Saddle surface

Hyperbolic paraboloid

A model of an elliptic hyperboloid of one sheet

A monkey saddle

A saddle surface is a smooth surface containing one or more saddle points. The term derives from the peculiar shape of historical horse saddles, which curve both up and down.

Classical examples of two-dimensional saddle surfaces in the Euclidean space are second order surfaces, the hyperbolic paraboloid z = x² − y² (which is often referred to as the saddle surface or "the standard saddle surface") and hyperboloid of one sheet.

Saddle surfaces have negative Gaussian curvature which distinguish them from convex/elliptical surfaces which have positive Gaussian curvature. A classical third-order saddle surface is the monkey saddle. The Pringles potato chip or crisp is an everyday example of a hyperbolic paraboloid shape.

Ph.D. thesis of Grigori Perelman was devoted to saddle surfaces, its title is "Saddle surfaces in Euclidean spaces".

References

• Hilbert, David; Cohn-Vossen, Stephan (1952). Geometry and the Imagination (2nd ed.). Chelsea. ISBN 0-8284-1087-9.