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Handkerchief Surface

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HandkerchiefSurface

A surface given by the parametric equations

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

The handkerchief surface has stationary points summarized in the table below, where the type of point can be found using the second derivative test.

(x_0,y_0)z_(uu)z_(uu)z_(vv)-z_(uv)^2point
(-4,0)-448local maximum
(0,0)4-16saddle point

Its first fundamental form has coefficients

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

and its second fundamental form has coefficients

e=(2(u+2))/(sqrt(1+u^2(u+4)^2+16v^2+2u(3u-4)v^2+v^3))
(7)
f=(2v)/(sqrt(1+u^2(u+4)^2+16v^2+2u(3u-4)v^2+v^3))
(8)
g=(2(u-2))/(sqrt(1+u^2(u+4)^2+16v^2+2u(3u-4)v^2+v^3)).
(9)

The Gaussian curvature can be given implicitly by

K(x,y,z)=(4(-4+x^2-y^2))/((1+16x^2+8x^3+x^4+16y^2-8xy^2+6x^2y^2+y^4)^2).
(10)

See also

Crossed Trough, Monkey Saddle

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References

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 948-949, 1997.

Referenced on Wolfram|Alpha

Handkerchief Surface

Cite this as:

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

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