Whitney Umbrella

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   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.