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Serpentine curve

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This article is about the mathematical concept. For the design feature, see Serpentine shape.
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A serpentine curve is a curve whose 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.}

Equivalently, it has a parametric representation

x = a cot ⁡ ( t ) {\displaystyle x=a\cot(t)} , y = b sin ⁡ ( t ) cos ⁡ ( t ) , {\displaystyle y=b\sin(t)\cos(t),}

or functional representation

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

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


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