Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have no idea how this equation: \begin{equation} (x^2 + y^2 - 1)^3 - x^2 y^3 = 0 \end{equation}

Produces this picture:

enter image description here

Can someone provide a general explanation of plotting this function?

share|cite|improve this question
The answer depends on what kind of software you wish to use. The picture you provide looks as if it has been generated by Mathematica. The command would be ContourPlot[f[x,y],{x,-2,2},{y,-2,2}]. – Eckhard Dec 22 '12 at 14:54
@Eckhard: I'm asking of some explanation like this. – m0nhawk Dec 22 '12 at 15:01
In general, plotting implicit functions is done by algorithms such as marching squares. – Peter Sheldrick Dec 22 '12 at 19:00
up vote 3 down vote accepted

The solution set is obviously symmetric with respect to the $y$-axis. Therefore we may assume $x\geq 0$. In the domain $\{(x,y)\in {\mathbb R}^2\ |\ x\geq0\}$ the equation is equivalent with $$x^2+ y^2 -1=x^{2/3} y\ ,$$ which can easily be solved for $y$: $$y={1\over2}\bigl(x^{2/3}\pm\sqrt{x^{4/3}+4(1-x^2)}\bigr)\ .$$ Now plot this, taking both branches of the square root into account. You might have to numerically solve the equation $x^{4/3}+4(1-x^2)=0$ in order to get the exact $x$-interval.

share|cite|improve this answer

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.