Whitney umbrella

Section of the surface

In mathematics, the Whitney umbrella (or Whitney's 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.

Formulas

Whitney's umbrella can be given by the parametric equations in Cartesian coordinates

${\begin{aligned}x(u,v)&=uv\\y(u,v)&=u\\z(u,v)&=v^{2}\end{aligned}}$

where the parameters u and v range over the real numbers. It is also given by the implicit equation

$x^{2}=y^{2}z$

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 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.

See also

• cross-cap
• Right conoid
• Ruled surface

References

• "Whitney's Umbrella". The Topological Zoo. The Geometry Center. Retrieved 2006-03-08. {{cite web}}: Cite has empty unknown parameter: |accessyear= (help); External link in |publisher= (help) (Images and movies of the Whitney umbrella.)