differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilin ...

, a smooth

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...

in three dimensions has a parabolic point when the

Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . F ...

is zero. Typically such points lie on a curve called the parabolic line which separates the surface into regions of positive and negative Gaussian curvature. Points on the parabolic line give rise to folds on the

Gauss map In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere ''S''2. Namely, given a surface ''X'' lying in R3, the Gauss map is a continuous map ''N'': ''X'' → ''S''2 such that '' ...

: where a

ridge A ridge or a mountain ridge is a geographical feature consisting of a chain of mountains or hills that form a continuous elevated crest for an extended distance. The sides of the ridge slope away from the narrow top on either side. The line ...

crosses a parabolic line there is a cusp of the Gauss map. Ian R. Porteous (2001) ''Geometric Differentiation'', Chapter 11 Ridges and Ribs, pp 182–97, Cambridge University Press .

References

Differential geometry of surfaces Surfaces {{differential-geometry-stub