Serpentine curve

This article is about the mathematical concept. For the design feature, see Serpentine shape.

A serpentine curve is a curve whose Cartesian equation is of the form

$x^{2}y+a^{2}y-abx=0,\quad ab>0$

Its functional representation is

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

Its parametric equation for $0<t<\pi$ is

$x=a\cot(t)$

$y=b\sin(t)\cos(t)$

Its parametric equation for $-\pi /2<t<\pi /2$ is

$x=a\tan(t)$

$y=b\sin(t)\cos(t)$

It has a maximum at $x=a$ and a minimum at $x=-a$, given that

$y'={\frac {ab\left(a-x\right)\left(a-x\right)}{\left(a^{2}+x^{2}\right)^{2}}}=0$

The minimum and maximum points are at $\pm b/2$, which are independent of $a$.

The inflection points are at $x=\pm {\sqrt {3}}a$, given that

$y''={\frac {2abx\left(x^{2}-3a^{2}\right)}{\left(x^{2}+a^{2}\right)^{3}}}=0$

In the parametric representation, its curvature is given by

$\kappa (t)=-{\frac {2ab\cot t\left(\cot ^{2}t-3\right)}{\left(b^{2}\cos ^{2}\left(2t\right)+a^{2}\csc ^{4}4\right)^{3/2}}}$

An alternate parametric representation:

$\kappa (t)={\frac {2abx\left(x^{2}-3a^{2}\right)}{\left(x^{2}+a^{2}\right)^{3}\left(1+{\frac {\left(a^{3}b-abx^{2}\right)^{2}}{\left(x^{2}+a^{2}\right)^{4}}}\right)^{3/2}}}$

A generalization of the curve is given by the flipped curve when $a=2$, resulting in the flipped curve equation

$y^{2}\left(x^{2}+1\right)^{2}=x^{2}$

which is equivalent to a serpentine curve with the parameters $a=1,b=\pm 1$.

History

L'Hôpital and Huygens had studied the curve in 1692, which was then named by Newton and classified as a cubic curve in 1701.

Visual appearance

The serpentine curve for a = b = 1.