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Whitney Umbrella


WhitneysUmbrella

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

x=uv
(1)
y=u
(2)
z=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

E=1+v^2
(4)
F=uv
(5)
G=u^2+4v^2,
(6)

and the second fundamental form are

e=0
(7)
f=(2v)/(sqrt(u^2+4v^2+4v^4))
(8)
g=-(2u)/(sqrt(u^2+4v^2+4v^4)),
(9)

giving area element

 dA=sqrt(u^2+4v^2(1+v^2)),
(10)

and Gaussian curvature and mean curvature

K=-(4v^2)/((u^2+4v^2+4v^4)^2)
(11)
H=-(u(1+3v^2))/((u^2+4v^2+4v^4)^(3/2)).
(12)

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

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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.Update a linkGeometry Center. "Whitney's Umbrella." http://www.geom.umn.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.

Cite this as:

Barile, Margherita and Weisstein, Eric W. "Whitney Umbrella." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WhitneyUmbrella.html

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