mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the monkey saddle is the

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...

defined by the equation :

z = x^3 - 3xy^2, \,

or in

cylindrical coordinates A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...

z = \rho^3 \cos(3\varphi).

It belongs to the class of

saddle surface In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the functi ...

s, and its name derives from the observation that a

saddle The saddle is a supportive structure for a rider of an animal, fastened to an animal's back by a girth. The most common type is equestrian. However, specialized saddles have been created for oxen, camels and other animals. It is not kno ...

for a

monkey Monkey is a common name that may refer to most mammals of the infraorder Simiiformes, also known as the simians. Traditionally, all animals in the group now known as simians are counted as monkeys except the apes, which constitutes an incomple ...

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 In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are eq ...

with zero

Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . F ...

at the origin, while the curvature is strictly negative at all other points. One can relate the rectangular and cylindrical equations using

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

x+iy = r e^:

= r^3\cos(3\varphi).

By replacing 3 in the cylindrical equation with any integer one can create a saddle with depressions. 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 in the Smelt petal. Shape_petal

Horse saddle

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 In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function ...

, a local minimum or maximum in every direction of the ''xy''-plane. In contrast, the monkey saddle has a stationary

point of inflection In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of ...

in every direction.

References

External links

* {{MathWorld , urlname=MonkeySaddle , title=Monkey Saddle Multivariable calculus Surfaces