## Whitney Umbrella

Additional images: Anaglyph, Anaglyph Wireframe, Parallel Stereo

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.