Serpentine Curve

A curve named and studied by Newton in 1701 and contained in his classification of cubic curves. It had been studied earlier by L'Hospital and Huygens in 1692 (MacTutor Archive).

The curve is given by the Cartesian equation

(1)

It has parametric equations

			(2)
			(3)

for or

			(4)
			(5)

for .

The curve has a maximum at and a minimum at , where

(6)

Interestingly, the minimum and maximum values are , which are independent of .

And inflection points at , where

(7)

In the parametric representation, the curvature is given by

(8)