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Monkey Saddle


MonkeySaddle

A surface which a monkey can straddle with both legs and his tail. A simple Cartesian equation for such a surface is

 z=x(x^2-3y^2),
(1)

which can also be given by the parametric equations

x(u,v)=u
(2)
y(u,v)=v
(3)
z(u,v)=u^3-3uv^2.
(4)

The monkey saddle has a single stationary point as summarized in the table below. While the second derivative test is not sufficient to classify this stationary point, it turns out to be a saddle point.

(x_0,y_0)z_(uu)z_(uu)z_(vv)-z_(uv)^2point
(0,0)20saddle point

The coefficients of the first fundamental form of the monkey saddle are

E=1+9(u^2-v^2)^2
(5)
F=-18uv(u^2-v^2)
(6)
G=1+36u^2v^2
(7)

and the second fundamental form coefficients are

e=(6u)/(sqrt(1+9(u^2+v^2)^2))
(8)
f=-(6v)/(sqrt(1+9(u^2+v^2)^2))
(9)
g=-(6u)/(sqrt(1+9(u^2+v^2)^2)),
(10)

giving Riemannian metric

 ds^2=[1+(3u^2-3v^2)^2]du^2-2[18uv(u^2-v^2)]dudv+(1+36u^2v^2)dv^2,
(11)

area element

 dA=sqrt(1+9(u^2+v^2)^2)du ^ dv,
(12)

and Gaussian and mean curvatures

K=-(36(u^2+v^2))/([1+9(u^2+v^2)^2]^2)
(13)
H=(27u(-u^4+2u^2v^2+3v^4))/([1+9(u^2+v^2)^2]^(3/2))
(14)

(Gray 1997). The Gaussian curvature can be written implicitly as

 K(x,y,z)=-(36a^4(x^2+y^2))/((a^4+9x^4+18x^2y^2+9y^4)^2),
(15)

so every point of the monkey saddle except the origin has negative Gaussian curvature.


See also

Crossed Trough, Handkerchief Surface, Partial Derivative

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References

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 365, 1969.Gray, A. "Monkey Saddle." Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 299-301, 382-383, and 408, 1997.Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, p. 202, 1999.

Cite this as:

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

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