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)

References

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

Geometry Center. ``Whitney's Umbrella.'' http://www.geom.umn.edu/zoo/features/whitney/.

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