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Devil's Curve

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DevilsCurve

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

y^4-a^2y^2=x^4-b^2x^2,
(1)

equivalent to

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

the polar equation is

r^2(sin^2theta-cos^2theta)=a^2sin^2theta-b^2cos^2theta,
(3)

and the parametric equations are

x=costsqrt((a^2sin^2t-b^2cos^2t)/(sin^2t-cos^2t))
(4)
y=sintsqrt((a^2sin^2t-b^2cos^2t)/(sin^2t-cos^2t)).
(5)

The curve illustrated above corresponds to parameters a^2=1 and b^2=2.

It has a crunode at the origin.

Devil's curve animation

For a/b<1, the cental hourglass is horizontal, for a/b>1, it is vertical, and as it passes through a=b, the curve changes to a circle.

ElectricMotor

A special case of the Devil's curve is the so-called "electric motor curve":

y^2(y^2-96)=x^2(x^2-100)
(6)

(Cundy and Rollett 1989).

See also

Barbell Graph, Butterfly Curve, Dumbbell Curve, Eight Curve, Lemniscate, Piriform Curve, Pitchfork Bifurcation, Teardrop Curve

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References

--. Nouvelle Annales, p. 317, 1858.Cramer, G. Introduction a l'analyse des lignes courbes algébriques. Geneva, p. 19, 1750.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 with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 92-93, 1997.Lacroix, S. F. Traité du calcul différentiel et intégral, Vol. 1. Paris, p. 391, 1810.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.Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, p. 328, 1958.

Cite this as:

Weisstein, Eric W. "Devil's Curve." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DevilsCurve.html

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