In

^{2} , a horizontal translation 5 units to the right would be represented by ''T''((''x'', ''y'')) = (''x'' + 5, ''y''). Now we must connect this transformation notation to an algebraic notation. Consider the point (''a'', ''b'') on the original parabola that moves to point (''c'', ''d'') on the translated parabola. According to our translation, ''c'' = ''a'' + 5 and ''d'' = ''b''. The point on the original parabola was ''b'' = ''a''^{2}. Our new point can be described by relating ''d'' and ''c'' in the same equation. ''b'' = ''d'' and ''a'' = ''c'' − 5.
So ''d'' = ''b'' = ''a''^{2} = (''c'' − 5)^{2}. Since this is true for all the points on our new parabola the new equation is ''y'' = (''x'' − 5)^{2}.

Robot manipulators: mathematics, programming, and control : the computer control of robot manipulators

MIT Press, Cambridge, MA To translate an object by a

Translation Transform

at

Geometric Translation (Interactive Animation)

at Math Is Fun

Understanding 2D Translation

an

Understanding 3D Translation

by Roger Germundsson,

Transformations of Graphs: Horizontal Translations

(2006, January 1). BioMath: Transformation of Graphs. Retrieved April 29, 2014 {{DEFAULTSORT:Translation (Geometry) Euclidean symmetries Elementary geometry Transformation (function) Functions and mappings

Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandria
Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic
Coptic may refer to:
Afro-Asia
* Copts, an ethnoreligious group mainly in the area of modern ...

, a translation is a geometric transformation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

that moves every point of a figure, shape or space by the same distance
Distance is a numerical measurement
'
Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

in a given direction. A translation can also be interpreted as the addition of a constant vector
Vector may refer to:
Biology
*Vector (epidemiology)
In epidemiology
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...

to every point, or as shifting the origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...

of the coordinate system
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

. In a Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...

, any translation is an isometry
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

.
As a function

If $\backslash mathbf$ is a fixed vector, known as the ''translation vector'', and $\backslash mathbf$ is the initial position of some object, then the translation function $T\_$ will work as $T\_(\backslash mathbf)=\backslash mathbf+\backslash mathbf$. If $T$ is a translation, then theimage
An image (from la, imago) is an artifact that depicts visual perception
Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...

of a subset $A$ under the function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

$T$ is the translate of $A$ by $T$. The translate of $A$ by $T\_$ is often written $A+\backslash mathbf$.
Horizontal and vertical translations

Ingeometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

, 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
A Cartesian coordinate system (, ) in a plane
Plane or planes may refer to:
* Airplane
An airplane or aeroplane (informally plane) is a fixed-wing aircraft
A fixed-wing aircraft is a heavier-than-air flying machine
Early fly ...

.
Often, vertical translations are considered for the graph of a function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

. If ''f'' is any function of ''x'', then the graph of the function ''f''(''x'') + ''c'' (whose values are given by adding a constant
Constant or The Constant may refer to:
Mathematics
* Constant (mathematics)
In mathematics, the word constant can have multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other Value (mathematics ...

''c'' to the values of ''f'') may be obtained by a vertical translation of the graph of ''f''(''x'') by distance ''c''. For this reason the function ''f''(''x'') + ''c'' is sometimes called a vertical translate of ''f''(''x''). For instance, the antiderivative
In calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zer ...

s of a function all differ from each other by a constant of integration
In calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. Th ...

and are therefore vertical translates of each other.
In function graphing
In mathematics, the graph of a Function (mathematics), function f is the set of ordered pairs (x, y), where f(x) = y. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in two-dimensional space ...

, a horizontal translation is a transformation
Transformation may refer to:
Science and mathematics
In biology and medicine
* Metamorphosis, the biological process of changing physical form after birth or hatching
* Malignant transformation, the process of cells becoming cancerous
* Transf ...

which results in a graph that is equivalent to shifting the base graph left or right in the direction of the ''x''-axis. A graph is translated ''k'' units horizontally by moving each point on the graph ''k'' units horizontally.
For the base function ''f''(''x'') and a constant
Constant or The Constant may refer to:
Mathematics
* Constant (mathematics)
In mathematics, the word constant can have multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other Value (mathematics ...

