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{{Redirect|C-space|the art gallery|C-Space, Beijing}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
{{other uses|PCI configuration space}}


In [[classical mechanics]], the parameters that define the configuration of a system are called ''[[generalized coordinates]],'' and the vector space defined by these coordinates is called the '''configuration space''' of the [[physical system]]. It is often the case that these parameters satisfy mathematical constraints, which means that the set of actual configurations of the system is a manifold in the space of generalized coordinates. This [[manifold]] is called the '''configuration manifold''' of the system.
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


==Configuration spaces in physics==
Registered users will be able to choose between the following three rendering modes:
The configuration space of a single particle moving in ordinary [[Euclidean space|Euclidean 3-space]] is just '''R'''<sup>3</sup>. For ''n'' particles the configuration space is '''R'''<sup>3''n''</sup>, or possibly the subspace where no two positions are equal. More generally, one can regard the configuration space of ''n'' particles moving in a manifold ''M'' as the [[function space]] ''M''<sup>''n''</sup>.


To take account of both position and momenta one moves to the [[cotangent bundle]] of the configuration manifold. This larger manifold is called the [[phase space]] of the system. In short, a configuration space is typically "half" of (see [[Lagrangian distribution]]) a [[phase space]] that is constructed from a function space.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


In [[quantum mechanics]] one [[path integral formulation|formulation]] emphasises 'histories' as configurations.
<!--'''PNG''' (currently default in production)
===Robotics===
:<math forcemathmode="png">E=mc^2</math>
In Robotics, ''configuration space'' generally refers to the set of positions reachable by a robot's [[end-effector]] considered to be a rigid body in three dimensional space.<ref>John J. Craig, '''Introduction to Robotics: Mechanics and Control''', 3rd Ed. Prentice-Hall, 2004</ref> Thus, the positions of the end-effector of a robot can be identified with the group of spatial rigid transformations, often denoted SE(3). 


The joint parameters of the robot are used as generalized coordinates to define its configurations.  The set of joint parameter values is called the ''joint space''.  The robot's [[forward kinematics|forward]] and [[inverse kinematics]] equations define mappings between its configurations and its end-effector positions, or between joint space and configuration space.  Robot [[motion planning]] uses these mappings to find a path in joint space that provides a desired path in the configuration space of the end-effector.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


== Configuration spaces in mathematics ==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
[[File:Moebius Surface 1 Display Small.png|thumb|The configuration space of 2 not necessarily distinct points on the circle is the [[orbifold]] <math>T^2/S_2,</math> which is the [[Möbius strip]].]]
<!-- Deleted image removed: [[File:Topological III by Robert R. Wilson, at Harvard University.JPG|upright|thumb|The configuration space of 3 not necessarily distinct points on the circle <math>T^3/S_3,</math> is the above orbifold.]] -->


In [[mathematics]] a '''configuration space''' refers to a broad family of constructions closely related to the '''[[state space]]''' notion in physics.  The most common notion of '''configuration space''' in mathematics <math>C_n X</math> is the set of ''n''-element subsets of a [[topological space]] <math>X</math>.  This set is given a [[topological space|topology]] by considering it as the [[quotient space|quotient]] <math>C_n X = F_n X / \Sigma_n</math> where <math>F_n X = \{(x_1,\cdots,x_n) \in X^n : x_i \neq x_j \forall \ i \neq j \}</math> and <math>\Sigma_n</math> is the [[symmetric group]] acting by permuting the coordinates of <math>F_n X</math>.  Typically, <math>C_n X</math> is called the configuration space of ''n'' unordered points in <math>X</math> and <math>F_n X</math> is called the configuration space of ''n'' ordered or coloured points in <math>X</math>; the space of ''n'' ordered not necessarily distinct points is simply <math>X^n.</math>
==Demos==


If the original space is a manifold, the configuration space of ''distinct,'' unordered points is also a manifold, while the configuration space of ''not necessarily distinct'' unordered points is instead an [[orbifold]].
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


Configuration spaces are related to [[braid theory]], where the [[braid group]] is considered as the [[fundamental group]] of the space <math>C_n \Bbb R^2</math>.


A configuration space is a type of [[classifying space]] or (fine) [[moduli space]]. In particular, there is a universal bundle <math> \pi\colon E_n\to C_n </math> which is a subbundle of the trivial bundle <math> C_n\times X^n\to C_n</math>, and which has the property that the fiber over each point <math> p\in C_n</math> is the ''n'' element subset of <math> X_n </math> classified by ''p''.
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


The homotopy type of configuration spaces is not [[homotopy invariant]] – for example, that the spaces <math>F_n \Bbb R^m</math> are not homotopic for any two distinct values of <math>m</math>. For instance, <math>F_n\Bbb R</math> is not connected, <math>F_n\Bbb R^2</math> is a <math>K(\pi,1)</math>, and <math>F_n \Bbb R^m</math> is simply connected for <math> m \geq 3</math>.
==Test pages ==


It used to be an open question whether there were examples of ''compact'' manifolds which were homotopic but had non-homotopic configuration spaces: such an example was found only in 2005 by Longini and Salvatore. Their example are two three-dimensional [[lens space]]s, and the configuration spaces of at least two points in them. That these configuration spaces are not homotopic was detected by [[Massey product]]s in their respective universal covers.<ref>{{citation|last1= Salvatore| first1=Paolo| last2=Longoni| first2=Riccardo| title=Configuration spaces are not homotopy invariant| journal= Topology |volume= 44| year=2005|issue= 2|pages=375&ndash;380|doi= 10.1016/j.top.2004.11.002}}</ref>
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


==See also==
*[[Inputtypes|Inputtypes (private Wikis only)]]
*[[Feature space]] (topic in pattern recognition)
*[[Url2Image|Url2Image (private Wikis only)]]
*[[Parameter space]]
==Bug reporting==
*[[Phase space]]
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
*[[State space (physics)]]
 
== References ==
<references/>
 
==External links==
* [http://www.overcomingbias.com/2008/04/conf-space.html Intuitive Explanation of Classical Configuration Spaces].
*''[http://ford.ieor.berkeley.edu/cspace'' Interactive Visualization of the C-space for a Robot Arm with Two Rotational Links] from [[UC Berkeley]].
* [http://www.youtube.com/watch?v=SBFwgR4K1Gk&list=UUswRb5tFvit2fXAiZtwpYuA&index=1&feature=plcp Configuration Space Visualization] from [[Free University of Berlin]]
{{DEFAULTSORT:Configuration Space}}
[[Category:Classical mechanics]]
[[Category:Manifolds]]
[[Category:Topology]]
 
[[ca:Espai de configuració]]
[[cs:Konfigurační prostor]]
[[de:Konfigurationsraum]]
[[es:Espacio de configuración]]
[[fr:Espace de configuration]]
[[ko:짜임새 공간]]
[[it:Spazio delle configurazioni]]
[[nl:Configuratieruimte]]
[[pt:Espaço de configuração]]
[[ru:Пространство конфигураций]]
[[zh:位形空间]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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