Translation (geometry)

Translation (geometry)

In Euclidean geometry, a translation is moving every point a constant distance in a specified direction. It is one of the rigid motions (other rigid motions include rotation and reflection). A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. A translation operator is an operator T_mathbf{delta} such that T_mathbf{delta} f(mathbf{v}) = f(mathbf{v}+mathbf{delta}).

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

If "T" is a translation, then the image of a subset "A" under the function "T" is the translate of "A" by "T". The translate of "A" by "T"v is often written "A" + v.

In an Euclidean space, any translation is an isometry. The set of all translations forms the translation group "T", which is isomorphic to the space itself, and a normal subgroup of Euclidean group "E"("n" ). The quotient group of "E"("n" ) by "T" is isomorphic to the orthogonal group "O"("n" )::"E"("n" ) "/ T" ≅ "O"("n" ).

Matrix representation

Since a translation is an affine transformation but not a linear transformation, homogeneous coordinates 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 = ("w""x", "w""y", "w""z") using 4 homogeneous coordinates as w = ("w""x", "w""y", "w""z", 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:

: T_{mathbf{v = egin{bmatrix}1 & 0 & 0 & v_x \0 & 1 & 0 & v_y \0 & 0 & 1 & v_z \0 & 0 & 0 & 1 end{bmatrix}. !

As shown below, the multiplication will give the expected result:: T_{mathbf{v mathbf{p} =egin{bmatrix}1 & 0 & 0 & v_x \0 & 1 & 0 & v_y \0 & 0 & 1 & v_z \0 & 0 & 0 & 1end{bmatrix}egin{bmatrix}p_x \ p_y \ p_z \ 1end{bmatrix}=egin{bmatrix}p_x + v_x \ p_y + v_y \ p_z + v_z \ 1end{bmatrix}= mathbf{p} + mathbf{v} . !

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

Similarly, the product of translation matrices is given by adding the vectors:: T_{mathbf{uT_{mathbf{v = T_{mathbf{u}+mathbf{v . ! Because addition of vectors is commutative, multiplication of translation matrices is therefore also commutative (unlike multiplication of arbitrary matrices).

See also

* Translation (physics)
* Translational symmetry
* Transformation matrix

External links

* [ Translation Transform] at cut-the-knot
* [ Geometric Translation (Interactive Animation)] at Math Is Fun
* [ Understanding 2D Translation] and [ Understanding 3D Translation] by Roger Germundsson, The Wolfram Demonstrations Project.

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Translation (disambiguation) — Translation, translate, or translator may refer to: * translation, conversion of text from one language to another * technical translation, translation of technical texts from one language to anotherIn science and mathematics: * translation… …   Wikipedia

  • Translation (physics) — In physics, translation is movement that changes the position of an object, as opposed to rotation. For example, according to Whittaker:cite book |title=A Treatise on the Analytical Dynamics of Particles and Rigid Bodies |author=Edmund Taylor… …   Wikipedia

  • Translation operator — The translation operator can refer to these things:* An operator which effects a geometric translation. See Translation (geometry) * An alternative name for the displacement operator in quantum optics …   Wikipedia

  • geometry — /jee om i tree/, n. 1. the branch of mathematics that deals with the deduction of the properties, measurement, and relationships of points, lines, angles, and figures in space from their defining conditions by means of certain assumed properties… …   Universalium

  • Translation plane — In mathematics, a translation plane is a particular kind of projective plane, as considered as a combinatorial object. [Projective Planes [ pjc/pps/pps2.pdf On projective planes] ] In a projective plane, scriptstyle p… …   Wikipedia

  • Geometry shader — Shader Un shader (anglais, du verbe to shade : ombrager ou estomper, nuancer) est un programme[Quoi ?] utilisé en image de synthèse pour paramétrer une partie du processus de rendu réalisé par une carte graphique ou un moteur de rendu… …   Wikipédia en Français

  • Vertical translation — In function graphing, a vertical translation is a related graph which, for every point ( x , y ); has a y value which differs from another graph, by exactly some constant c . For example, the antiderivatives of a family are vertical translations… …   Wikipedia

  • History of geometry — Geometry (Greek γεωμετρία ; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre modern mathematics, the other being the study of numbers. Classic geometry… …   Wikipedia

  • Euclidean geometry — geometry based upon the postulates of Euclid, esp. the postulate that only one line may be drawn through a given point parallel to a given line. [1860 65] * * * Study of points, lines, angles, surfaces, and solids based on Euclid s axioms. Its… …   Universalium

  • Molecular geometry — Geometry of the water molecule Molecular geometry or molecular structure is the three dimensional arrangement of the atoms that constitute a molecule. It determines several properties of a substance including its reactivity, polarity, phase of… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”