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 = 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.

In mathematics , parabolic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional parabolic coordinate system in the perpendicular z -direction.

Hence, the coordinate surfaces are confocal parabolic cylinders. Parabolic cylindrical coordinates have found many applications, e.g., the potential theory of edges.

 

A quadratic surface given by the equation: