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An aspect of my work led to a plane curve with implicit equation $$ x^2+y^2 = 3 (y/2)^{2/3} + 1 $$ Actually, I started with the parametrization below and derived from it the equation above: \begin{eqnarray} x(t) &=& t (3-2 t^2) \\ y(t) &=& 2(1-t^2)^{3/2} \end{eqnarray}

Here is what it looks like:
           Plots

If this falls in some classical class of curves, and perhaps even has a name, I would like to reference it appropriately. Does anyone recognize this curve? Thanks!

Answered. By Sylvain Bonnot and Francesco Polizzi: It is a type of nephroid! Here's the Wikipedia image from the article they both cited:
          Wiki image

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It is almost a nephroid (en.wikipedia.org/wiki/Nephroid ), except I can't exactly make the reparametrization right now. –  Jan Jitse Venselaar May 31 '12 at 13:28
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2 Answers

up vote 8 down vote accepted

Your curve is a nephroid, see http://en.wikipedia.org/wiki/Nephroid.

The general equation of such a plane curve is $$(x^2+y^2-4a^2)^3=108a^4y^2.$$ Your example corresponds to the value $a=\frac{1}{2}$ of the parameter.

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Thanks, Francesco! –  Joseph O'Rourke May 31 '12 at 13:31
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Pretty curve...I think it is a Nephroid: http://en.wikipedia.org/wiki/Nephroid

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Thanks, Sylvain! –  Joseph O'Rourke May 31 '12 at 13:31
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