''k'', the function given by ''g''(''x'') = ''f''(''x'' − ''k''), can be sketched ''f''(''x'') shifted ''k'' units horizontally.
If function transformation was talked about in terms of geometric transformations it may be clearer why functions translate horizontally the way they do. When addressing translations on the Cartesian plane
A Cartesian coordinate system (, ) in a plane
Plane or planes may refer to:
* Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft
Arts, entertainment and media
*Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

it is natural to introduce translations in this type of notation:
:$(x,y)\backslash rightarrow(x+a,y+b)$
or
:$T(x,y)\; =\; (x+a,y+b)$
where $a$ and $b$ are horizontal and vertical changes respectively.
Example

Taking theparabola
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

''y'' = ''x''Application in classical physics

Inclassical physics
Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...

, translational motion is movement that changes the position of an object, as opposed to rotation
A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...

. For example, according to Whittaker:
A translation is the operation changing the positions of all points $(x,\; y,\; z)$ of an object according to the formula
:$(x,y,z)\; \backslash to\; (x+\backslash Delta\; x,y+\backslash Delta\; y,\; z+\backslash Delta\; z)$
where $(\backslash Delta\; x,\backslash \; \backslash Delta\; y,\backslash \; \backslash Delta\; z)$ is the same vector
Vector may refer to:
Biology
*Vector (epidemiology)
In epidemiology
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...

for each point of the object. The translation vector $(\backslash Delta\; x,\backslash \; \backslash Delta\; y,\backslash \; \backslash Delta\; z)$ common to all points of the object describes a particular type of displacement
Displacement may refer to:
Physical sciences
Mathematics and Physics
*Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path c ...

of the object, usually called a ''linear'' displacement to distinguish it from displacements involving rotation, called ''angular'' displacements.
When considering spacetime
In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...

, a change of time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

coordinate is considered to be a translation.
As an operator

The translation operator turns a function of the original position, $f(\backslash mathbf)$, into a function of the final position, $f(\backslash mathbf+\backslash mathbf)$. In other words, $T\_\backslash mathbf$ is defined such that $T\_\backslash mathbf\; f(\backslash mathbf)\; =\; f(\backslash mathbf+\backslash mathbf).$ This operator is more abstract than a function, since $T\_\backslash mathbf$ 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 acts on a wavefunction, which is studied in the field of quantum mechanics.As a group

The set of all translations forms the translation group $\backslash mathbb$, which is isomorphic to the space itself, and anormal subgroup
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

of Euclidean group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

$E(n)$. The quotient group
A quotient group or factor group is a math
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geome ...

of $E(n)$ by $\backslash mathbb$ is isomorphic to the orthogonal group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

$O(n)$:
:$E(n)/\backslash mathbb\backslash cong\; O(n)$
Because translation is commutative, the translation group is abelian. There are an infinite number of possible translations, so the translation group is an infinite group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

.
In the theory of relativity
The theory of relativity usually encompasses two interrelated theories by Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born , widely acknowledged to be one of the greatest physicists of all time ...

, due to the treatment of space and time as a single spacetime
In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...

