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In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes in orthogonal directions are all zero , but which is not a local extremum of the function. An example of a saddle point is when there is a critical point with a relative minimum along one axial direction and at a relative maximum along the crossing axis. However, a saddle point need not be in this form. For example, the function f = x 2 + y 3 {\displaystyle f=x^{2}+y^{3}} has a critical point at {\displaystyle } that is a saddle point since it is neither a relative maximum nor relative minimum, but it does not have a relative maximum or relative minimum in the y {\displaystyle y} -direction.
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 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.