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Hyperbolic paraboloid

A model of an elliptic hyperboloid of one sheet

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 crisp is an everyday example of a hyperbolic paraboloid shape.

A horse saddle

A monkey saddle