Looking for a family of astroids

Question

I'm wondering what's the formula for a family of curves. Specifically the astroid. A few requirements: There should be one main one and then a bunch of them nestled inside. At each of the cusp-points, all of them are exactly at the (0,1), (1,0), (-1, 0), (0, -1) points, and only there. I've tried a bunch of formulas in which I use varying degrees of offset, but that doesn't work. If you come up with a formula, it should be in parametric equation style. The reason I'm doing this is for a personal wood working project. I want to make something cool out of wood using a computer controlled wood router. Also, It would look cool.

Answer 1

Do you mean something like $$ x = |\cos t|^{p-1} \cos t,\qquad y = |\sin t|^{p-1} \sin t, $$ for $p > 2$, which parametrizes $|x|^{2/p} + |y|^{2/p} = 1$ if $0 \leq t \leq 2\pi$?

Here's a plot for $2.5 \leq p \leq 7$: