In mathematics, specifically in the field of singularity theory, the Whitney umbrella (also referred to as Whitney's umbrella) is a self-intersecting rectangle placed in three dimensions. It can be given by the parametric equations
- <math>x \quad = \quad uv</math>
- <math>y \quad = \quad u</math>
- <math>z \quad= \quad v^2</math>
(which is just the rectangle in 3-space.) or by the implicit equation
- <math>x^2=y^2z</math>
which also include the negative <math>z</math> axis (which is called the handle of the umbrella). It is a local model of a pinch point singularity, together with the fold they are the only stable singularity of maps <math>R^2 \to R^3</math>. It is named after the American mathematician Hassler Whitney.
See also
References
- Whitney's Umbrella. The Topological Zoo. The Geometry Center. Retrieved on 2006-03-08. (Images and movies of the Whitney umbrella.)
