View source for Helicoid

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A [[Ruled surface|ruled surface]] described by a straight line that rotates at a constant angular rate around a fixed axis, intersects the axis at a constant angle $\alpha$, and at the same time becomes gradually displaced at a constant rate $k$ along this axis. The equation of a helicoid in parametric form is
\[
x = \rho \cos t \quad y = \rho \sin t \quad z = \rho \arctan \alpha + kt\, .
\]
If $\alpha = \pi/2$ the helicoid is called straight or right (cf. Fig. 1), otherwise it is called oblique. However, it is common to use the word "helicoid" without specification for the straight helicoid. Every straight helicoid is a [[Minimal surface|minimal surface]], i.e. it has mean curvature zero. In fact the straight helicoid can be characterized by the property of being the only minimal ruled suface. More precisely we have the following

'''Theorem'''
If $\Sigma\subset \mathbb R^3$ is a connected $2$-dimensional ruled minimal surface, then it is a subset of a straight helicoid or a subset of a plane.

Locally the helicoid is isometric to the [[Catenoid|catenoid]]. In fact such local isometry can be achieved as endpoint of a one-parameter family of isometric deformations which are all minimal surfaces. 



<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h046880a.gif" />

Figure 1: Straight helicoid 

====References====
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|valign="top"|{{Ref|BG}}|| M. Berger,   B. Gostiaux,   "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988) {{MR|0917479}} {{ZBL|0629.53001}} 
|-
|valign="top"|{{Ref|Doc}}|| M.P. Do Carmo,   "Differential geometry of curves and surfaces" , Prentice-Hall  (1975)  pp. 214  {{MR|}} {{ZBL|0733.53001}} {{ZBL|0606.53002}} {{ZBL|0326.53001}}
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|valign="top"|{{Ref|ON}}|| B. O'Neill,   "Elementary differential geometry" , Acad. Press  (1966)  {{MR|}} {{ZBL|0971.53500}} {{ZBL|0974.53001}} 
|-
|valign="top"|{{Ref|Sp}}||   M. Spivak,   "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  pp. 218–219  {{MR|0532834}} {{MR|0532833}} {{MR|0532832}} {{MR|0532831}} {{MR|0532830}} {{MR|0394453}} {{MR|0394452}} {{MR|0372756}} {{MR|1537051}} {{MR|0271845}} {{MR|0267467}} {{ZBL|1213.53001}} {{ZBL|0439.53005}} {{ZBL|0439.53004}} {{ZBL|0439.53003}} {{ZBL|0439.53002}} {{ZBL|0439.53001}} {{ZBL|0306.53003}} {{ZBL|0306.53002}} {{ZBL|0306.53001}} {{ZBL|0202.52201}} {{ZBL|0202.52001}} 
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