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In mathematics, the monkey saddle is the surface defined by the equation
or in cylindrical coordinates
It belongs to the class of saddle surfaces, and its name derives from the observation that a saddle for a monkey would require two depressions for the legs and one for the tail. The point {\displaystyle } on the monkey saddle corresponds to a degenerate critical point of the function z {\displaystyle z} at {\displaystyle }. The monkey saddle has an isolated umbilical point with zero Gaussian curvature at the origin, while the curvature is strictly negative at all other points.
One can relate the rectangular and cylindrical equations using complex numbers x + i y = r e i φ : {\displaystyle x+iy=re^{i\varphi }:}