geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a Devil's curve, also known as the Devil on Two Sticks, is a

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...

defined in the

Cartesian plane A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

by an equation of the form :

y^2(y^2 - b^2) = x^2(x^2 - a^2).

The

polar equation In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...

of this curve is of the form :

r = \sqrt = \sqrt

. Devil's curves were discovered in 1750 by

Gabriel Cramer Gabriel Cramer (; 31 July 1704 – 4 January 1752) was a Genevan mathematician. He was the son of physician Jean Cramer and Anne Mallet Cramer. Biography Cramer showed promise in mathematics from an early age. At 18 he received his doctorate ...

, who studied them extensively. The name comes from the shape its central

lemniscate In algebraic geometry, a lemniscate is any of several figure-eight or -shaped curves. The word comes from the Latin "''lēmniscātus''" meaning "decorated with ribbons", from the Greek λημνίσκος meaning "ribbons",. or which alternativel ...

takes when graphed. The shape is named after the juggling game

diabolo The diabolo ( ; commonly misspelled ''diablo'') is a juggling or circus prop consisting of an axle () and two cups (hourglass/egg timer shaped) or discs derived from the Chinese yo-yo. This object is spun using a string attached to two hand ...

, which was named after the Devil and which involves two sticks, a string, and a spinning prop in the likeness of the lemniscate. For

, b, <, a,

, the central lemniscate, often called hourglass, is horizontal. For

, b, >, a,

it is vertical. Is

, b,  = , a,

, the shape becomes a circle. The vertical hourglass intersects the y-axis at

b,-b, 0

. The horizontal hourglass intersects the x-axis at

a,-a,0

Electric Motor Curve

A special case of the Devil's curve occurs at

\frac=\frac

, where the curve is called the electric motor curve.Mathematical Models, p. 71 (Cundy and Rollet. 1961)
/ref> It is defined by an equation of the form

y^2(y^2-96) = x^2(x^2-100)

. The name of the special case comes from the middle shape's resemblance to the coils of wire, which rotate from forces exerted by magnets surrounding it.

References

External links

*

{{DEFAULTSORT:Devil'S Curve Plane curves