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 on the monkey saddle corresponds to a degenerate critical point of the function at . 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
By replacing 3 in the cylindrical equation with any integer one can create a saddle with depressions. [1]
Another orientation of the monkey saddle is the Smelt petal defined by so that the z-axis of the monkey saddle corresponds to the direction in the Smelt petal.[2][3]
Smelt petal: x + y + z + xyz = 0
Horse saddleedit
The term horse saddle may be used in contrast to monkey saddle, to designate an ordinary saddle surface in which z(x,y) has a saddle point, a local minimum or maximum in every direction of the xy-plane. In contrast, the monkey saddle has a stationary point of inflection in every direction.
^J., Rimrott, F. P. (1989). Introductory Attitude Dynamics. New York, NY: Springer New York. p. 26. ISBN9781461235026. OCLC 852789976.{{cite book}}: CS1 maint: multiple names: authors list (link)
^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.
monkey, saddle, mathematics, monkey, saddle, surface, defined, equation, displaystyle, cylindrical, coordinates, displaystyle, varphi, monkey, saddleit, belongs, class, saddle, surfaces, name, derives, from, observation, that, saddle, monkey, would, require, d. In mathematics the monkey saddle is the surface defined by the equation z x 3 3 x y 2 displaystyle z x 3 3xy 2 or in cylindrical coordinates z r 3 cos 3 f displaystyle z rho 3 cos 3 varphi The monkey saddleIt 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 0 0 0 displaystyle 0 0 0 on the monkey saddle corresponds to a degenerate critical point of the function z x y displaystyle z x y at 0 0 displaystyle 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 i y r e i f displaystyle x iy re i varphi z x 3 3 x y 2 Re x i y 3 Re r 3 e 3 i f r 3 cos 3 f displaystyle z x 3 3xy 2 operatorname Re x iy 3 operatorname Re r 3 e 3i varphi r 3 cos 3 varphi By replacing 3 in the cylindrical equation with any integer k 1 displaystyle k geq 1 one can create a saddle with k displaystyle k depressions 1 Another orientation of the monkey saddle is the Smelt petal defined by x y z x y z 0 displaystyle x y z xyz 0 so that the z axis of the monkey saddle corresponds to the direction 1 1 1 displaystyle 1 1 1 in the Smelt petal 2 3 Smelt petal x y z xyz 0Horse saddle editThe term horse saddle may be used in contrast to monkey saddle to designate an ordinary saddle surface in which z x y has a saddle point a local minimum or maximum in every direction of the xy plane In contrast the monkey saddle has a stationary point of inflection in every direction References edit 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 J Rimrott F P 1989 Introductory Attitude Dynamics New York NY Springer New York p 26 ISBN 9781461235026 OCLC 852789976 a href Template Cite book html title Template Cite book cite book a CS1 maint multiple names authors list link 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 External links editWeisstein Eric W Monkey Saddle MathWorld Retrieved from https en wikipedia org w index php title Monkey saddle amp oldid 1158434839, wikipedia, wiki, book, books, library,
