# Parametrising the surface $x^2+z^2 = y^3(1-y)^3$

I am working on the following exercise:

Consider the surface given by the equation $$x^2+z^2 = y^3(1-y)^3$$. Where can a parametrisation $$z(x,y)$$ be introduced? Calculate $$\partial z / \partial_x$$ and $$\partial z / \partial_y$$.

REMARK: I found out that the surface looks like this here:

https://imaginary.org/gallery/herwig-hauser-classic

I reorded the equation to: $$z^2 = y^3(1-y)^3-x^2$$ , which implies

$$z = \pm\sqrt{y^3(1-y)^3-x^2}$$

So I think a parametrisation of the surface by $$z(x,y)$$ is not possible, every point $$(x,y)$$ gets mapped to two different $$z$$ unless $$y^3(1-y)^3-x^2 = 0.$$ Is there something I am missing in here?

• Not sure what the sentence "Where can a parametrisation $z(x, y)$ be introduced?" means: by restricting $z > 0$ say, then you can choose the $+$ sign in your formula and get a parametrisation? – NickD Oct 21 '19 at 21:55
• I suppose it is meant in the way that you can not just restrict to one sign for the root. – 3nondatur Oct 21 '19 at 21:57

No, you are missing nothing. For each $$(x,y,z)$$ in your surface, $$(x,y,-z)$$ belongs to the surface too. And both points have the same first and second coordinates. So, unless $$z=0$$, there are two points of the surface with the same first and second coordinates and therefore your surface is not of the form$$\left\{\bigl(x,y,z(x,y)\bigr)\,\middle|\,(x,y)\in D\right\}.$$for some subset $$D$$ of $$\mathbb R^2$$ and some function $$z\colon D\longrightarrow\mathbb R$$.
One can parameterize $$x^2+z^2=y^3(1-y)^3$$ on $$[0,1]^2$$ as $$\left((t(1-t))^{3/2}\cos(2\pi s),\,t,\,(t(1-t))^{3/2}\sin(2\pi s)\right)$$ As given, the radius in the $$y=\frac12$$ plane is $$\frac18$$:
However, as with a lot of orientable, closed surfaces, almost all lines that intersect the surface (entering the interior) will intersect the surface again (exiting the interior). Thus, the lines for most $$x$$ and $$y$$ will intersect the surface at more than one point. This prevents a complete parameterization of the form $$(x,y,z(x,y))$$.