# Whitney umbrella

In mathematics, the **Whitney umbrella** (or **Whitney's umbrella** and sometimes called a **Cayley umbrella**) is a self-intersecting 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, parallel to the axis of the parabola and lying on its perpendicular bisecting plane.

## Formulas

Whitney's umbrella can be given by the parametric equations in Cartesian coordinates
;
;
where the parameters *u* and *v* range over the real numbers. It is also given by the implicit equation

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.

## See also

## References

- Template:Cite web (Images and movies of the Whitney umbrella.)