Whitney umbrella

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[]

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).

Properties[]

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 are the only of maps from R² to R³.

It is named after the American mathematician Hassler Whitney.

In string theory, a 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[]

• Cross-cap
• Right conoid
• Ruled surface

References[]

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