a question about how to parametrize a surface in $R^3$

Question

Given a surface $$x^4/a^4+y^4/b^4+z^4/c^4=1$$,how can I parametrize the surface using X(u,v).

I tried to use $x=a\sqrt{cos(\theta)sin(\phi)}$,$y=b\sqrt{cos(\theta)sin(\phi)}$,and $z=c\sqrt{sin(\phi)}$,but turns out it couldn't include all the points of the surface,can some help me solve it?

Answer 1

It looks to me like there are three issues with your parametrization

• $x$,$y$, and $z$ will only be the same sign as $a$, $b$, and $c$, respectively.
• The given square roots are not defined unless their arguments are positive.
• You use $\cos\theta\sin\phi$ twice and $\sin\phi$ once, instead of something like $\cos\theta\cos\phi$, $\cos\theta\sin\phi$, and $\sin\theta$.

I would tweak this by setting

• $x = a \,\textrm{sign}(\cos\theta\cos\phi) \sqrt{|\cos\theta\cos\phi|}$
• $y = b \,\textrm{sign}(\cos\theta\sin\phi)\sqrt{|\cos\theta\sin\phi|}$
• $z = c \,\textrm{sign}(\sin\theta)\sqrt{|\sin\theta|}$

Here $\textrm{sign}(x)$ is equal to 1 for positive $x$, $-1$ for negative $x$, and $0$ for $x=0$. You can also write it as $\textrm{sign}(x) = x/|x|$ for nonzero $x$. For instance, if $\sin\theta$ is nonzero we would have $z = \dfrac{c \sin\theta}{\sqrt{|\sin\theta|}}$.