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Hyperbolic Helicoid

HyperbolicHelicoid

The surface with parametric equations

x=(sinhvcos(tauu))/(1+coshucoshv)
(1)
y=(sinhvsin(tauu))/(1+coshucoshv)
(2)
z=(coshvsinh(u))/(1+coshucoshv),
(3)

where tau is the torsion.

The coefficients of the first fundamental form are

E=(a^2[1-tau^2+(1+tau^2)cosh(2v)])/(2(1+coshucoshv)^2)
(4)
F=0
(5)
G=(a^2)/((1+coshucoshv)^2)
(6)

and those of the second fundamental form are

e=-(atausqrt(1-tau^2+(1+tau^2)cosh(2v))sinhusinhv)/(sqrt(2)(1+coshucoshv)^2)
(7)
f=(sqrt(2)atau)/((1+coshucoshv)sqrt(1-tau^2+(1+tau^2)cosh(2v)))
(8)
g=-(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

H=(sqrt(2)tausinhusinhv)/(asqrt(1-tau^2+(1+tau^2)cosh(2v))).
(10)

See also

Helicoid

Explore with Wolfram|Alpha

References

JavaView. "Classic Surfaces from Differential Geometry: Hyperbolic Helicoid." http://www-sfb288.math.tu-berlin.de/vgp/javaview/demo/surface/common/PaSurface_HyperbolicHelicoid.html.

Referenced on Wolfram|Alpha

Hyperbolic Helicoid

Cite this as:

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

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