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
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(1)
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equivalent to
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(2)
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the polar equation is
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(3)
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and the parametric equation s are
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(4)
|
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(5)
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The curve illustrated above corresponds to parameters 
and 
.
It has a crunode at the origin.
For 
, the cental hourglass is horizontal, for 
, it is vertical, and as it passes through 
, the curve changes to a circle.
For , the cental hourglass is horizontal, for , it is vertical, and as it passes through , the curve changes to a circle.
A special case of the Devil's curve is the so-called "electric motor curve":
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(6)
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(Cundy and Rollett 1989).
SEE ALSO:Barbell Graph, Butterfly Curve, Dumbbell Curve, Eight Curve, Lemniscate, Piriform, Pitchfork Bifurcation, Teardrop Curve
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.
