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 = [画像:-(a^2cos^3v)/(sqrt(a^2cos^2v+b^2sin^2v))]

(7)

f = [画像:(absin^2v)/(sqrt(a^2cos^2v+b^2sin^2v))]

(8)

g = [画像:-(a^2cosv)/(sqrt(a^2cos^2v+b^2sin^2v)).]

(9)

The Gaussian and mean curvatures are

K = [画像:(a^2cos^4v-b^2sin^4v)/((a^2cos^2v+b^2sin^2v)^2)]

(10)

M = [画像:-(b^2+a(a^2cos^2v+b^2sin^2v))/(2(a^2cos^2v+b^2sin^2v)^(3/2))cosv.]

(11)

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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.

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Corkscrew Surface

Cite this as:

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

Corkscrew Surface

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