Whitney umbrella

Section of the surface

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

$\begin{align} x & = uv \\ y & = u \\ z & = v^2 \end{align}$

(which is just the rectangle in 3-space) or by the implicit equation

x 2 = y 2 z

which also includes the negative $z$ axis (which is called the handle of the umbrella).

Whitney umbrella as a ruled surface

The Whitney umbrella is a ruled surface and a right conoid.The figure on the right shows how the Whitney unmbrella is generated by a family of lines.

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

[edit] See also

"Whitney's Umbrella". The Topological Zoo. The Geometry Center. http://www.geom.uiuc.edu/zoo/features/whitney/. Retrieved 2006-03-08. (Images and movies of the Whitney umbrella.)

This geometry-related article is a stub. You can help Wikipedia by expanding it.