Whitney umbrella
# Whitney umbrella

In geometry, the Whitney umbrella (or Whitney's umbrella, named after American mathematician Hassler Whitney, and sometimes called a Cayley umbrella) is a specific self-intersecting ruled surface placed in three dimensions. It is the union of all straight lines that pass through points of a fixed parabola and are perpendicular to a fixed straight line which is parallel to the axis of the parabola and lies on its perpendicular bisecting plane.

## Formulas

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

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

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

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

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

## Properties

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 $R^2$ to $R^3.$

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.

