrea |page=1}}</ref>

{{Quotation|If a body is moved from one position to another, and if the lines joining the initial and final points of each of the points of the body are a set of parallel straight lines of length ''ℓ'', so that the orientation of the body in space is unaltered, the displacement is called a ''translation parallel to the direction of the lines, through a distance ℓ''. |[[E. T. Whittaker]], ''[[A Treatise on the Analytical Dynamics of Particles and Rigid Bodies]]'', p. 1}}

A translation is the operation changing the positions of all points <math>(x, y, z)</math> of an object according to the formula

:<math>(x,y,z) \to (x+\Delta x,y+\Delta y, z+\Delta z)</math>

where <math>(\Delta x,\ \Delta y,\ \Delta z)</math> is the same [[Euclidean vector|vector]] for each point of the object. The translation vector <math>(\Delta x,\ \Delta y,\ \Delta z)</math> common to all points of the object describes a particular type of [[Displacement (vector)|displacement]] of the object, usually called a ''linear'' displacement to distinguish it from displacements involving rotation, called ''angular'' displacements.

When considering [[spacetime]], a change of [[time]] coordinate is considered to be a translation.