A serpentine curve is a curve whose Cartesian equation is of the form
x 2 y + a 2 y − a b x = 0 , a b > 0 {\displaystyle x^{2}y+a^{2}y-abx=0,\quad ab>0}
Its functional representation is
y = a b x x 2 + a 2 {\displaystyle y={\frac {abx}{x^{2}+a^{2}}}}
Its parametric equation for 0 < t < π {\displaystyle 0<t<\pi } is
x = a cot ( t ) {\displaystyle x=a\cot(t)}
y = b sin ( t ) cos ( t ) {\displaystyle y=b\sin(t)\cos(t)}
Its parametric equation for − π / 2 < t < π / 2 {\displaystyle -\pi /2<t<\pi /2} is
x = a tan ( t ) {\displaystyle x=a\tan(t)}
y = b sin ( t ) cos ( t ) {\displaystyle y=b\sin(t)\cos(t)}
It has a maximum at x = a {\displaystyle x=a} and a minimum at x = − a {\displaystyle x=-a} , given that
y ′ = a b ( a − x ) ( a − x ) ( a 2 + x 2 ) 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 ± b / 2 {\displaystyle \pm b/2} , which are independent of a {\displaystyle a} .
The inflection points are at x = ± 3 a {\displaystyle x=\pm {\sqrt {3}}a} , given that
y ″ = 2 a b x ( x 2 − 3 a 2 ) ( x 2 + a 2 ) 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
κ ( t ) = − 2 a b cot t ( cot 2 t − 3 ) ( b 2 cos 2 ( 2 t ) + a 2 csc 4 4 ) 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:
κ ( t ) = 2 a b x ( x 2 − 3 a 2 ) ( x 2 + a 2 ) 3 ( 1 + ( a 3 b − a b x 2 ) 2 ( x 2 + a 2 ) 4 ) 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 a = 2 {\displaystyle a=2} , resulting in the flipped curve equation
y 2 ( x 2 + 1 ) 2 = x 2 {\displaystyle y^{2}\left(x^{2}+1\right)^{2}=x^{2}}
which is equivalent to a serpentine curve with the parameters a = 1 , b = ± 1 {\displaystyle 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.
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