Editing Catastrophe theory (section)

==Potential functions of two active variables==
[[File:Hyperbolic umbilic focal surface.png|thumb|A surface with a hyperbolic umbilic and its focal surface. The hyperbolic umbilic catastrophe is just the upper part of this image.]]
[[File:Elliptical umbilic focal surface.png|thumb|right|A surface with an elliptical umbilic, and its focal surface. The elliptic umbilic catastrophe is just the upper part of this image.]]

Umbilic catastrophes are examples of corank 2 catastrophes. They can be observed in [[optics]] in the [[focal surface]]s created by light reflecting off a surface in three dimensions and are intimately connected with the geometry of nearly spherical surfaces: [[umbilical point]].
Thom proposed that the hyperbolic umbilic catastrophe modeled the breaking of a wave and the elliptical umbilic modeled the creation of hair-like structures.

===Hyperbolic umbilic catastrophe===
:<math>V = x^3 + y^3 + axy + bx +cy \, </math>

===Elliptic umbilic catastrophe===
:<math>V = \frac{x^3}{3} - xy^2 + a(x^2+y^2) + bx + cy \, </math>

===Parabolic umbilic catastrophe===
:<math>V = x^2y + y^4 + ax^2 + by^2 + cx + dy \, </math>