Whitney umbrella

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
Section of the surface

In mathematics, the Whitney umbrella (or Whitney's umbrella, named after American mathematician Hassler Whitney, and sometimes called a Cayley umbrella) is a specific 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[edit]

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[edit]

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[edit]

References[edit]

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