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Saddle surface

Hyperbolic paraboloid
A saddle surface is a [smooth surface] containing one or more [saddle point] s.

The term derives from the peculiar shape of historical [horse] [saddle] s, 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=x^2-y^2 (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

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