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In
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, a translation is a
geometric transformation In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often b ...
that moves every point of a figure, shape or space by the same
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
in a given direction. A translation can also be interpreted as the addition of a constant
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
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, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
. In a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, any translation is an
isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' ...
.


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 is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
of a subset A under the function 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 (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, 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 is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
. Often, vertical translations are considered for the
graph of a function In mathematics, the graph of a 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 and thus form a subs ...
. If ''f'' is any function of ''x'', then the graph of the function ''f''(''x'') + ''c'' (whose values are given by adding a constant ''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, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
s of a function all differ from each other by a
constant of integration In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connecte ...
and are therefore vertical translates of each other. In function graphing, a horizontal translation is a transformation 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 In mathematics, a parent function is the simplest function of a family of functions that preserves the definition (or shape) of the entire family. For example, for the family of quadratic functions having the general form : y = ax^2 + bx + c\,, th ...
''f''(''x'') and a constant ''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 is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
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, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
''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 Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
. 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 most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
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 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 a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
, 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(\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, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
of
Euclidean group In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations ...
E(n) . The
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...
of E(n) by \mathbb is isomorphic to the
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
O(n): :E(n)/\mathbb\cong O(n) Because translation is
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, the translation group is
abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...
. There are an infinite number of possible translations, so the translation group is an infinite group. In the
theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...
, due to the treatment of space and time as a single
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
, translations can also refer to changes in the time coordinate. For example, the
Galilean group In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rota ...
and the
Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
include translations with respect to time.


Lattice groups

One kind of
subgroup In group theory, a branch of mathematics, given a 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, ''H'' is a subgroup ...
of the three-dimensional translation group are the lattice groups, which are infinite groups, 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 of objects, together with a set of operations that can be applied to ...
generates the entire group.


Matrix representation

A translation is an
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generall ...
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 solut ...
using
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometr ...
to represent a translation of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
with
matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
: 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 most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
\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, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, 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 In mathematics, a translation of axes in two dimensions is a mapping from an ''xy''-Cartesian coordinate system to an ''x'y-Cartesian coordinate system in which the ''x axis is parallel to the ''x'' axis and ''k'' units away, and the ''y ...
.


Translational symmetry

An object that looks the same before and after translation is said to have
translational symmetry In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equati ...
. A common example is a
periodic function A periodic function is a 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 throughout science to des ...
, which is an
eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of a translation operator.


Applications


Vehicle dynamics

For describing
vehicle dynamics For motorized vehicles, such as automobiles, aircraft, and watercraft, vehicle dynamics is the study of vehicle motion, e.g., how a vehicle's forward movement changes in response to driver inputs, propulsion system outputs, ambient conditions, air ...
(or movement of any
rigid body In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
), including ship dynamics and aircraft dynamics, it is common to use a mechanical model consisting of six
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
, which includes translations along three reference axes, as well as rotations about those three axes. These translations are often called: * Surge, translation along the longitudinal axis (forward or backwards) * Sway, translation along the transverse axis (from side to side) * Heave, translation along the
vertical axis A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
(to move up or down). The corresponding rotations are often called: * roll, about the longitudinal axis * pitch, about the transverse axis * yaw, about the vertical axis.


See also

*
Advection In the field of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is al ...
*
Parallel transport In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent b ...
*
Rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \ ...
*
Scaling (geometry) In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a '' scale factor'' that is the same in all directions. The result of uniform scaling is simil ...
* Transformation matrix *
Translational symmetry In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equati ...


External links


Translation Transform
at
cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a 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 Electronics and Math ...

Geometric Translation (Interactive Animation)
at Math Is Fun
Understanding 2D Translation
an
Understanding 3D Translation
by Roger Germundsson, The Wolfram Demonstrations Project.


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