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In [[mathematics]], a '''translation plane''' is a particular kind of [[projective plane]], as considered as a combinatorial object.<ref>Projective Planes [http://www.maths.qmul.ac.uk/~pjc/pps/pps2.pdf On projective planes]</ref>
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


In a projective plane, <math>\scriptstyle p</math> represents a point, and <math>\scriptstyle L</math> represents a line. A central [[collineation]] with center <math>\scriptstyle p</math> and axis <math>\scriptstyle L</math> is a collineation fixing every point on <math>\scriptstyle L</math> and every line through <math>\scriptstyle p</math>. It is called an "elation" if <math>\scriptstyle p</math> is on <math>\scriptstyle L</math>, otherwise it is called a "homology". The central collineations with centre <math>\scriptstyle p</math> and axis <math>\scriptstyle L</math> form a group.<ref>Geometry [http://www.math.uni-kiel.de/geometrie/klein/math/geometry/translation.html Translation Plane] Retrieved on June 13, 2007</ref>
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A projective plane <math>\scriptstyle \Pi</math> is called a translation plane if there exists a line <math>\scriptstyle L</math> such that the group of elations with axis <math>\scriptstyle L</math> is transitive on the affine plane Π<sub>l</sub> (the [[Affine geometry|affine]] derivative of Π).
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== Relationship to spreads ==
'''MathML'''
Translation planes are related to spreads in finite projective spaces by the André/Bruck-Bose construction.<ref>{{cite web|url=http://www-ma4.upc.es/~simeon/bblpsympspread.pdf|title=Symplectice Spreads|last=Ball|first=Simeon|author2=John Bamberg |author3=Michel Lavrauw |author4=Tim Penttila |date=2003-09-15|publisher=[[Polytechnic University of Catalonia]]|accessdate=2008-10-08}}</ref> A spread of <math>\scriptstyle PG(3, q) </math> is a set of ''q''<sup>2</sup>&nbsp;+&nbsp;1 lines, with no two intersecting. Equivalently, it is a partition of the points of <math>\scriptstyle PG(3, q) </math> into lines.
:<math forcemathmode="mathml">E=mc^2</math>


Given a spread <math>\scriptstyle S</math> of <math>\scriptstyle PG(3, q) </math>, the André/Bruck-Bose construction<sup>1</sup> produces a translation plane <math>\scriptstyle \pi(S)</math> of order ''q''<sup>2</sup> as follows: Embed <math>\scriptstyle PG(3, q) </math> as a hyperplane of <math>\scriptstyle PG(4, q) </math>. Define an incidence structure <math>\scriptstyle A(S)</math> with "points," the points of <math>\scriptstyle PG(4, q) </math> not on <math>\scriptstyle PG(3, q) </math> and "lines" the planes of <math>\scriptstyle PG(4, q) </math> meeting <math>\scriptstyle PG(3, q) </math> in a line of <math>\scriptstyle S</math>. Then <math>\scriptstyle A(S)</math> is a translation affine plane of order ''q''<sup>2</sup>. Let <math>\scriptstyle \pi(S)</math> be the projective completion of <math>\scriptstyle A(S)</math>.<ref>{{cite book
<!--'''PNG'''  (currently default in production)
  | last =André  | first =Johannes  | authorlink =  | title = Über nicht-Dessarguessche Ebenen mit transitiver Translationsgruppe  | publisher =  | year =1954  | location =  | pages =156–186  | url =  | doi =  | id =  }}</ref><ref>{{cite book
:<math forcemathmode="png">E=mc^2</math>
  | last =Bruck  | first = R. H. | authorlink = Richard Bruck|author2=R. C. Bose  | title = The Construction of Translation Planes from Projective Spaces  | publisher =  | year =1964  | location =  | pages = 85–102  | url =  | doi =  | id =  }}</ref>


==References==
'''source'''
{{Reflist}}
:<math forcemathmode="source">E=mc^2</math> -->


==Further reading==
<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].
* Mauro Biliotti, Vikram Jha, Norman L. Johnson (2001) ''Foundations of Translation Planes'', [[Marcel Dekker]] ISBN 0-8247-0609-9 .


==External links==
==Demos==
*[http://www.library.tuiasi.ro/ipm/vol13no34/pure.html  Foundations_of_Translation_Planes]
*[http://www-math.ucdenver.edu/~wcherowi/courses/m6221/pglc3a.html Lecture Notes on Projective Geometry]
*[http://mellinger.umwblogs.org/publications/ Publications of Keith Mellinger]


{{DEFAULTSORT:Translation Plane}}
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
[[Category:Projective geometry]]
 
 
* 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.
 
==Test pages ==
 
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
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==Bug reporting==
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 .

Latest revision as of 22:52, 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

E=mc2


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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