, translations can also refer to changes in the time coordinate. For example, the Galilean group
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succe ...

and the Poincaré group
The Poincaré group, named after Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French
French (french: français(e), link=no) may refer to:
* Something of, from, or related to ...

include translations with respect to time.
Lattice groups

One kind ofsubgroup
In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...

of the three-dimensional translation group are the lattice groups, which are infinite group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

s, but unlike the translation groups, are finitely generated. That is, a finite generating set
In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set (mathematics), set of objects, together with a set of Operation (mathe ...

generates the entire group.
Matrix representation

A translation is anaffine transformation
In Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...

with ''no'' fixed points. Matrix multiplications ''always'' have the origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...

as a fixed point. Nevertheless, there is a common workaround
A workaround is a bypass of a recognized problem or limitation in a system or policy. A workaround is typically a temporary fix that implies that a genuine solution to the problem is needed. But workarounds are frequently as creative as true soluti ...

using homogeneous coordinates
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

to represent a translation of a vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

with matrix multiplication
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

: Write the 3-dimensional vector $\backslash mathbf=(v\_x,\; v\_y,\; v\_z)$ using 4 homogeneous coordinates as $\backslash mathbf=(v\_x,\; v\_y,\; v\_z,\; 1)$.Richard Paul, 1981Robot manipulators: mathematics, programming, and control : the computer control of robot manipulators

MIT Press, Cambridge, MA To translate an object by a

vector
Vector may refer to:
Biology
*Vector (epidemiology)
In epidemiology
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...

$\backslash mathbf$, each homogeneous vector $\backslash mathbf$ (written in homogeneous coordinates) can be multiplied by this translation matrix:
: $T\_\; =\; \backslash begin\; 1\; \&\; 0\; \&\; 0\; \&\; v\_x\; \backslash \backslash \; 0\; \&\; 1\; \&\; 0\; \&\; v\_y\; \backslash \backslash \; 0\; \&\; 0\; \&\; 1\; \&\; v\_z\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; 1\; \backslash end$
As shown below, the multiplication will give the expected result:
: $T\_\; \backslash mathbf\; =\; \backslash begin\; 1\; \&\; 0\; \&\; 0\; \&\; v\_x\; \backslash \backslash \; 0\; \&\; 1\; \&\; 0\; \&\; v\_y\backslash \backslash \; 0\; \&\; 0\; \&\; 1\; \&\; v\_z\backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; 1\; \backslash end\; \backslash begin\; p\_x\; \backslash \backslash \; p\_y\; \backslash \backslash \; p\_z\; \backslash \backslash \; 1\; \backslash end\; =\; \backslash begin\; p\_x\; +\; v\_x\; \backslash \backslash \; p\_y\; +\; v\_y\; \backslash \backslash \; p\_z\; +\; v\_z\; \backslash \backslash \; 1\; \backslash end\; =\; \backslash mathbf\; +\; \backslash mathbf$
The inverse of a translation matrix can be obtained by reversing the direction of the vector:
: $T^\_\; =\; T\_\; .\; \backslash !$
Similarly, the product of translation matrices is given by adding the vectors:
: $T\_T\_\; =\; T\_\; .\; \backslash !$
Because addition of vectors is commutative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

, multiplication of translation matrices is therefore also commutative (unlike multiplication of arbitrary matrices).
Translation of axes

While geometric translation is often viewed as an active process 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

An object that looks the same before and after translation is said to havetranslational symmetry
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

. A common example is periodic function
A periodic function is a Function (mathematics), function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used th ...

s, which are eigenfunction
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s of the translation operator.
See also

*Advection
In the field of physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy ...

* Parallel transport
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

* Rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation (mathematics), rotation in Euclidean space. For example, using the convention below, the matrix
:R = \begin
\cos \theta & -\sin \theta \\
\sin \th ...

* Scaling (geometry)
In Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's metho ...

* Transformation matrix
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and ...

* Translational symmetry
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

External links

Translation Transform

at

cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electron ...

Geometric Translation (Interactive Animation)

at Math Is Fun

Understanding 2D Translation

an

Understanding 3D Translation

by Roger Germundsson,

The Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hoste ...

.
References

*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/jmathbTransformations of Graphs: Horizontal Translations

(2006, January 1). BioMath: Transformation of Graphs. Retrieved April 29, 2014 {{DEFAULTSORT:Translation (Geometry) Euclidean symmetries Elementary geometry Transformation (function) Functions and mappings