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Translation (geometry)
 
 
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In Euclidean geometry
Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
, a translation is moving every point a constant distance in a specified direction. It is one of the rigid motion
Euclidean group

In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometry associated with the Euclidean Metric , are called Euclidean moves....
s (other rigid motions include rotation and reflection). A translation can also be interpreted as the addition of a constant vector
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
 to every point, or as shifting the origin
Origin (mathematics)

In mathematics, the origin of a Euclidean space is a special Point , usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space....
 of the coordinate system
Coordinate system

In mathematics and its applications, a coordinate system is a system for assigning an n-tuple of numbers or scalar to each Point in an n-dimensional space....
. A translation operator is an operator
Operator

In mathematics, an operator is a function which operates on another function. Often, an "operator" is a function which acts on functions to produce other functions ; or it may be a generalization of such a function, as in linear algebra, where some of the terminology reflects the origin of the subject in operations on the functions which ar...
  such that

If v is a fixed vector, then the translation Tv will work as Tv(p) = p + v.

If T is a translation, then the image
Image (mathematics)

In mathematics, the image of a set under a given function is the set of all possible function outputs when taking each element of the set, successively, as the function's argument....
 of a subset A under the function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 T is the translate of A by T.






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Encyclopedia


In Euclidean geometry
Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
, a translation is moving every point a constant distance in a specified direction. It is one of the rigid motion
Euclidean group

In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometry associated with the Euclidean Metric , are called Euclidean moves....
s (other rigid motions include rotation and reflection). A translation can also be interpreted as the addition of a constant vector
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
 to every point, or as shifting the origin
Origin (mathematics)

In mathematics, the origin of a Euclidean space is a special Point , usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space....
 of the coordinate system
Coordinate system

In mathematics and its applications, a coordinate system is a system for assigning an n-tuple of numbers or scalar to each Point in an n-dimensional space....
. A translation operator is an operator
Operator

In mathematics, an operator is a function which operates on another function. Often, an "operator" is a function which acts on functions to produce other functions ; or it may be a generalization of such a function, as in linear algebra, where some of the terminology reflects the origin of the subject in operations on the functions which ar...
  such that

If v is a fixed vector, then the translation Tv will work as Tv(p) = p + v.

If T is a translation, then the image
Image (mathematics)

In mathematics, the image of a set under a given function is the set of all possible function outputs when taking each element of the set, successively, as the function's argument....
 of a subset A under the function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 T is the translate of A by T. The translate of A by Tv is often written A + v.

In an Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
, any translation is an isometry
Isometry

In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
. The set of all translations forms the translation group T, which is isomorphic to the space itself, and a normal subgroup
Normal subgroup

In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup. Normal subgroups are important because they can be used to construct quotient groups from a given group ....
 of Euclidean group
Euclidean group

In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometry associated with the Euclidean Metric , are called Euclidean moves....
 E(n ). The quotient group
Quotient group

In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that "collapses" the normal subgroup N to the identity element....
 of E(n ) by T is isomorphic to the orthogonal group
Orthogonal group

In mathematics, the orthogonal group of degree n over a field F is the group of n-by-n orthogonal matrix with entries from F, with the group operation that of matrix multiplication....
 O(n ):
E(n ) / T ? O(n ).


Matrix representation


Since a translation is an affine transformation
Affine transformation

In geometry, an affine transformation or affine map or an affinity between two vector spaces consists of a linear transformation followed by a translation :...
 but not a linear transformation
Linear transformation

In mathematics, a linear map is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication....
, homogeneous coordinates
Homogeneous coordinates

In mathematics, homogeneous coordinates, introduced by August Ferdinand M?bius in his 1827 work Der barycentrische Calc?l, allow affine transformations to be easily represented by a matrix....
 are normally used to represent the translation operator by a matrix and thus to make it linear. Thus we write the 3-dimensional vector w = (wx, wy, wz) using 4 homogeneous coordinates as w = (wx, wy, wz, 1).

To translate an object by a vector v, each homogeneous vector p (written in homogeneous coordinates) would need to be multiplied by this translation matrix:



As shown below, the multiplication will give the expected result:


The inverse of a translation matrix can be obtained by reversing the direction of the vector:


Similarly, the product of translation matrices is given by adding the vectors:
Because addition of vectors is commutative, multiplication of translation matrices is therefore also commutative (unlike multiplication of arbitrary matrices).

See also


  • Translation (physics)
    Translation (physics)

    In physics, translation is movement that changes the displacement of an object, as opposed to rotation. For example, according to Whittaker:...
  • Translational symmetry
    Translational symmetry

    In geometry, a translation "slides" an object by a a: Ta = p + a.In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation....
  • Transformation matrix
    Transformation matrix

    In linear algebra, linear transformations can be represented by matrix . If T is a linear transformation mapping Rn to Rm and x is a column vector with n entries, then...


External links

  • at cut-the-knot
    Cut-the-knot

    Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics....
  • at Math Is Fun
  • and by Roger Germundsson, The Wolfram Demonstrations Project.