# .

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

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

History

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