Whitney umbrella

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Section of the surface

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.


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).


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.

See also