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
{ x ( u , v ) = u v , y ( u , v ) = u , z ( u , v ) = v 2 , {\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
x 2 − y 2 z = 0. {\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'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 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.)
