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Shoe Surface

ShoeSurface

A surface given by the parametric equations

x(u,v)=u
(1)
y(u,v)=v
(2)
z(u,v)=1/3u^3-1/2v^2.
(3)

The coefficients of the coefficients of the first fundamental form are

E=1+u^4
(4)
F=-u^2v
(5)
G=1+v^2,
(6)

and the second fundamental form coefficients are

e=(2u)/(sqrt(1+u^4+v^2))
(7)
f=0
(8)
g=-1/(sqrt(1+u^4+v^2)),
(9)

giving area element

dA=sqrt(-(2u)/(sqrt(1+u^4+v^2)))du ^ dv,
(10)

and Gaussian and mean curvatures

K=-(2u)/((1+u^4+v^2)^2)
(11)
H=(2u(1+v^2)-u^4-1)/(2(1+u^4+v^2)^(3/2)).
(12)

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References

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 965, 1997.

Cite this as:

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

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