A serpentine curve is a curve whose Cartesian equation is of the form
Its functional representation is
Its parametric equation for is
Its parametric equation for is
It has a maximum at and a minimum at , given that
The minimum and maximum points are at , which are independent of .
The inflection points are at , given that
In the parametric representation, its curvature is given by
An alternate parametric representation:
A generalization of the curve is given by the flipped curve when , resulting in the flipped curve equation
which is equivalent to a serpentine curve with the parameters .
L'Hôpital and Huygens had studied the curve in 1692, which was then named by Newton and classified as a cubic curve in 1701.