A saddle surface is a smooth surface containing one or more saddle point

Saddle point

In mathematics, a saddle point is a point in the domain of a function that is a stationary point but not a local extremum. The name derives from the fact that in two dimensions the surface resembles a saddle that curves up in one direction, and curves down in a different direction...

s.

The term derives from the peculiar shape of historical horse

Horse

The horse is one of two extant subspecies of Equus ferus, or the wild horse. It is a single-hooved mammal belonging to the taxonomic family Equidae. The horse has evolved over the past 45 to 55 million years from a small multi-toed creature into the large, single-toed animal of today...

 saddle

Saddle

A saddle is a supportive structure for a rider or other load, fastened to an animal's back by a girth. The most common type is the equestrian saddle designed for a horse, but specialized saddles have been created for camels and other creatures...

s, which curve both up and down.

Classical examples of two-dimensional saddle surfaces in the Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 are second order surfaces, the hyperbolic paraboloid  (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

Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...

 which distinguish them from convex/elliptical surfaces which have positive Gaussian curvature.

A classical third-order saddle surface is the monkey saddle

Monkey saddle

In mathematics, the monkey saddle is the surface defined by the equation z = x^3 - 3xy^2. \, It belongs to the class of saddle surfaces and its name derives from the observation that a saddle for a monkey requires three depressions: two for the legs, and one for the tail...

.

The Pringles

Pringles

Pringles is a brand of potato and wheat based snacks originally developed by Procter & Gamble. Pringles are sold in more than 140 countries and have yearly sales of more than...

 potato chip or crisp is an everyday example of a hyperbolic paraboloid shape.

Ph.D. thesis of Grigori Perelman

Grigori Perelman

Grigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture...

was devoted to saddle surfaces, its title is "Saddle surfaces in Euclidean spaces".