Serpentine curve

A serpentine curve is a curve whose equation is of the form
 * $$x^2y+a^2y-abx=0, \quad ab > 0.$$

Equivalently, it has a parametric representation
 * $$x=a\cot(t)$$, $$y=b\sin (t)\cos(t),$$

or functional representation
 * $$y=\frac{abx}{x^2+a^2}.$$

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

History
Serpentine curves were studied by L'Hôpital and Huygens, and named and classified by Newton.