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.

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

. Solving for x, we get

x=\frac

History

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

Visual appearance