Serpentine curve

A serpentine curve is a curve whose Cartesian equation is of the form^[1]

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

Its functional representation is

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

Its parametric equation for ${\displaystyle 0<t<\pi }$ is

${\displaystyle x=a\cot(t)}$

${\displaystyle y=b\sin(t)\cos(t)}$

Its parametric equation for ${\displaystyle -\pi /2<t<\pi /2}$ is^[2]

${\displaystyle x=a\tan(t)}$

${\displaystyle y=b\sin(t)\cos(t)}$

It has a maximum at ${\displaystyle x=a}$ and a minimum at ${\displaystyle x=-a}$, given that

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

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

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

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

In the parametric representation, its curvature is given by^[2]

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

An alternate parametric representation:^[3]

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

A generalization of the curve is given by the flipped curve when ${\displaystyle a=2}$, resulting in the flipped curve equation^[4]

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

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

L'Hôpital and Huygens had studied the curve in 1692, which was then named by Newton and classified as a cubic curve in 1701.^[2]