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Cornucopia

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Cornucopia

The surface given by the parametric equations

x=e^(bv)cosv+e^(av)cosucosv
(1)
y=e^(bv)sinv+e^(av)cosusinv
(2)
z=e^(av)sinu.
(3)

For a=b=1, the coefficients of the first fundamental form are

E=e^(2v)
(4)
F=-e^(2v)sinu
(5)
G=2e^(2v)cos^2(1/2u)(3+cosu),
(6)

and of the second fundamental form are

e=(e^v(1+cosu)sec^2(1/2u))/(2sqrt(2))
(7)
f=(e^v(1+cosu)sec^2(1/2u)sinu)/(2sqrt(2))
(8)
g=2sqrt(2)e^vsin^4(1/2u).
(9)

The Gaussian and mean curvatures are given by

K=(e^(-2v)cosu)/(2(1+cosu))
(10)
H=(e^(-v))/(4sqrt(2))(4-2/(1+cosu)),
(11)

and the principal curvatures are

kappa_1=(e^(-v))/(sqrt(2))
(12)
kappa_2=(e^(-v)cosu)/(sqrt(2)(1+cosu)).
(13)

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References

von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 304, 1993.

Cite this as:

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

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