Whitney Umbrella

whitney umbrella 001
The Whitney Umbrella is famous for its pinch-point singularity. In the animation a second such singularity is created.
whitney umbrella st
whitney umbrella st
whitney umbrella sw
whitney umbrella sw
   Whitney Umbrella Surface Parametric Equations

 Strictly speaking, the Whitney umbrella is the algebraic
surface defined by the implicit equation:

    x^2 - y^2*z == 0

(See this in the Implicit Surface submenu). It has two
algebraic components, a two-dimensional piece that
is given parametrically by the equations:

  x := u * v
  y := u
  z := v * v

and a one-dimensional piece, namely the entire z-axis
(the handle of the umbrella). The two-dimensional part
is important in the theory of singularities. The point at
the end of the line of self-intersections is a singularity
of a type called a pinch point. This is a canonical
representative of the only stable singularity for
mappings from R^2 to R^3.

RuledSurfaces.pdf