Serpentine curve

 ( Cubic Curves ) 

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

$x^2y+a^2y-abx=0,\; \backslash quad\; ab\; >\; 0$

Its functional representation is

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

Its parametric equation for is

$x=a\backslash cot(t)$

$y=b\backslash sin(t)\backslash cos(t)$

Its parametric equation for is

$x=a\backslash tan(t)$

$y=b\backslash sin(t)\backslash cos(t)$

It has a maximum at $x\; =\; a$ and a minimum at $x\; =\; -a$, given that

$y\text{'}\; =\; \backslash frac\{ab\backslash left(a-x\backslash right)\backslash left(a-x\backslash right)\}\{\backslash left(a^2+x^2\backslash right)^2\}\; =\; 0$

The minimum and maximum points are at $\backslash pm\; b\; /\; 2$, which are independent of $a$.

The inflection points are at $x\; =\; \backslash pm\; \backslash sqrt\{3\}a$, given that

$y\text{'}\text{'}\; =\; \backslash frac\{2abx\backslash left(x^2\; -\; 3a^2\backslash right)\}\{\backslash left(x^2+a^2\backslash right)^3\}\; =\; 0$

In the parametric representation, its curvature is given by

$\backslash kappa(t)\; =\; -\backslash frac\{2ab\; \backslash cot\; t\; \backslash left(\backslash cot^2\; t\; -\; 3\backslash right)\}\{\backslash left(b^2\; \backslash cos^2\; \backslash left(2t\backslash right)\; +\; a^2\; \backslash csc^4\; 4\backslash right)^\{3/2\}\}$

An alternate parametric representation:

$\backslash kappa(t)\; =\; \backslash frac\{2abx\backslash left(x^2\; -\; 3a^2\; \backslash right)\}\{\backslash left(x^2\; +\; a^2\; \backslash right)^3\; \backslash left(1\; +\; \backslash frac\{\backslash left(a^3\; b\; -\; abx^2\backslash right)^2\}\{\backslash left(x^2\; +\; a^2\backslash right)^4\}\backslash right)^\{3/2\}\}$

A generalization of the curve is given by the Flipped Curve when $a\; =\; 2$, resulting in the flipped curve equation

$y^2\backslash left(x^2+1\backslash right)^2\; =\; x^2$

which is equivalent to a serpentine curve with the parameters $a\; =\; 1,\; b\; =\; \backslash pm1$.

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History L'Hôpital and Huygens had studied the curve in 1692, which was then named by Isaac Newton and classified as a cubic curve in 1701.

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Visual appearance

Categories: Cubic Curves

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