Serpentine curve

For serpentine shapes in architecture and design, see Serpentine shape.

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

x^{2}y+a^{2}y-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}}}.

History

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

The serpentine curve for a = b = 1.

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