# Serpentine curve

For serpentine shapes in architecture and design, see Serpentine shape.

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

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

## History

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

## Visual appearance

The serpentine curve for a = b = 1.