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.