Editing Gauss map (section)

==Cusps of the Gauss map==
[[File:Cusp of the Gauss map.png|thumb|A surface with a parabolic line and its Gauss map. A ridge passes through the parabolic line giving rise to a cusp on 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 [[Catastrophe theory#Fold catastrophe|fold catastrophe]].<ref>{{Cite journal |author1=McCrory, Clint |author2=Shifrin, Theodore |year=1984 |title=Cusps of the projective Gauss map |journal=Journal of Differential Geometry |volume=19 |pages=257–276|doi=10.4310/JDG/1214438432 |s2cid=118784720 }}</ref> This fold may contain [[cusp (singularity)|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:
#The surface has a bi-tangent plane
#A [[ridge (differential geometry)|ridge]] crosses a parabolic line
#at the closure of the set of inflection points of the [[asymptotic curve]]s of the surface.
There are two types of cusp: ''elliptic cusp'' and ''hyperbolic cusps''.