Editing Translation (geometry)

{{Short description|Planar movement within a Euclidean space without rotation}}
[[File:Traslazione OK.svg|right|thumb|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 geometry|distance]] in a given [[direction (geometry)|direction]]. A translation can also be interpreted as the addition of a constant [[vector space|vector]] to every point, or as shifting the [[Origin (mathematics)|origin]] of the [[coordinate system]]. In a [[Euclidean space]], any translation is an [[isometry]].

==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.

==As an operator==
{{main|Shift operator}}

The [[Shift operator|translation operator]] turns a function of the original position, <math>f(\mathbf{v})</math>, into a function of the final position, <math>f(\mathbf{v}+\mathbf{\delta})</math>. In other words, <math>T_\mathbf{\delta}</math> is defined such that <math>T_\mathbf{\delta} f(\mathbf{v}) = f(\mathbf{v}+\mathbf{\delta}).</math> This [[operator (mathematics)|operator]] is more abstract than a function, since <math>T_\mathbf{\delta}</math> defines a relationship between two functions, rather than the underlying vectors themselves. The translation operator can act on many kinds of functions, such as when the [[Translation operator (quantum mechanics)|translation operator acts on a wavefunction]], which is studied in the field of quantum mechanics.

==As a group==
{{see also|Translation operator (quantum mechanics)#Translation group}}
The set of all translations forms the '''translation group''' <math>\mathbb{T} </math>, which is isomorphic to the space itself, and a [[normal subgroup]] of [[Euclidean group]] <math> E(n) </math>. The [[quotient group]] of <math>E(n) </math> by <math>\mathbb{T} </math> is isomorphic to the group of rigid motions which fix a particular origin point, the [[orthogonal group]] <math> O(n)</math>:
:<math>E(n)/\mathbb{T}\cong O(n) </math>

Because translation is [[commutative]], the translation group is [[Abelian group|abelian]]. There are an infinite number of possible translations, so the translation group is an [[infinite group]].

In the [[theory of relativity]], due to the treatment of space and time as a single [[spacetime]], translations can also refer to changes in the [[Coordinate time|time coordinate]]. For example, the [[Galilean group]] and the [[Poincaré group]] include translations with respect to time.

===Lattice groups===
{{main|Lattice (group)}}

One kind of [[subgroup]] of the three-dimensional translation group are the [[Lattice (group)|lattice groups]], which are [[infinite group]]s, but unlike the translation groups, are [[Finitely generated group|finitely generated]]. That is, a finite [[Generating set of a group|generating set]] generates the entire group.

==Matrix representation==<!-- This section is linked from [[Affine transformation]] -->
A translation is an [[affine transformation]] with ''no'' [[fixed point (mathematics)|fixed point]]s. Matrix multiplications ''always'' have the [[origin (mathematics)|origin]] as a fixed point. Nevertheless, there is a common [[workaround]] using [[homogeneous coordinates]] to represent a translation of a [[vector space]] with [[matrix multiplication]]: Write the 3-dimensional vector <math>\mathbf{v}=(v_x, v_y, v_z) </math> using 4 homogeneous coordinates as <math>\mathbf{v}=(v_x, v_y, v_z, 1) </math>.<ref>Richard Paul, 1981, [https://books.google.com/books?id=UzZ3LAYqvRkC Robot manipulators: mathematics, programming, and control : the computer control of robot manipulators], MIT Press, Cambridge, MA</ref>

To translate an object by a [[Vector (geometry)|vector]] <math>\mathbf{v} </math>, each homogeneous vector <math>\mathbf{p} </math> (written in homogeneous coordinates) can be multiplied by this '''translation matrix''':

: <math> T_{\mathbf{v}} = 
\begin{bmatrix}
1 & 0 & 0 & v_x \\
0 & 1 & 0 & v_y \\
0 & 0 & 1 & v_z \\
0 & 0 & 0 & 1
\end{bmatrix}
</math>
As shown below, the multiplication will give the expected result:
: <math> T_{\mathbf{v}} \mathbf{p} =
\begin{bmatrix}
1 & 0 & 0 & v_x \\
0 & 1 & 0 & v_y\\
0 & 0 & 1 & v_z\\
0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
p_x \\ p_y \\ p_z \\ 1
\end{bmatrix}
=
\begin{bmatrix}
p_x + v_x \\ p_y + v_y \\ p_z + v_z \\ 1
\end{bmatrix}
= \mathbf{p} + \mathbf{v} </math>

The inverse of a translation matrix can be obtained by reversing the direction of the vector:
: <math> T^{-1}_{\mathbf{v}} = T_{-\mathbf{v}} . \! </math>

Similarly, the product of translation matrices is given by adding the vectors:
: <math> T_{\mathbf{v}}T_{\mathbf{w}} = T_{\mathbf{v}+\mathbf{w}} . \! </math>
Because addition of vectors is [[commutative]], multiplication of translation matrices is therefore also commutative (unlike multiplication of arbitrary matrices).

==Translation of axes==
{{main|Translation of axes}}

While geometric translation is often viewed as an [[active transformation]] that changes the position of a geometric object, a similar result can be achieved by a [[passive transformation]] that moves the coordinate system itself but leaves the object fixed. The passive version of an active geometric translation is known as a ''[[translation of axes]]''.

==Translational symmetry==
{{main|Translational symmetry}}

An object that looks the same before and after translation is said to have [[translational symmetry]]. A common example is a [[periodic function]], which is an [[eigenfunction]] of a translation operator.

