Devil's Curve

$\begin{figure}\begin{center}\BoxedEPSF{devils_curve.epsf}\end{center}\end{figure}$

The devil's curve was studied by G. Cramer in 1750 and Lacroix in 1810 (MacTutor Archive). It appeared in Nouvelles Annales in 1858. The Cartesian equation is

$\begin{displaymath} y^4-a^2y^2=x^4-b^2x^2, \end{displaymath}$

(1)

equivalent to

$\begin{displaymath} y^2(y^2-a^2)=x^2(x^2-b^2), \end{displaymath}$

(2)

the polar equation is

$\begin{displaymath} r^2(\sin^2\theta-\cos^2\theta)=a^2\sin^2\theta-b^2\cos^2\theta, \end{displaymath}$

(3)

and the parametric equations are

$\displaystyle x$	$\textstyle =$	$\displaystyle \cos t\sqrt{a^2\sin^2 t-b^2\cos^2 t\over\sin^2 t-\cos^2 t}$	(4)
$\displaystyle y$	$\textstyle =$	$\displaystyle \sin t\sqrt{a^2\sin^2 t-b^2\cos^2 t\over\sin^2 t-\cos^2 t}.$	(5)

A special case of the Devil's curve is the so-called Electric Motor Curve:

$\begin{figure}\begin{center}\BoxedEPSF{ElectricMotor.epsf}\end{center}\end{figure}$

$\begin{displaymath} y^2(y^2-96)=x^2(x^2-100) \end{displaymath}$

(6)

(Cundy and Rollett 1989).

See also Electric Motor Curve

References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 71, 1989.

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, p. 71, 1993.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 151-152, 1972.

MacTutor History of Mathematics Archive. ``Devil's Curve.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Devils.html.