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[[Image:Helicoid.svg|right|thumb|350px|A helicoid with α=1, -1≤ρ≤1 and -π≤θ≤π.]]
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


The '''helicoid''', after the [[Plane (geometry)|plane]] and the [[catenoid]], is the third [[minimal surface]] to be known. It was first discovered by [[Jean Baptiste Meusnier]] in 1776. Its [[Nomenclature|name]] derives from its similarity to the [[helix]]: for every [[Point (geometry)|point]] on the helicoid there is a helix contained in the helicoid which passes through that point. Since it is considered that the planar range extends through negative and positive infinity, close observation shows the appearance of two parallel or mirror planes in the sense that if the slope of one plane is traced, the co-plane can be seen to be bypassed or skipped, though in actuality the co-plane is also traced from the opposite perspective.
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.


The helicoid is also a [[ruled surface]] (and a [[right conoid]]), meaning that it is a trace of a line. Alternatively, for any point on the surface, there is a line on the surface passing through it.  Indeed, [[Eugène Charles Catalan|Catalan]] proved in 1842 that the helicoid and the plane were the only ruled minimal surfaces.<ref>Elements of the Geometry and Topology of Minimal Surfaces in Three-dimensional Space
Registered users will be able to choose between the following three rendering modes:
By [[A. T. Fomenko]], A. A. Tuzhilin
Contributor A. A. Tuzhilin
Published by AMS Bookstore, 1991
ISBN 0-8218-4552-7, ISBN 978-0-8218-4552-3, p.33</ref>


The helicoid and the [[catenoid]] are parts of a family of helicoid-catenoid minimal surfaces.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


The helicoid is shaped like [[Archimedes' screw]], but extends infinitely in all directions. It can be described by the following [[parametric equation]]s in [[Cartesian coordinates]]:
<!--'''PNG'''  (currently default in production)
:<math> x = \rho \cos (\alpha \theta), \ </math>
:<math forcemathmode="png">E=mc^2</math>
:<math> y = \rho \sin (\alpha \theta), \ </math>
:<math> z = \theta, \ </math>
where ''ρ'' and ''θ'' range from negative [[infinity]] to [[positive number|positive]] infinity, while ''α'' is a constant.  If ''α'' is positive then the helicoid is right-handed as shown in the figure; if negative then left-handed.


The helicoid has [[principal curvature]]s <math>\pm 1/(1+ \rho ^2) \ </math>.  The sum of these quantities gives the [[mean curvature]] (zero since the helicoid is a [[minimal surface]]) and the product gives the [[Gaussian curvature]].
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


The helicoid is [[homeomorphism|homeomorphic]] to the plane <math> \mathbb{R}^2 </math>. To see this, let alpha decrease [[continuous function|continuous]]ly from its given value down to [[0 (number)|zero]]. Each intermediate value of ''α'' will describe a different helicoid, until ''α = 0'' is reached and the helicoid becomes a vertical [[plane (mathematics)|plane]].
<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].


Conversely, a plane can be turned into a helicoid by choosing a line, or ''axis'', on the plane then twisting the plane around that axis.
==Demos==


For example, if we take h as the maxium value at z and R the radium, the area of the surface is π[R√(R²+h²)+h2*ln(R +√(R²+h²)/h)]
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


==Helicoid and catenoid==
[[File:Helicatenoid.gif|thumb|256px|Animation showing the transformation of a helicoid into a catenoid.]]
The helicoid and the [[catenoid]] are locally isometric surfaces, see discussion there.


==See also==
* accessibility:
*[[Dini's surface]]
** 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]]
*[[Right conoid]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
*[[Ruled surface]]
** [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.


==Notes==
==Test pages ==
<references />


==External links==
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[http://chamicewicz.com/p5/helicoid/ Interactive 3D Helicoid plotter using Processing (with code)]
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


[[Category:Geometric shapes]]
*[[Inputtypes|Inputtypes (private Wikis only)]]
[[Category:Minimal surfaces]]
*[[Url2Image|Url2Image (private Wikis only)]]
[[Category:Surfaces]]
==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 .
[[ar:سطح حلزوني]]
[[de:Minimalfläche#Die Wendelfläche]]
[[et:Helikoid]]
[[fr:Hélicoïde]]
[[it:Elicoide]]
[[hu:Csavarfelület]]
[[nl:Helicoïde]]
[[pl:Helikoida]]
[[ru:Геликоид]]
[[sl:Helikoid]]
[[uk:Гелікоїд]]

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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