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 R^2 to R^3. It is given by the parametric equations
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
and the second fundamental form are
giving area element
| dA=sqrt(u^2+4v^2(1+v^2)), |
(10)
|
and Gaussian curvature and mean curvature
Note that the ruled cubic surface given by the equation:
| x^2-y^2z=0 |
(13)
|
is the union of Whitney umbrella and the ray x=y=0, z<0, called the handle of the Whitney umbrella.
Portions of this entry contributed by Margherita Barile
Explore with Wolfram|Alpha
More things to try:
References
Apéry, F. Models of the Real Projective Plane. Braunschweig, Germany: Vieweg, pp. 62-63, 1987.Francis, G. K. A Topological Picturebook. New York: Springer-Verlag, pp. 8-9, 1987.Geometry Center. "Whitney's Umbrella." http://www.geom.uiuc.edu/zoo/features/whitney/.Gray, A. "The Whitney Umbrella." Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 311 and 401-402, 1997.Referenced on Wolfram|Alpha
Whitney UmbrellaCite this as:
Barile, Margherita and Weisstein, Eric W. "Whitney Umbrella." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WhitneyUmbrella.html