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

\( x^2y+a^2y-abx=0, \quad ab > 0. \)

Equivalently, it has a parametric representation

\( x=a\cot(t), y=b\sin (t)\cos(t), \)

or functional representation

\( y=\frac{abx}{x^2+a^2}. \)


Serpentine curves were studied by L'Hôpital and Huygens, and named and classified by Newton.

External links

MathWorld – Serpentine Equation

Mathematics Encyclopedia

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