Serpentine curve

A serpentine curve is a curve whose equation is of the form

${\displaystyle x^{2}y+a^{2}y-abx=0,\quad ab>0.}$

Equivalently, it has a parametric representation

${\displaystyle x=a\cot(t)}$, ${\displaystyle y=b\sin(t)\cos(t),}$

or functional representation

${\displaystyle y={\frac {abx}{x^{2}+a^{2}}}.}$

The curve has an inflection point at the origin. It has local extrema at ${\displaystyle x=\pm a}$, with a maximum value of ${\displaystyle y=b/2}$ and a minimum value of ${\displaystyle y=-b/2}$.

Solving for x, we get

${\displaystyle x={\frac {(x^{2}+a^{2})y}{ab}}}$