Whitney umbrella

In mathematics, the Whitney umbrella (or Whitney's umbrella and sometimes called a Cayley 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.

Whitney's umbrella can be given by the parametric equations in Cartesian coordinates ${\displaystyle x(u,v)=uv}$; ${\displaystyle y(u,v)=u}$; ${\displaystyle z(u,v)=v^{2}}$ where the parameters u and v range over the real numbers. It is also given by the implicit equation

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