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[parent] asymptote of Lamé's cubic (Example)

We will show that the Lamé's cubic

$\displaystyle x^3+y^3 = a^3,$ (1)

where $ a$ is a positive constant, has the line
$\displaystyle y = \underbrace{-x}_{g(x)}$
as its asymptote.

Because the equation (1) of the curve is symmetric with respect to $ x$ and $ y$, the curve is symmetric about the line $ y = x$. From the solved form

$\displaystyle y = \underbrace{\sqrt[3]{a^3-x^3}}_{f(x)}$ (2)

of (1) we see that every real value of $ x$ gives one point of the curve.

\begin{pspicture}(-4,-4)(4,4) \psaxes[Dx=9,Dy=9]{->}(0,0)(-3.5,-3.5)(3.5,3.5) \r... ...\rput(0.5,-4){Lam\'e's cubic\, $y = \sqrt[3]{a^3-x^3}$\, (blue)} \end{pspicture}

The difference $ \Delta = f(x)\!-\!g(x)$ represents the distance of a point $ (x,\,y)$ of the curve and the point of the asserted asymptote $ y = -x$ with the same abscissa $ x$. We multiply the numerator and denominator with the expression $ (\sqrt[3]{a^3-x^3})^2-x\sqrt[3]{a^3-x^3}+x^2$ for being able to utilise the polynomial formula

$\displaystyle (u+v)(u^2-uv+v^2) = u^3+v^3,$
getting
$\displaystyle \Delta$ $\displaystyle = f(x)\!-\!g(x)$    
  $\displaystyle = \frac{\sqrt[3]{a^3-x^3}+x}{1}$    
  $\displaystyle = \frac{(\sqrt[3]{a^3-x^3})^3+x^3}{(\sqrt[3]{a^3-x^3})^2-x\sqrt[3]{a^3-x^3}+x^2}$    
  $\displaystyle = \frac{a^3}{(\sqrt[3]{a^3-x^3})^2-x\sqrt[3]{a^3-x^3}+x^2}.$    

Thus, $ \displaystyle \Delta \to \frac{a^3}{\infty+\infty+\infty} = 0$ when $ \vert x\vert \to \infty$ (see the improper limits). According to the definition of asymptote, the line $ y = -x$ is asymptote of Lamé's cubic.



"asymptote of Lamé's cubic" is owned by pahio.
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See Also: hyperbola, witch of Agnesi

Also defines:  Lamé's cubic

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Cross-references: improper limits, polynomial, expression, denominator, numerator, abscissa, distance, difference, point, real, symmetric about, symmetric, curve, equation, asymptote, line, positive
There is 1 reference to this entry.

This is version 4 of asymptote of Lamé's cubic, born on 2008-03-14, modified 2008-03-16.
Object id is 10403, canonical name is AsymptoteOfLamesCubic.
Accessed 302 times total.

Classification:
AMS MSC: 26C05 (Real functions :: Polynomials, rational functions :: Polynomials: analytic properties, etc.)
 53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)
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