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
(1)
equivalent to
(2)
the polar equation is
(3)
and the parametric equations are
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 .
A special case of the Devil's curve is the so-called "electric motor curve":
(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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