Whitney umbrella

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Section of the surface
Section of the surface

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

x \quad = \quad uv
y \quad = \quad u
z \quad= \quad v^2

(which is just the rectangle in 3-space.)

or by the implicit equation

x2 = y2z

which also include the negative z 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 R^2 \to R^3.

It is named after the American mathematician Hassler Whitney.

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