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Corkscrew Surface

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CorkscrewSurface

The corkscrew surface, sometimes also called the twisted sphere (Gray 1997, p. 477), is a surface obtained by extending a sphere along a diameter and then twisting. It can be specified parametrically as

x=acosucosv
(1)
y=asinucosv
(2)
z=asinv+bu.
(3)

The coefficients of the first fundamental form are

E=b^2+a^2cos^2v
(4)
F=abcosv
(5)
G=a^2,
(6)

and those of the second fundamental form are

e=-(a^2cos^3v)/(sqrt(a^2cos^2v+b^2sin^2v))
(7)
f=(absin^2v)/(sqrt(a^2cos^2v+b^2sin^2v))
(8)
g=-(a^2cosv)/(sqrt(a^2cos^2v+b^2sin^2v)).
(9)

The Gaussian and mean curvatures are

K=(a^2cos^4v-b^2sin^4v)/((a^2cos^2v+b^2sin^2v)^2)
(10)
M=-(b^2+a(a^2cos^2v+b^2sin^2v))/(2(a^2cos^2v+b^2sin^2v)^(3/2))cosv.
(11)

See also

Projective Plane, Sphere, Twisted Sphere

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References

Gray, A. "The Corkscrew Surface." Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 477-478, 1997.

Cite this as:

Weisstein, Eric W. "Corkscrew Surface." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CorkscrewSurface.html

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