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Whitney Umbrella

A surface which can be interpreted as a self-intersecting Rectangle in 3-D. It is given by the parametric equations

$\displaystyle x$ $\textstyle =$ $\displaystyle uv$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle u$ (2)
$\displaystyle z$ $\textstyle =$ $\displaystyle v^2$ (3)

for $u,v\in [-1,1]$. The center of the ``plus'' shape which is the end of the line of self-intersection is a Pinch Point. The coefficients of the first Fundamental Form are
$\displaystyle E$ $\textstyle =$ $\displaystyle 0$ (4)
$\displaystyle F$ $\textstyle =$ $\displaystyle {2v\over\sqrt{u^2+4v^2+4v^4}}$ (5)
$\displaystyle G$ $\textstyle =$ $\displaystyle -{2u\over\sqrt{u^2+4v^2+4v^4}},$ (6)

and the coefficients of the second Fundamental Form are
$\displaystyle e$ $\textstyle =$ $\displaystyle 1+v^2$ (7)
$\displaystyle f$ $\textstyle =$ $\displaystyle uv$ (8)
$\displaystyle g$ $\textstyle =$ $\displaystyle u^2+4v^2,$ (9)

giving Gaussian Curvature and Mean Curvature
$\displaystyle K$ $\textstyle =$ $\displaystyle -{4v^2\over(u^2+4v^2+4v^4)^2}$ (10)
$\displaystyle H$ $\textstyle =$ $\displaystyle -{u(1+3v^2)\over(u^2+4v^2+4v^4)^{3/2}}.$ (11)


Francis, G. K. A Topological Picturebook. New York: Springer-Verlag, pp. 8-9, 1987.

Geometry Center. ``Whitney's Umbrella.''

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 225 and 309-310, 1993.

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© 1996-9 Eric W. Weisstein