Whitney umbrella

Whitney's umbrella can be given by the parametric equations in Cartesian coordinates

${\displaystyle \left\{{\begin{aligned}x(u,v)&=uv,\\y(u,v)&=u,\\z(u,v)&=v^{2},\end{aligned}}\right.}$

where the parameters u and v range over the real numbers. It is also given by the implicit equation

${\displaystyle x^{2}-y^{2}z=0.}$

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

Whitney umbrella as a ruled surface, generated by a moving straight line

Whitney umbrella made with a single string inside a plastic cube

Whitney's umbrella is a ruled surface and a right conoid. It is important in the field of singularity theory, as a simple local model of a pinch point singularity. The pinch point and the fold singularity are the only stable local singularities of maps from R² to R³.

It is named after the American mathematician Hassler Whitney.

In string theory, a Whitney brane is a D7-brane wrapping a variety whose singularities are locally modeled by the Whitney umbrella. Whitney branes appear naturally when taking Sen's weak coupling limit of F-theory.

• Cross-cap
• Right conoid
• Ruled surface

• "Whitney's Umbrella". The Topological Zoo. The Geometry Center. Retrieved 2006-03-08. (Images and movies of the Whitney umbrella.)