Intensive and extensive properties - Revision history

en>Mlweems: correct typo in index (should be j instead of i) for equation that follows from Euler's homogenous function theorem

2015-01-01T17:44:38Z

correct typo in index (should be j instead of i) for equation that follows from Euler's homogenous function theorem

← Older revision		Revision as of 19:44, 1 January 2015
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en>Hhhippo: Reverted 1 edit by 207.35.65.36 (talk) to last revision by Aarruunn. (TW)

2014-02-23T21:50:36Z

Reverted 1 edit by 207.35.65.36 (talk) to last revision by Aarruunn. (TW)

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75.102.112.98 at 16:36, 26 January 2014

2014-01-26T16:36:56Z

← Older revision		Revision as of 18:36, 26 January 2014
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	~~Roberto~~ is ~~what~~'s written to ~~his birth certificate but nonetheless~~ , ~~he never really liked that name~~. ~~Managing people~~ is ~~literally where his primary profit coming in comes~~ from. [http://~~www~~.~~Guardian~~.~~co.uk~~/~~search~~?q=~~Base~~+~~jumping Base jumping~~] is ~~something~~ that ~~she has been doing for months~~. [http://~~Www~~.~~squidoo~~.com/~~search/results~~?q=~~Massachusetts Massachusetts] has always been his everyday living place~~ and ~~his spouse~~ and ~~kids loves it~~. Go to ~~his website~~ to ~~come across out more: http:~~//~~prometeu~~.~~net~~<br><br>~~Visit my blog~~: [http://~~prometeu~~.~~net clash of clans triche~~]		<!-- content from article translation (physics) has been merged with this article {{Merge from\|Translation (physics)\|date=December 2010}}-->
			<!-- added references {{unreferenced\|date=December 2007}}-->
			[[Image:TraslazioneOK.png\|right\|thumb\|A translation moves every point of a figure or a space by the same amount in a given direction.]]
			[[Image:Simx2=traslOK.png\|right\|thumb\|A [[reflection (mathematics)\|reflection]] against an axis followed by a reflection against a second axis parallel to the first one results in a total motion which is a translation.]]

			In [[Euclidean geometry]], a '''translation''' is a function that moves every point a constant distance in a specified direction.<ref>{{cite book\|authors=Osgood, William F. & Graustein, William C.\|title=Plane and solid analytic geometry\|publisher=The Macmillan Company\|year=1921\|page=330\|url=http://books.google.com/books?id=mxOBAAAAMAAJ&pg=PA330}}</ref> A translation can be described as a [[Euclidean group\|rigid motion]]: other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant [[vector space\|vector]] to every point, or as shifting the [[Origin (mathematics)\|origin]] of the [[coordinate system]]. A '''translation operator''' is an [[operator (mathematics)\|operator]] <math>T_\mathbf{\delta}</math> such that <math>T_\mathbf{\delta} f(\mathbf{v}) = f(\mathbf{v}+\mathbf{\delta}).</math>

			If '''v''' is a fixed vector, then the translation ''T''<sub>'''v'''</sub> will work as ''T''<sub>'''v'''</sub>('''p''') = '''p''' + '''v'''.

			If ''T'' is a translation, then the [[image (mathematics)\|image]] of a subset ''A'' under the [[function (mathematics)\|function]] ''T'' is the '''translate''' of ''A'' by ''T''. The translate of ''A'' by ''T''<sub>'''v'''</sub> is often written ''A'' + '''v'''.

			In a [[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 ==<!-- This section is linked from [[Affine transformation]] -->
			A translation is an [[affine transformation]] with ''no'' [[fixed point (mathematics)\|fixed point]]s. Matrix multiplications ''always'' have the [[origin (mathematics)\|origin]] as a fixed point. Nevertheless, there is a common [[workaround]] using [[homogeneous coordinates]] to represent a translation of a [[vector space]] with [[matrix multiplication]]: Write the 3-dimensional vector '''w''' = (''w''<sub>''x''</sub>, ''w''<sub>''y''</sub>, ''w''<sub>''z''</sub>) using 4 homogeneous coordinates as '''w''' = (''w''<sub>''x''</sub>, ''w''<sub>''y''</sub>, ''w''<sub>''z''</sub>, 1).<ref> Richard Paul, 1981, [http://books.google.com/books?id=UzZ3LAYqvRkC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Robot manipulators: mathematics, programming, and control : the computer control of robot manipulators], MIT Press, Cambridge, MA</ref>