== Translations of a graph {{anchor|Horizontal|Vertical}} ==
{{redirect|Vertical translation|the concept in physics|Vertical separation}}

[[File:Translated graph of a function.png|thumb|upright=1.25|Compared to the graph {{math|1=''y'' = ''f''(''x'')}}, the graph {{math|1=''y'' = ''f''(''x''&thinsp;&minus;&thinsp;''a'')}} has been translated horizontally by {{mvar|a}}, while the graph {{math|1=''y'' = ''f''(''x'')&thinsp;+&thinsp;b}} has been translated vertically by {{mvar|b}}.]]

The [[graph of a function|graph]] of a [[real function]] {{mvar|f}}, the set of points {{tmath|(x, f(x)) }}, is often pictured in the [[real coordinate plane]] with {{mvar|x}} as the horizontal coordinate and {{tmath|1= y = f(x)}} as the vertical coordinate.

Starting from the graph of {{mvar|f}}, a '''horizontal translation''' means [[function composition|composing]] {{mvar|f}} with a function {{tmath|x \mapsto x - a}}, for some constant number {{mvar|a}}, resulting in a graph consisting of points {{tmath|(x, f(x - a)) }}. Each point {{tmath|(x, y)}} of the original graph corresponds to the point {{tmath|(x + a, y)}} in the new graph, which pictorially results in a horizontal shift.

A '''vertical translation''' means composing the function {{tmath|y \mapsto y + b}} with {{mvar|f}}, for some constant {{mvar|b}}, resulting in a graph consisting of the points {{tmath|\bigl(x, f(x) + b\bigr) }}. Each point {{tmath|(x, y)}} of the original graph corresponds to the point {{tmath|(x, y + b)}} in the new graph, which pictorially results in a vertical shift.<ref>{{citation|title=Nonlinear Filters for Image Processing|series=SPIE/IEEE series on imaging science & engineering|volume=59|first1=Edward R.|last1=Dougherty|first2=Jaakko|last2=Astol|publisher=SPIE Press|year=1999|isbn=9780819430335|page=169|url=https://books.google.com/books?id=4PV-sTF6qJQC&pg=PA169}}.</ref>

For example, taking the [[quadratic function]] {{tmath|1= y = x^2}}, whose graph is a [[parabola]] with vertex at {{tmath|(0, 0)}}, a horizontal translation 5 units to the right would be the new function {{tmath|1= y = (x - 5)^2 = x^2 - 10x + 25}} whose vertex has coordinates {{tmath|(5, 0)}}. A vertical translation 3 units upward would be the new function {{tmath|1= y = x^2 + 3}} whose vertex has coordinates {{tmath|(0, 3)}}.

The [[antiderivative]]s of a function all differ from each other by a [[constant of integration]] and are therefore vertical translates of each other.<ref>{{citation|title=Single Variable Calculus: Early Transcendentals|first1=Dennis|last1=Zill|first2=Warren S.|last2=Wright|publisher=Jones & Bartlett Learning|year=2009|isbn=9780763749651|page=269|url=https://books.google.com/books?id=0n0iPYKLo74C&pg=PA269}}.</ref>

== Applications ==

For describing [[vehicle dynamics]] (or movement of any [[rigid body]]), including [[ship motions|ship dynamics]] and [[aircraft principal axes|aircraft dynamics]], it is common to use a mechanical model consisting of six [[Degrees of freedom (mechanics)|degrees of freedom]], which includes translations along three reference axes (as well as rotations about those three axes). These translations are often called [[Ship motions#Surge|''surge'']], [[Ship motions#Sway|''sway'']], and [[Ship motions#Heave|''heave'']].

==See also==
{{col div|colwidth=40em}}
* [[2D computer graphics#Translation]]
* [[Advection]]
* [[Change of basis]]
* [[Parallel transport]]
* [[Rotation matrix]]
* [[Scaling (geometry)]]
* [[Transformation matrix]]
* [[Translational symmetry]]
{{colend}}

==References==
{{reflist}}

==Further reading==
*Zazkis, R., Liljedahl, P., & Gadowsky, K. Conceptions of function translation: obstacles, intuitions, and rerouting. Journal of Mathematical Behavior, 22, 437-450. Retrieved April 29, 2014, from www.elsevier.com/locate/jmathb
*[http://www.biology.arizona.edu/biomath/tutorials/transformations/horizontaltranslations.html Transformations of Graphs: Horizontal Translations]. (2006, January 1). BioMath: Transformation of Graphs. Retrieved April 29, 2014

==External links==
{{Commons category|Translation (geometry)}}
* [http://www.cut-the-knot.org/Curriculum/Geometry/Translation.shtml Translation Transform] at [[cut-the-knot]]
* [http://www.mathsisfun.com/geometry/translation.html Geometric Translation (Interactive Animation)] at Math Is Fun
* [http://demonstrations.wolfram.com/Understanding2DTranslation/ Understanding 2D Translation] and [http://demonstrations.wolfram.com/Understanding3DTranslation/ Understanding 3D Translation] by Roger Germundsson, [[The Wolfram Demonstrations Project]].

{{Computer graphics}}
{{DEFAULTSORT:Translation (Geometry)}}

[[Category:Euclidean symmetries]]
[[Category:Elementary geometry]]
[[Category:Transformation (function)]]
[[Category:Functions and mappings]]