Whitney Umbrella

A surface which can be interpreted as a self-intersecting rectangle in three dimensions. The Whitney umbrella is the only stable singularity of mappings from to . It is given by the parametric equations

			(1)
			(2)
			(3)

for . 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

			(4)
			(5)
			(6)

and the second fundamental form are

			(7)
			(8)
			(9)

giving area element

(10)

and Gaussian curvature and mean curvature

			(11)
			(12)

Note that the ruled cubic surface given by the equation:

(13)

is the union of Whitney umbrella and the ray , , called the handle of the Whitney umbrella.