			To translate an object by a [[Vector (geometry)\|vector]] '''v''', each homogeneous vector '''p''' (written in homogeneous coordinates) can be multiplied by this '''translation matrix''':

			: <math> T_{\mathbf{v}} =
			\begin{bmatrix}
			1 & 0 & 0 & v_x \\
			0 & 1 & 0 & v_y \\
			0 & 0 & 1 & v_z \\
			0 & 0 & 0 & 1
			\end{bmatrix}
			</math>
			As shown below, the multiplication will give the expected result:
			: <math> T_{\mathbf{v}} \mathbf{p} =
			\begin{bmatrix}
			1 & 0 & 0 & v_x \\
			0 & 1 & 0 & v_y\\
			0 & 0 & 1 & v_z\\
			0 & 0 & 0 & 1
			\end{bmatrix}
			\begin{bmatrix}
			p_x \\ p_y \\ p_z \\ 1
			\end{bmatrix}
			=
			\begin{bmatrix}
			p_x + v_x \\ p_y + v_y \\ p_z + v_z \\ 1
			\end{bmatrix}
			= \mathbf{p} + \mathbf{v} </math>

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

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

			==Translations in physics==
			In [[physics]], '''translation''' (Translational motion) is movement that changes the [[displacement (vector)\|position]] of an object, as opposed to [[rotation]]. For example, according to Whittaker:<ref name=Whittaker>{{cite book \|title=A Treatise on the Analytical Dynamics of Particles and Rigid Bodies \|author=Edmund Taylor Whittaker \|isbn=0-521-35883-3 \|publisher=Cambridge University Press \|year=1988 \|url=http://books.google.com/books?id=epH1hCB7N2MC&pg=PA4&dq=rigid+bodies+translation#PPA1,M1 \|edition=Reprint of fourth edition of 1936 with foreword by William McCrea \|page=1}}</ref>

			{{Quotation\|If a body is moved from one position to another, and if the lines joining the initial and final points of each of the points of the body are a set of parallel straight lines of length ''ℓ'', so that the orientation of the body in space is unaltered, the displacement is called a ''translation parallel to the direction of the lines, through a distance ℓ''. \|E.T. Whittaker: ''A Treatise on the Analytical Dynamics of Particles and Rigid Bodies'', p. 1}}

			A translation is the operation changing the positions of all points ''(x, y, z)'' of an object according to the formula

			:<math>(x,y,z) \to (x+\Delta x,y+\Delta y, z+\Delta z)</math>

			where <math>(\Delta x,\ \Delta y,\ \Delta z)</math> is the same [[Euclidean vector\|vector]] for each point of the object. The translation vector <math>(\Delta x,\ \Delta y,\ \Delta z)</math> common to all points of the object describes a particular type of [[Displacement (vector)\|displacement]] of the object, usually called a ''linear'' displacement to distinguish it from displacements involving rotation, called ''angular'' displacements.

			A translation of space (or time) should not be confused with a translation of an object. Such translations have no [[Fixed point (mathematics)\|fixed points]].

			== See also ==
			* [[Translational symmetry]]
			* [[Transformation matrix]]
			* [[Rotation matrix]]
			* [[Scaling (geometry)]]
			* [[Advection]]

			==External links==
			{{Commons category\|Translation (geometry)}}
			* [http://www.cut-the-knot.org/Curriculum/Geometry/Translation.shtml Translation Transform] at [[cut-the-knot]]
			* [http://www.mathsisfun.com/geometry/translation.html Geometric Translation (Interactive Animation)] at Math Is Fun
			* [http://demonstrations.wolfram.com/Understanding2DTranslation/ Understanding 2D Translation] and [http://demonstrations.wolfram.com/Understanding3DTranslation/ Understanding 3D Translation] by Roger Germundsson, [[The Wolfram Demonstrations Project]].

			==References==
			{{reflist}}

			{{DEFAULTSORT:Translation (Geometry)}}

			[[Category:Euclidean symmetries]]
			[[Category:Transformation (function)]]

en>Incnis Mrsi: Reverted edits by 99.25.218.69 (talk) to last version by 99.245.92.28

2012-08-29T09:15:48Z

Reverted edits by 99.25.218.69 (talk) to last version by 99.245.92.28

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