Devil's Curve
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)
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and the parametric equations are
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 CurveExplore with Wolfram|Alpha
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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." https://mathshistory.st-andrews.ac.uk/Curves/Devils/.Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, p. 328, 1958.Referenced on Wolfram|Alpha
Devil's CurveCite this as:
Weisstein, Eric W. "Devil's Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DevilsCurve.html