Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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:

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

share|improve this question
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

2 Answers 2

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.

share|improve this answer
Thanks, Francesco! –  Joseph O'Rourke May 31 '12 at 13:31

Pretty curve...I think it is a Nephroid: http://en.wikipedia.org/wiki/Nephroid

share|improve this answer
Thanks, Sylvain! –  Joseph O'Rourke May 31 '12 at 13:31

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.