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 of 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.
A horse saddle
A monkey saddle
