Gauss Map - Cusps of The Gauss Map

Cusps of The Gauss Map

The Gauss map reflects many properties of the surface: when the surface has zero Gaussian curvature, (that is along a parabolic line) the Gauss map will have a fold catastrophe. This fold may contain cusps and these cusps were studied in depth by Thomas Banchoff, Terence Gaffney and Clint McCrory. Both parabolic lines and cusp are stable phenomena and will remain under slight deformations of the surface. Cusps occur when:

1. The surface has a bi-tangent plane
2. A ridge crosses a parabolic line
3. at the closure of the set of inflection points of the asymptotic curves of the surface.

There are two types of cusp: elliptic cusp and hyperbolic cusps.