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In
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 ...

coordinate system
. 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 \mathbf is a fixed vector, known as the ''translation vector'', and \mathbf is the initial position of some object, then the translation function T_ will work as T_(\mathbf)=\mathbf+\mathbf. If T is a translation, then the
image 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+\mathbf .


Horizontal and vertical translations

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 figures. A mat ...

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 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 ...

graph of a function
. 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 ...

constant of integration
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'') ...

Cartesian plane
it is natural to introduce translations in this type of notation: :(x,y)\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 the
parabola 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 ...

parabola
''y'' = ''x''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.


Application in classical physics

In
classical 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 ...

rotation
. 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) \to (x+\Delta x,y+\Delta y, z+\Delta z) where (\Delta x,\ \Delta y,\ \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 (\Delta x,\ \Delta y,\ \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 ...

time
coordinate is considered to be a translation.


As an operator

The translation operator turns a function of the original position, f(\mathbf), into a function of the final position, f(\mathbf+\mathbf). In other words, T_\mathbf is defined such that T_\mathbf f(\mathbf) = f(\mathbf+\mathbf). This operator is more abstract than a function, since T_\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 \mathbb , which is isomorphic to the space itself, and a
normal 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 \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)/\mathbb\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 of
subgroup 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 an
affine 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 ...

matrix multiplication
: Write the 3-dimensional vector \mathbf=(v_x, v_y, v_z) using 4 homogeneous coordinates as \mathbf=(v_x, v_y, v_z, 1) .Richard Paul, 1981
Robot 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 ...
\mathbf , each homogeneous vector \mathbf (written in homogeneous coordinates) can be multiplied by this translation matrix: : T_ = \begin 1 & 0 & 0 & v_x \\ 0 & 1 & 0 & v_y \\ 0 & 0 & 1 & v_z \\ 0 & 0 & 0 & 1 \end As shown below, the multiplication will give the expected result: : T_ \mathbf = \begin 1 & 0 & 0 & v_x \\ 0 & 1 & 0 & v_y\\ 0 & 0 & 1 & v_z\\ 0 & 0 & 0 & 1 \end \begin p_x \\ p_y \\ p_z \\ 1 \end = \begin p_x + v_x \\ p_y + v_y \\ p_z + v_z \\ 1 \end = \mathbf + \mathbf The inverse of a translation matrix can be obtained by reversing the direction of the vector: : T^_ = T_ . \! Similarly, the product of translation matrices is given by adding the vectors: : T_T_ = T_ . \! 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 have
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 ...
. 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 ...

periodic function
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 ...

Parallel transport
*
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 ...

Transformation matrix
*
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/jmathb
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