Devil's curve

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^[1]

Devil's curve for a = 0.8 and b = 1.

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

$${\displaystyle y^{2}(y^{2}-b^{2})=x^{2}(x^{2}-a^{2})}$$

The polar equation of this curve is of the form

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

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

Electric Motor Curve

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

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

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.