It belongs to the class of saddle surfaces, and its name derives from the observation that a saddle for a monkey would require three depressions: two for the legs and one for the tail. The point (0,0,0) on the monkey saddle corresponds to a degenerate critical point of the function z(x,y) at (0, 0). 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.
To show that the monkey saddle has three depressions, one can write the equation for z using complex numbers as
It follows that z(tx,ty) = t3 z(x,y) for t ≥ 0, so the surface is determined by z on the unit circle. Parameterizing this by eiφ, with φ ∈ [0, 2π], one can see that on the unit circle, z(φ) = cos 3φ, so z has three depressions. By replacing 3 with any integer k ≥ 1, one can create a saddle with k depressions.
The term horse saddle is used, in contrast to monkey saddle, to designate a saddle point that is a minimax, that is to say a local minimum or maximum depending on the intersecting plane used. The monkey saddle has just a point of inflection. To see this, consider a line y = kx. Along this direction the surface becomes simply z = (1 − 3k2)x3, lacking any critical points.
- Peckham, S.D. (2011) Monkey, starfish and octopus saddles, Proceedings of Geomorphometry 2011, Redlands, CA, pp. 31-34, http://geomorphometry.org/Peckham2011b.
- J., Rimrott, F. P. (1989). Introductory Attitude Dynamics. New York, NY: Springer New York. p. 26. ISBN 9781461235026. OCLC 852789976.
- Chesser, H.; Rimrott, F.P.J. (1985). Rasmussen, H. (ed.). "Magnus Triangle and Smelt Petal". CANCAM '85: Proceedings, Tenth Canadian Congress of Applied Mechanics, June 2-7, 1985, the University of Western Ontario, London, Ontario, Canada.