Whitney umbrella

In

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

Cartesian coordinates

\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.

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's umbrella is a

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.

References

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

Formulas

Properties

See also

References