Whitney umbrella

In mathematics, specifically in the field of singularity theory, the Whitney umbrella (also referred to as Whitney's umbrella) is a self-intersecting rectangle placed in three dimensions.

It can be given by the parametric equations

:$x\; quad\; =\; quad\; uv$:$y\; quad\; =\; quad\; u$:$z\; quad=\; quad\; v^2$(which is just the rectangle in 3-space.)

or by the implicit equation: $x^2=y^2z$which also include the negative $z$ axis (which is called the "handle" of the umbrella).

It is a local model of a pinch point singularity, together with the fold they are the only stable local singularity of maps $R^2\; o\; R^3$.

It is named after the American mathematician Hassler Whitney.

See also

* cross-cap

References

* cite web
author=
year=
url=http://www.geom.uiuc.edu/zoo/features/whitney/
title=Whitney's Umbrella
format=
work=The Topological Zoo
publisher= [http://www.geom.uiuc.edu/ The Geometry Center]
accessdate=2006-03-08
accessyear= (Images and movies of the Whitney umbrella.)