Serpentine curve

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

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

Equivalently, it has a parametric representation

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

or functional representation

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

The curve has an inflection point at the origin. It has local extrema at $x=\pm a$, with a maximum value of $y=b/2$ and a minimum value of $y=-b/2$.

History

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

Visual appearance

The serpentine curve for a = b = 1.

External links

• MathWorld – Serpentine Equation
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