Can you give a piecewise smooth parametrization of the astroid

Question

The astroid is given by $\phi(\theta) = (cos^3(\theta),sin^3(\theta))$. It is not smooth as the derivative of the function at $0, \frac{\pi}{2}, \frac{3\pi}{2}$ and $2\pi$ is $0$. However, is it possible to come up with a piecewise smooth parametrization? I have tried multiple attempts but I have no idea how to proceed. Is it even possible?

Answer 1

I think you wonder if one fourth of the astroid is part of a graph of a smooth function. In other words, is the function $y(x)$ given in implicit form as $x^{2/3}+y^{2/3}=a^{2/3}$ smooth around $x=a$, $x\le a$.. So $y=(a^{2/3}-x^{2/3})^{3/2}$. What do you think?

Added:

One piece of the astroid is the graph of the function $y(x)=(a^{2/3}-x^{2/3})^{3/2}$ on $[0,a]$. We have $$y'(x)=-\frac{\sqrt{a^{2/3}-x^{2/3}}}{x^{1/3}}\\ y''(x)=\frac{a^{2/3}}{3 x^{4/3}\sqrt{a^{2/3}-x^{2/3}}} $$

on $(0,a)$. We see that $y$ is a $C^1$ function on $(0,a]$, but not a $C^{2}$ function.