Monkey Saddle

A Surface which a monkey can straddle with both his two legs and his tail. A simply 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 coefficients of the first and second Fundamental Forms of the monkey saddle are given by

$$e = {6u\over\sqrt{1+9u^4+18u^2v^2+9v^4}}$$ (5)

$$f = -{6v\over\sqrt{1+9u^4+18u^2v^2+9v^4}}$$ (6)

$$g = -{6u\over\sqrt{1+9u^4+18u^2v^2+9v^4}}$$ (7)

$$E = 1+9(u^2-v^2)^2$$ (8)

$$F = -18uv(u^2-v^2)$$ (9)

$$G = 1+36u^2v^2,$$ (10)

giving Riemannian Metric

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

Area Element

$$dA=\sqrt{1+9u^4+18u^2v^2+9v^4}\,du\wedge dv,$$ (12)

and Gaussian and Mean Curvatures

$$K = -{36(u^2+v^2)\over(1+9u^4+18u^2v^2+9v^4)^2}$$ (13)

$$H = {27u(-u^4+2u^2v^2+3v^4)\over(1+9u^4+18u^2v^2+9v^4)^{3/2}}$$ (14)

(Gray 1993). Every point of the monkey saddle except the origin has Negative Gaussian Curvature.

See also Crossed Trough, Partial Derivative

References

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 365, 1969.

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 213-215, 262-263, and 288-289, 1993.

Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, p. 202, 1952.