Hyperbolic Helicoid

HyperbolicHelicoid

The surface with parametric equations

x = [画像:(sinhvcos(tauu))/(1+coshucoshv)]

(1)

y = [画像:(sinhvsin(tauu))/(1+coshucoshv)]

(2)

z = [画像:(coshvsinh(u))/(1+coshucoshv),]

(3)

where tau is the torsion.

The coefficients of the first fundamental form are

E = [画像:(a^2[1-tau^2+(1+tau^2)cosh(2v)])/(2(1+coshucoshv)^2)]

(4)

F =

(5)

G = [画像:(a^2)/((1+coshucoshv)^2)]

(6)

and those of the second fundamental form are

e = [画像:-(atausqrt(1-tau^2+(1+tau^2)cosh(2v))sinhusinhv)/(sqrt(2)(1+coshucoshv)^2)]

(7)

f = [画像:(sqrt(2)atau)/((1+coshucoshv)sqrt(1-tau^2+(1+tau^2)cosh(2v)))]

(8)

g = [画像:-(sqrt(2)atausinhusinhv)/((1+coshucoshv)sqrt(1-tau^2+(1+tau^2)cosh(2v))).]

(9)

The Gaussian curvature is a somewhat complicated, but the mean curvature is given by

[画像: H=(sqrt(2)tausinhusinhv)/(asqrt(1-tau^2+(1+tau^2)cosh(2v))). ]

(10)

See also

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References

JavaView. "Classic Surfaces from Differential Geometry: Hyperbolic Helicoid." http://www.javaview.de/demo/surface/common/PaSurface_HyperbolicHelicoid.html.

Referenced on Wolfram|Alpha

Hyperbolic Helicoid

Cite this as:

Weisstein, Eric W. "Hyperbolic Helicoid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HyperbolicHelicoid.html

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