Monkey saddle

{{short description|1=Mathematical surface defined by z = x³ – 3xy²}}

Image:Monkey Saddle Surface (Shaded).png

In mathematics, the monkey saddle is the surface defined by the equation

:$z\; =\; x^3\; -\; 3xy^2,\; \backslash ,$

or in cylindrical coordinates

:$z\; =\; \backslash rho^3\; \backslash cos(3\backslash varphi).$

It belongs to the class of saddle surfaces, and its name derives from the observation that a saddle used by a monkey would require two depressions for its legs and one for its tail. The point {{tmath|(0,0,0)}} on the monkey saddle corresponds to a degenerate critical point of the function {{tmath|z(x,y)}} at {{tmath|(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.

One can relate the rectangular and cylindrical equations using complex numbers $x+iy\; =\; r\; e^\{i\backslash varphi\}:$

:$z\; =\; x^3\; -\; 3xy^2\; =\; \backslash operatorname\{Re\}\; [(x+iy)^3]\; =\; \backslash operatorname\{Re\}[r^3\; e^\{3i\backslash varphi\}]\; =\; r^3\backslash cos(3\backslash varphi).$

By replacing 3 in the cylindrical equation with any integer {{tmath|k \geq 1,}} one can create a saddle with {{tmath|k}} depressions.

Peckham, S.D. (2011) Monkey, starfish and octopus saddles, Proceedings of Geomorphometry 2011, Redlands, CA, pp. 31-34, https://www.researchgate.net/publication/256808897_Monkey_Starfish_and_Octopus_Saddles

Another orientation of the monkey saddle is the Smelt petal defined by $x+y+z+xyz=0,$ so that the z-axis of the monkey saddle corresponds to the direction {{tmath|(1,1,1)}} in the Smelt petal.{{Cite book|last=J.|first=Rimrott, F. P.|title=Introductory Attitude Dynamics|date=1989|publisher=Springer New York|isbn=9781461235026|location=New York, NY|pages=26|oclc=852789976}}{{Cite journal|last=Chesser|first=H.|last2=Rimrott|first2=F.P.J.|date=1985|editor-last=Rasmussen|editor-first=H.|title=Magnus Triangle and Smelt Petal|journal=CANCAM '85: Proceedings, Tenth Canadian Congress of Applied Mechanics, June 2-7, 1985, the University of Western Ontario, London, Ontario, Canada}}

Another function, which has not three but four areas - in each quadrant of the $\backslash mathbb\; R^2$, in which the function goes to minus infinity, is given by $z\; =\; x^4\; -\; 6x^2y^2\; +\; y^4$.

File:Shape_petal.svg