The name derives from the fact that the prototypical example in two dimensions is a surface that curves up in one direction, and curves down in a different direction, resembling a riding saddle or a mountain pass between two peaks forming a landform saddle. In terms of contour lines, a saddle point in two dimensions gives rise to a contour map ( graph or trace) with a pair of lines intersecting at the point. However typical ordnance survey maps for example don’t generally show such intersections as there is no reason such critical points in nature will just happen to coincide with typical integer multiple contour spacing. Instead dead space with 4 sets of contour lines approaching and veering away surround a basic saddle point with opposing high pair and opposing low pair in orthogonal directions. The critical contour lines generally don’t intersect perpendicularly.
In the most general terms, a saddle point for a smooth function (whose graph is a curve, surface or hypersurface) is a stationary point such that the curve/surface/etc. in the neighborhood of that point is not entirely on any side of the tangent space at that point.
A saddle surface is a smooth surface containing one or more saddle points.
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.
In optimization subject to equality constraints, the first-order conditions describe a saddle point of the Lagrangian.
In dynamical systems, if the dynamic is given by a differentiable map f then a point is hyperbolic if and only if the differential of ƒ n (where n is the period of the point) has no eigenvalue on the (complex) unit circle when computed at the point. Then a saddle point is a hyperbolic periodic point whose stable and unstable manifolds have a dimension that is not zero.
A saddle point of a matrix is an element which is both the largest element in its column and the smallest element in its row.