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}$0}"/>

Its functional representation is

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

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

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

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

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

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

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

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

${\displaystyle y'={\frac {ab\left(a-x\right)\left(a-x\right)}{\left(a^{2}+x^{2}\right)^{2}}}=0}${\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 ${\displaystyle \pm b/2}${\displaystyle \pm b/2}, which are independent of ${\displaystyle a}${\displaystyle a}.

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

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

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

${\displaystyle \kappa (t)={\frac {2abx\left(x^{2}-3a^{2}\right)}{\left(x^{2}+a^{2}\right)^{3}\left(1+{\frac {\left(a^{3}b-abx^{2}\right)^{2}}{\left(x^{2}+a^{2}\right)^{4}}}\right)^{3/2}}}}${\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 ${\displaystyle a=2}${\displaystyle a=2}, resulting in the flipped curve equation^[4]

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

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