Devil's curve

2-dimensional curve

{\displaystyle a} — Devil's curve with $a$ ranging from 0 to 1 and b = 1 (with the curve colour going from blue to red).

In geometry, a Devil's curve, also known as the Devil on Two Sticks, is a curve defined in the Cartesian plane by an equation of the form

y^{2}(y^{2}-b^{2})=x^{2}(x^{2}-a^{2}).

^[1]

The polar equation of this curve is of the form

r={\sqrt {\frac {b^{2}\sin ^{2}\theta -a^{2}\cos ^{2}\theta }{\sin ^{2}\theta -\cos ^{2}\theta }}}={\sqrt {\frac {b^{2}-a^{2}\cot ^{2}\theta }{1-\cot ^{2}\theta }}}

Devil's curves were discovered in 1750 by Gabriel Cramer, who studied them extensively.^[2]

The name comes from the shape its central lemniscate takes when graphed. The shape is named after the juggling game diabolo, which was named after the Devil^[3] and which involves two sticks, a string, and a spinning prop in the likeness of the lemniscate.^[4]

For $|b|>|a|$ , the central lemniscate, often called hourglass, is horizontal. For $|b|<|a|$ it is vertical. If $|b|=|a|$ , the shape becomes a circle. The vertical hourglass intersects the y-axis at $b,-b,0$ . The horizontal hourglass intersects the x-axis at $a,-a,0$ .

Electric Motor Curve

A special case of the Devil's curve occurs at ${\frac {a^{2}}{b^{2}}}={\frac {25}{24}}$ , where the curve is called the electric motor curve.^[5] It is defined by an equation of the form

$y^{2}(y^{2}-96)=x^{2}(x^{2}-100)$ .

The name of the special case comes from the middle shape's resemblance to the coils of wire, which rotate from forces exerted by magnets surrounding it.

References

^ "Devil's Curve". Wolfram MathWorld.
^ Introduction a l'analyse des lignes courbes algébriques, p. 19 (Genova, 1750).
^ "Diabolo Patent". Retrieved 16 July 2013.
^ Wassenaar, Jan. "devil's curve". www.2dcurves.com. Retrieved 2018-02-26.
^ Mathematical Models, p. 71 (Cundy and Rollet. 1961)