Editing Helicoid

{{short description|Mathematical shape}}
[[Image:Helicoid.svg|right|thumb|350px|A helicoid with ''α''&nbsp;=&nbsp;1, −1&nbsp;≤&nbsp;''ρ''&nbsp;≤&nbsp;1 and −{{pi}}&nbsp;≤&nbsp;''θ''&nbsp;≤&nbsp;{{pi}}.]]

The '''helicoid''', also known as '''helical surface''', is a smooth [[Surface (differential geometry)|surface]] embedded in [[three-dimensional space]]. It is the surface traced by an infinite line that is simultaneously being rotated and lifted along its [[Rotation around a fixed axis|fixed]] axis of rotation. It is the third [[minimal surface]] to be known, after the [[Plane (geometry)|plane]] and the [[catenoid]].

==Description==
It was described by [[Euler]] in 1774 and 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.

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 surface]]s.<ref>{{Cite journal |last=Catalan |first=Eugène |date=1842 |title=Sur les surfaces réglées dont l'aire est un minimum |url=http://www.numdam.org/item/JMPA_1842_1_7__203_0.pdf |journal=Journal de mathématiques pures et appliquées |language=fr |volume=7 |pages=203 - 211}}</ref><ref>''Elements of the Geometry and Topology of Minimal Surfaces in Three-dimensional Space''
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> 

A helicoid is also a [[Translation surface (differential geometry)|translation surface]] in the sense of differential geometry.

The helicoid and the [[catenoid]] are parts of a family of helicoid-catenoid minimal surfaces.

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]]:
:<math> x = \rho \cos (\alpha \theta), \ </math>
:<math> y = \rho \sin (\alpha \theta), \ </math>
:<math> z = \theta, \ </math>
where {{math|''ρ''}} and {{math|''θ''}} range from negative [[infinity]] to [[positive number|positive]] infinity, while {{math|''α''}} is a constant. If {{math|''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 \alpha /(1+ \alpha^2 \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]].

The helicoid is [[homeomorphism|homeomorphic]] to the plane <math> \mathbb{R}^2 </math>.  To see this, let {{math|''α''}} decrease [[continuous function|continuous]]ly from its given value down to [[0 (number)|zero]].  Each intermediate value of {{math|''α''}} will describe a different helicoid, until {{math|1=''α'' = 0}} is reached and the helicoid becomes a vertical [[plane (mathematics)|plane]].

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.

If a helicoid of radius {{math|''R''}} revolves by an angle of {{math|''θ''}} around its axis while rising by a height {{math|''h''}}, the area of the surface is given by<ref>{{MathWorld|id=Helicoid|access-date=2020-06-08}}</ref>

:<math>\frac{\theta}{2} \left[R \sqrt{R^2+c^2}+c^2 \ln \left(\frac{R  + \sqrt{R^2+c^2}} c\right) \right],
\ c = \frac{h}{\theta}.</math>

==Helicoid and catenoid==
[[File:Helicatenoid.gif|thumb|256px|Animation showing the local isometry of a helicoid segment and a catenoid segment.]]
The helicoid and the [[catenoid]] are locally isometric surfaces; see [[Catenoid#Helicoid transformation]].

==See also==
*[[Generalized helicoid]]
*[[Dini's surface]]
*[[Right conoid]]
*[[Ruled surface]]

==Notes==
<references />

==External links==
{{commonscat|Helicoids}}
* {{springer|title=Helicoid|id=p/h046880}}
* [http://www.princeton.edu/~rvdb/WebGL/helicoid.html WebGL-based Interactive 3D Helicoid]

{{Minimal surfaces}}

[[Category:Geometric shapes]]
[[Category:Minimal surfaces]]
[[Category:Surfaces]]