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{{Original research|date=June 2010}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


[[Image:Hyperbolic coordinates.svg|thumb|400px|right|Hyperbolic coordinates plotted on the Cartesian plane: ''u'' in blue and ''v'' in red.]]
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.


In [[mathematics]], '''hyperbolic coordinates''' are a method of locating points in Quadrant I of the [[Cartesian plane]]{{Why?|date=May 2010}}
Registered users will be able to choose between the following three rendering modes:


:<math>\{(x, y) \ :\  x > 0,\ y > 0\ \} = Q\ \!</math >.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


Hyperbolic coordinates take values in the hyperbolic plane defined as:
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


:<math>HP = \{(u, v) : u \in \mathbb{R}, v > 0 \}</math>.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


These coordinates in ''HP'' are useful for studying logarithmic comparisons of [[direct proportion]] in ''Q'' and measuring deviations from direct proportion.
<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].


For <math>(x,y)</math> in <math>Q</math> take
==Demos==


:<math>u = -\frac{1}{2} \ln \left( \frac{y}{x} \right)</math>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


and


:<math>v = \sqrt{xy}</math>.
* 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.


Sometimes the parameter <math>u</math> is called [[hyperbolic angle]] and v the [[geometric mean]].
==Test pages ==


The inverse mapping is
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]]


:<math>x = v e^u ,\quad y = v e^{-u}</math>.
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
This is a [[continuous mapping]], but not an [[analytic function]].
==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 .
==Quadrant model of hyperbolic geometry==
 
The correspondence
 
:<math>Q \leftrightarrow HP</math>
 
affords the [[hyperbolic geometry]] structure to ''Q'' that is erected on ''HP'' by [[hyperbolic motion]]s. The ''hyperbolic lines'' in ''Q'' are [[Line (mathematics)#Ray|rays]] from the origin or [[petal]]-shaped [[curve]]s leaving and re-entering the origin. The left-right shift in ''HP'' corresponds to a [[squeeze mapping]] applied to ''Q''. Note that hyperbolas in ''Q'' do ''not'' represent [[geodesic]]s in this model.
 
If one only considers the [[Euclidean topology]] of the plane and the topology inherited by ''Q'', then the lines bounding ''Q'' seem close to ''Q''. Insight from the [[metric space]] ''HP'' shows that the [[open set]] ''Q'' has only the [[origin (mathematics)|origin]] as boundary when viewed as the quadrant model of the hyperbolic plane. Indeed, consider rays from the origin in ''Q'', and their images, vertical rays from the boundary ''R'' of ''HP''. Any point in ''HP'' is an infinite distance from the point ''p'' at the foot of the perpendicular to ''R'', but a sequence of points on this perpendicular may tend in the direction of ''p''. The corresponding sequence in ''Q'' tends along a ray toward the origin. The old Euclidean boundary of ''Q'' is irrelevant to the quadrant model.
 
==Applications in physical science==
Physical unit relations like:
* ''V'' = ''I R''  : [[Ohm's law]]
* ''P'' = ''V I''  : [[Electrical power]]
* ''P V'' = ''k T''  :  [[Ideal gas law]]
* ''f'' λ = ''c'' : [[Sine wave]]s
all suggest looking carefully at the quadrant. For example, in [[thermodynamics]] the [[isothermal process]] explicitly follows the hyperbolic path and [[work (thermodynamics)|work]] can be interpreted as a hyperbolic angle change. Similarly, an [[isobaric process#Variable density viewpoint|isobaric process]] may trace a hyperbola in the quadrant of absolute temperature and gas density.
 
For hyperbolic coordinates in the [[Theory of relativity]] see the History section below.
 
==Statistical applications==
*Comparative study of [[population density]] in the quadrant begins with selecting a reference nation, region, or [[urban density|urban]] area whose population and area are taken as  the point (1,1).
*Analysis of the [[legislator|elected representation]] of regions in a [[representative democracy]] begins with selection of a standard for comparison: a particular represented group, whose magnitude and slate magnitude (of representatives) stands at (1,1) in the quadrant.
 
==Economic applications==
There are many natural applications of hyperbolic coordinates in [[economics]]:
* Analysis of currency [[exchange rate]] fluctuation:
The unit currency sets <math>x = 1</math>. The price currency corresponds to <math>y</math>. For
 
:<math>0 < y < 1</math>
 
we find <math>u > 0</math>, a positive hyperbolic angle. For a ''fluctuation'' take a new price
 
:<math>0 < z < y</math>.
 
Then the change in ''u'' is:
 
:<math>\Delta u = \frac{1}{2} \log \left( \frac{y}{z} \right)</math>.
 
Quantifying exchange rate fluctuation through hyperbolic angle provides an objective, symmetric, and consistent [[measure (mathematics)|measure]]. The quantity <math>\Delta u</math> is the length of the left-right shift in the hyperbolic motion view of the currency fluctuation.
* Analysis of inflation or deflation of prices of a [[basket of consumer goods]].
* Quantification of change in marketshare in [[duopoly]].
* Corporate [[stock split]]s versus stock buy-back.
 
==History==
While the geometric mean is an ancient concept, the hyperbolic angle is contemporary with the development of [[logarithm]], the latter part of the seventeenth century. [[Gregoire de Saint-Vincent]], [[Marin Mersenne]], and [[Alphonse Antonio de Sarasa]] evaluated the quadrature of the hyperbola as a function having properties now familiar for the logarithm. The exponential function, the hyperbolic sine, and the hyperbolic cosine followed. As [[complex function]] theory referred to [[infinite series]] the circular functions sine and cosine seemed to absorb the hyperbolic sine and cosine as depending on an imaginary variable. In the nineteenth century [[biquaternion]]s came into use and exposed the alternative complex plane called [[split-complex number]]s where the hyperbolic angle is raised to a level equal to the classical angle. In English literature biquaternions were used to model [[spacetime]] and show its symmetries. There the hyperbolic angle parameter came to be called [[rapidity]]. For relativists, examining the quadrant as the possible future between oppositely directed photons, the geometric mean parameter is [[time|temporal]].
 
In relativity the focus is on the 3-dimensional [[hypersurface]] in the future of spacetime where various velocities arrive after a given [[proper time]]. Scott Walter<ref>Walter (1999) page 6</ref> explains that in November 1907 [[Herman Minkowski]] alluded to a well-known three-dimensional hyperbolic geometry while speaking to the Göttingen Mathematical Society, but not to a four-dimensional one.<ref>Walter (1999) page 8</ref>
In tribute to [[Wolfgang Rindler]], the author of the standard introductory university-level textbook on relativity, hyperbolic coordinates of spacetime are called [[Rindler coordinates]].
 
==References==
<references/>
*David Betounes (2001) ''Differential Equations: Theory and Applications'', page 254, Springer-TELOS, ISBN 0-387-95140-7 .
*Scott Walter (1999). [http://www.univ-nancy2.fr/DepPhilo/walter/papers/nes.pdf "The non-Euclidean style of Minkowskian relativity"]. Chapter 4 in: Jeremy J. Gray (ed.), ''The Symbolic Universe: Geometry and Physics 1890-1930'', pp.&nbsp;91–127. [[Oxford University Press]]. ISBN 0-19-850088-2.
 
{{Orthogonal coordinate systems}}
 
[[Category:Coordinate systems]]
[[Category:Hyperbolic geometry]]
 
[[ar:نظام إحداثيات قطعي زائدي]]
[[pt:Coordenadas hiperbólicas]]
[[zh:雙曲坐標系]]

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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