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In [[geometry]], a '''triangulation''' is a subdivision of a geometric object into [[simplex|simplices]]. In particular, in the plane it is a subdivision into [[triangle]]s, hence the name.  Triangulation of a three-dimensional volume would involve subdividing it into [[tetrahedra]] ("pyramids" of various shapes and sizes) packed together.  
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


In most instances, the triangles of a triangulation are required to meet edge-to-edge and vertex-to-vertex.
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.


Different types of triangulation may be defined, depending both on what geometric object is to be subdivided and on how the subdivision is determined.
Registered users will be able to choose between the following three rendering modes:
* A triangulation ''T'' of <math>\mathbb{R}^{n+1}</math> is a subdivision of <math>\mathbb{R}^{n+1}</math> into (''n''&nbsp;+&nbsp;1)-dimensional simplices such that any two simplices in ''T'' intersect in a common face (a simplex of any lower dimension) or not at all, and any [[bounded set]] in <math>\mathbb{R}^{n+1}</math> intersects only [[finite set|finite]]ly many simplices in ''T''. That is, it is a locally finite [[simplicial complex]] that covers the entire space.
* A [[point set triangulation]], i.e., a triangulation of a [[discrete space|discrete]] set of points <math>P\subset\mathbb{R}^{n+1}</math>, is a subdivision of the [[convex hull]] of the points into simplices such that any two simplices intersect in a common face or not at all and such that the set of vertices of the simplices coincides with <math>P</math>. Frequently used and studied point set triangulations include the [[Delaunay triangulation]] (for points in general position, the set of simplices that are circumscribed by an open ball that contains no input points) and the [[minimum-weight triangulation]] (the point set triangulation minimizing the sum of the edge lengths).
* In [[cartography]], a [[triangulated irregular network]] is a point set triangulation of a set of two-dimensional points together with elevations for each point. Lifting each point from the plane to its elevated height lifts the triangles of the triangulation into three-dimensional surfaces, which form an approximation of a three-dimensional landform.
* A [[polygon triangulation]] is a subdivision of a given [[polygon]] into triangles meeting edge-to-edge, again with the property that the set of triangle vertices coincides with the set of vertices of the polygon. Polygon triangulations may be found in [[linear time]] and form the basis of several important geometric algorithms, including a simple solution to the [[art gallery problem]]. The [[constrained Delaunay triangulation]] is an adaptation of the Delaunay triangulation from point sets to polygons or, more generally, to [[planar straight-line graph]]s.
* A [[Surface triangulation|triangulation of a surface]] consists of a net of triangles with points on a given surface covering the surface partly or totally.
* In the [[finite element method]], triangulations are often used as the mesh underlying a computation. In this case, the triangles must form a subdivision of the domain to be simulated, but instead of restricting the vertices to input points, it is allowed to add additional [[Steiner point]]s as vertices. In order to be suitable as finite element meshes, a triangulation must have well-shaped triangles, according to criteria that depend on the details of the finite element simulation; for instance, some methods require that all triangles be right or acute, forming [[nonobtuse mesh]]es. Many meshing techniques are known, including [[Delaunay refinement]] algorithms such as [[Chew's second algorithm]] and [[Ruppert's algorithm]].
* In more general topological spaces, [[Triangulation (topology)|triangulations]] of a space generally refer to simplicial complexes that are [[homeomorphic]] to the space.


The concept of a triangulation may also be generalized somewhat to subdivisions into shapes related to triangles. In particular, a [[pseudotriangulation]] of a point set is a partition of the convex hull of the points into pseudotriangles, polygons that like triangles have exactly three convex vertices. As in point set triangulations, pseudotriangulations are required to have their vertices at the given input points.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


==External links==
<!--'''PNG''' (currently default in production)
* {{mathworld | urlname = SimplicialComplex | title = Simplicial complex}}
:<math forcemathmode="png">E=mc^2</math>
* {{mathworld | urlname = Triangulation | title = Triangulation}}


[[Category:Triangulation (geometry)| ]]
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->
 
<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].
 
==Demos==
 
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
 
 
* 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:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]
 
*[[Inputtypes|Inputtypes (private Wikis only)]]
*[[Url2Image|Url2Image (private Wikis only)]]
==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 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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