In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a saddle point is a point in the domain

Domain (mathematics)

In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...

 of a function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 that is a stationary point

Stationary point

In mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....

 but not a local extremum. The name derives from the fact that in two dimensions the surface

Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

 resembles a 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...

 that curves up in one direction, and curves down in a different direction (like a horse saddle or a mountain pass

Mountain pass

A mountain pass is a route through a mountain range or over a ridge. If following the lowest possible route, a pass is locally the highest point on that route...

). In terms of contour line

Contour line

A contour line of a function of two variables is a curve along which the function has a constant value. In cartography, a contour line joins points of equal elevation above a given level, such as mean sea level...

s, a saddle point can be recognized, in general, by a contour that appears to intersect itself.