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In [[mathematics]], the term '''''hyperbolic triangle''''' has more than one meaning.
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


[[Image:Uniform tiling 73-t2.png|thumb|right|200px|A tiling of the hyperbolic plane with hyperbolic triangles – the [[order-7 triangular tiling]].]]
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.


==Hyperbolic geometry==
Registered users will be able to choose between the following three rendering modes:
In [[hyperbolic geometry]], a '''hyperbolic triangle''' is a figure in the hyperbolic plane, analogous to a triangle in Euclidean geometry, consisting of three sides and three angles.  The relations among the angles and sides are analogous to those of [[spherical trigonometry]]; they are most conveniently stated if the lengths are measured in terms of a special unit of length analogous to a [[radian]]. In terms of the  [[Gaussian curvature]] ''K'' of the plane this unit is given by


::<math>R=\frac{1}{\sqrt{-K}}.</math>
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


In all the trig formulas stated below the sides ''a'', ''b'', and ''c'' must be measured in this unit.  In a hyperbolic triangle the sum of the angles ''A'', ''B'', ''C'' (respectively opposite to the side with the corresponding letter) is strictly less than a [[straight angle]]. The difference is often called the [[defect (geometry)|defect]] of the triangle.  The area of a hyperbolic triangle is equal to its defect multiplied by the square of&nbsp;''R'':
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


::<math>(\pi-A-B-C) R^2{}{}.\!</math>
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


The corresponding theorem in [[spherical geometry]] is [[Girard's theorem]] first  proven by [[Johann Heinrich Lambert]].
<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].


===Right triangles===
==Demos==


If ''C'' is a right angle then:
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


*The '''sine''' of angle A is the ratio of the '''hyperbolic sine''' of the side opposite the angle to the '''hyperbolic sine''' of the [[hypotenuse]].
:: <math>\sin A=\frac{\textrm{sinh(opposite)}}{\textrm{sinh(hypotenuse)}}=\frac{\sinh a}{\,\sinh c\,}.\,</math>
*The '''cosine''' of angle A is the ratio of the '''hyperbolic tangent''' of the adjacent leg to the '''hyperbolic tangent''' of the hypotenuse.
:: <math>\cos A=\frac{\textrm{tanh(adjacent)}}{\textrm{tanh(hypotenuse)}}=\frac{\tanh b}{\,\tanh c\,}.\,</math>
*The '''tangent''' of angle A is the ratio of the '''hyperbolic tangent''' of the opposite leg to the '''hyperbolic sine''' of the adjacent leg.
:: <math>\tan A=\frac{\textrm{tanh(opposite)}}{\textrm{sinh(adjacent)}}=\frac{\tanh a}{\,\sinh b\,}.\,</math>


The hyperbolic sine, cosine, and tangent are [[hyperbolic functions]] which are analogous to the standard trigonometric functions.
* 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.


===Oblique triangles===
==Test pages ==


Whether ''C'' is a right angle or not, the following relationships hold.
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]]


There is a [[hyperbolic law of cosines|law of cosines]]:
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
:: <math>\cosh c=\cosh a\cosh b-\sinh a\sinh b \cos C,\,</math>
==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 .
its dual:
 
:: <math>\cos C= -\cos A\cos B+\sin A\sin B \cosh c,\,</math>
 
a law of sines:
 
:<math>\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c},</math>
 
and a four-parts formula:
 
:: <math>\cos C\cosh a=\sinh a\coth b-\sin C\cot B.\,</math>
 
===Ideal triangles===
 
If a pair of sides is asymptotic they may be said to form an angle of zero.  In [[projective geometry]], they meet at an '''ideal vertex''' on the circle at infinity.  If all three are vertices are ideal, then the resulting figure is called an '''[[ideal triangle]]'''. An ideal hyperbolic triangle has an angle sum of 0°, a property it has in common with the triangular area in the Euclidean plane bounded by three tangent circles.
 
==Euclidean geometry==
[[Image:Hyperbolic sector.svg|200px|right]]
In the foundations of the [[hyperbolic function]]s sinh, cosh and tanh, a '''hyperbolic triangle''' is a [[right triangle]] in the [[first quadrant of the Cartesian plane]]
:<math>\{(x,y):x,y \in \mathbb R\},</math>
with one [[vertex (geometry)|vertex]] at the origin, base on the diagonal ray ''y''&nbsp;=&nbsp;''x'', and third vertex on the [[hyperbola]]  
 
:<math>xy=1.\,</math>
 
The length of the base of such a triangle is
:<math>\sqrt 2 \cosh u,\,</math>
and the [[altitude (triangle)|altitude]] is
:<math>\sqrt 2 \sinh u,\,</math>
where ''u'' is the appropriate [[hyperbolic angle]].
 
==See also==
* [[Hyperbolic law of cosines]]
* [[Pair of pants]]
* [[Triangle group]]
 
==References==
{{inline|date=November 2011}}
* [[Augustus De Morgan]] (1849) [http://books.google.com/books?id=7UwEAAAAQAAJ Trigonometry and Double Algebra], Chapter VI: "On the connection of common and hyperbolic trigonometry".
*{{citation|first=Wilson|last=Stothers|title=Hyperbolic geometry|url=http://www.maths.gla.ac.uk/~wws/cabripages/hyperbolic/hyperbolic0.html|publisher=[[University of Glasgow]]|year=2000}}, interactive instructional website.
* Svetlana Katok, ''Fuchsian Groups'' (1992), University of Chicago Press, Chicago ISBN 0-226-42583-5 ''(Provides a brief but simple, easily readable review in chapter 1.)''
 
{{DEFAULTSORT:Hyperbolic Triangle}}
[[Category:Hyperbolic geometry]]
 
[[it:Triangolo iperbolico]]
[[pl:Trójkąt asymptotyczny]]
[[pt:Triângulo hiperbólico]]
[[sv:Hyperbolisk triangel]]

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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