differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

, 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 ...

in three dimensions has a parabolic point when the

Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For ...

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: where a

ridge A ridge is a long, narrow, elevated geomorphologic landform, structural feature, or a combination of both separated from the surrounding terrain by steep sides. The sides of a ridge slope away from a narrow top, the crest or ridgecrest, wi ...

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 Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

References

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