Recognize this plane curve? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T06:28:43Z http://mathoverflow.net/feeds/question/98482 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98482/recognize-this-plane-curve Recognize this plane curve? Joseph O'Rourke 2012-05-31T13:00:49Z 2012-05-31T13:46:28Z <p>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: <code>\begin{eqnarray} x(t) &amp;=&amp; t (3-2 t^2) \\ y(t) &amp;=&amp; 2(1-t^2)^{3/2} \end{eqnarray}</code></p> <p>Here is what it looks like: <br /> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img src="http://cs.smith.edu/~orourke/MathOverflow/KnightsVisorImplicitEqn.jpg" alt="Plots" /> <br /></p> <p>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!</p> <p><b>Answered</b>. By Sylvain Bonnot and Francesco Polizzi: It is a type of <em>nephroid</em>! Here's the Wikipedia image from the article they both cited: <br /> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<img src="http://upload.wikimedia.org/wikipedia/commons/0/02/EpitrochoidOn2.gif" alt="Wiki image"></p> http://mathoverflow.net/questions/98482/recognize-this-plane-curve/98483#98483 Answer by Sylvain Bonnot for Recognize this plane curve? Sylvain Bonnot 2012-05-31T13:20:50Z 2012-05-31T13:20:50Z <p>Pretty curve...I think it is a Nephroid: <a href="http://en.wikipedia.org/wiki/Nephroid" rel="nofollow">http://en.wikipedia.org/wiki/Nephroid</a></p> http://mathoverflow.net/questions/98482/recognize-this-plane-curve/98484#98484 Answer by Francesco Polizzi for Recognize this plane curve? Francesco Polizzi 2012-05-31T13:26:21Z 2012-05-31T13:26:21Z <p>Your curve is a <em>nephroid</em>, see <a href="http://en.wikipedia.org/wiki/Nephroid" rel="nofollow">http://en.wikipedia.org/wiki/Nephroid</a>.</p> <p>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.</p>