A surface which a monkey can straddle with both legs and his tail. A simple Cartesian equation for such a surface is
(1)
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which can also be given by the parametric equations
(2)
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(3)
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(4)
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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.
point | |||
2 | 0 | saddle point |
The coefficients of the first fundamental form of the monkey saddle are
(5)
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(6)
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(7)
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and the second fundamental form coefficients are
(8)
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(9)
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(10)
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giving Riemannian metric
(11)
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(12)
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and Gaussian and mean curvatures
(13)
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(14)
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(Gray 1997). The Gaussian curvature can be written implicitly as
(15)
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so every point of the monkey saddle except the origin has negative Gaussian curvature.