Serpentine Curve

$\begin{figure}\begin{center}\BoxedEPSF{serpentine_curve.epsf}\end{center}\end{figure}$

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

$\begin{displaymath} y(x)={abx\over x^2+a^2} \end{displaymath}$

(1)

and parametric equations

$\displaystyle x(t)$	$\textstyle =$	$\displaystyle a\cot t$	(2)
$\displaystyle y(t)$	$\textstyle =$	$\displaystyle b\sin t\cos t.$	(3)

The curve has a Maximum at and a Minimum at , where

$\begin{displaymath} y'(x)={ab(a-x)(a+x)\over(a^2+x^2)^2}=0, \end{displaymath}$

(4)

and inflection points at $x=\pm\sqrt{3}\,a$ , where

$\begin{displaymath} y''(x)={2abx(x^2-3a^2)\over(x^2+a^2)^3}=0. \end{displaymath}$

(5)

The Curvature is given by

$\displaystyle \kappa(x)$	$\textstyle =$	$\displaystyle {2abx(x^2-3a^2)\over(x^2+a^2)^3\left[{1+{(a^3b-abx^2)^2\over(x^2+a^2)^4}}\right]^{3/2}}$	(6)
$\displaystyle \kappa(t)$	$\textstyle =$	$\displaystyle -{4\sqrt{2}\,ab[2\cos(2t)-1]\cot t\csc^2 t\over \{b^2[1+\cos(4t)]+2a^2\csc^4 t\}^{3/2}}.$	(7)

References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 111-112, 1972.