TOPICS
Search

Elliptic Helicoid

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
EllipticalHelicoid

A generalization of the helicoid to the parametric equations

x(u,v)=avcosu
(1)
y(u,v)=bvsinu
(2)
z(u,v)=cu.
(3)

In this parametrization, the surface has first fundamental form coefficients

E=c^2+v^2(a^2sin^2u+b^2cos^2u)
(4)
F=(b^2-a^2)vcosusinu
(5)
G=a^2cos^2u+b^2sin^2u
(6)

and second fundamental form coefficients

e=0
(7)
f=(sqrt(2)abc)/(sqrt((a^2+b^2)c^2+2a^2b^2v^2+(a^2-b^2)c^2cos(2u)))
(8)
g=0.
(9)

The Gaussian and mean curvatures are given by

K=-(4a^2b^2c^2)/([(a^2+b^2)c^2+2a^2b^2v^2+(a^2-b^2)c^2cos(2u)]^2)
(10)
H=-(sqrt(2)ab(a^2-b^2)cvsin(2u))/([(a^2+b^2)c^2+2a^2b^2v^2+(a^2-b^2)c^2cos(2u)]^(3/2)).
(11)

See also

Helicoid

Explore with Wolfram|Alpha

References

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 422, 1997.

Cite this as:

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