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

${\displaystyle y^{2}(y^{2}-b^{2})=x^{2}(x^{2}-a^{2}).}$^[1]

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 ${\displaystyle a,-a,0}$.