Editing Saddle point (section)

{{short description|Critical point on a surface graph which is not a local extremum}}
{{About|the mathematical property|the peninsula in the Antarctic|Saddle Point|the type of landform and general uses of the word "saddle" as a technical term|Saddle (landform)}}

[[File:Saddle point.svg|thumb|right|300px|A saddle point (in red) on the graph of {{math|1=''z'' = ''x''{{sup|2}} − ''y''{{sup|2}}}} ([[Paraboloid#Hyperbolic_paraboloid|hyperbolic paraboloid]])]]

In [[mathematics]], a '''saddle point''' or '''minimax point'''<ref>Howard Anton, Irl Bivens, Stephen Davis (2002): ''Calculus, Multivariable Version'', p.&nbsp;844.</ref> is a [[Point (geometry)|point]] on the [[surface (mathematics)|surface]] of the [[graph of a function]] where the [[slope]]s (derivatives) in [[orthogonal]] directions are all zero (a [[Critical point (mathematics)|critical point]]), but which is not a [[local extremum]] of the function.<ref>{{cite book |last = Chiang |first = Alpha C. |author-link=Alpha Chiang |title = Fundamental Methods of Mathematical Economics |url = https://archive.org/details/fundamentalmetho0000chia_h4v2 |url-access = registration |location=New York |publisher = [[McGraw-Hill]] |edition=3rd |year=1984 |isbn = 0-07-010813-7 |page = [https://archive.org/details/fundamentalmetho0000chia_h4v2/page/312 312] }}</ref>  An example of a saddle point is when there is a critical point with a relative [[minimum]] along one axial direction (between peaks) and a relative [[maxima and minima|maximum]] along the crossing axis.  However, a saddle point need not be in this form.  For example, the function <math>f(x,y) = x^2 + y^3</math> has a critical point at <math>(0, 0)</math> 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 <math>y</math>-direction.
 
[[File:Gordena Jackson, Saddle, 1935-1942, NGA 26642.jpg|thumb|right|150px|A riding saddle]]
The name derives from the fact that the prototypical example in two dimensions is a [[surface (mathematics)|surface]] that ''curves up'' in one direction, and ''curves down'' in a different direction, resembling a riding [[saddle]]. In terms of [[contour line]]s, a saddle point in two dimensions gives rise to a contour map with, in principle, a pair of lines intersecting at the point. Such intersections are rare in contour maps drawn with discrete contour lines, such as ordnance survey maps, as the height of the saddle point is unlikely to coincide with the integer multiples used in such maps. Instead, the saddle point appears as a blank space in the middle of four sets of contour lines that approach and veer away from it. For a basic saddle point, these sets occur in pairs, with an opposing high pair and an opposing low pair positioned in orthogonal directions. The critical contour lines generally do not have to intersect orthogonally.

[[File:Saddle_Point_between_maxima.svg|thumb|300px|right|Saddle point between two hills (the intersection of the figure-eight {{mvar|z}}-contour)]]
[[File:Julia set for z^2+0.7i*z.png|thumb|right|300px|Saddle point on the contour plot is the point where level curves cross ]]