Helicoid

A 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 , and at the same time becomes gradually displaced at a constant rate along this axis.

Figure: h046880a

If , the helicoid is called straight or right (cf. Fig.); if , it is called oblique. The equation of a helicoid in parametric form is

Comments

Every straight helicoid is a minimal surface (it is then sometimes called minimal). See [a2], [a1]. Moreover, a ruled surface which is minimal is necessarily a part of a right helicoid.

References

[a1]	M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1975) pp. 214 Zbl 0733.53001 Zbl 0606.53002 Zbl 0326.53001
[a2]	M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish pp. 218–219 MR0532834 MR0532833 MR0532832 MR0532831 MR0532830 MR0394453 MR0394452 MR0372756 MR1537051 MR0271845 MR0267467 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
[a3]	B. O'Neill, "Elementary differential geometry" , Acad. Press (1966) Zbl 0971.53500 Zbl 0974.53001
[a4]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French) MR0917479 Zbl 0629.53001