Whitney Umbrella
A surface which can be interpreted as a self-intersecting Rectangle in 3-D. It is given by the parametricequations
 |  |  | (1) |
 | |  | (2) |
 | |  | (3) |
for
. The center of the ``plus'' shape which is the end of the line of self-intersection is aPinch Point. The coefficients of the first Fundamental Form are | |  | (4) |
 | |  | (5) |
 | |  | (6) |
and the coefficients of the second Fundamental Form are
 | |  | (7) |
 | | | (8) |
 | |  | (9) |
giving Gaussian Curvature and Mean Curvature
 | |  | (10) |
 | |  | (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.
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