Why is the plot of $\left|x\right|^{\frac{2}{3}}+\left|y\right|^{\frac{2}{3}}=4^{\frac{2}{3}}$ an astroid.

Question

I was exploring the implicit function $\left|x\right|^{n}+\left|y\right|^n = 1$ and noticed that when $n$ is between 0 and 1 the plot looks similar to an astroid (The resulting curve of tracing a point that is rolling inside a circle with a radius that is 4 times greater) and I noticed that at exactly $n=2/3$ the 2 plots were the same. Is there any reason for this? Desmos demonstration

Answer 1

The astroid is a special case of a hypocycloid. If you take $k=4$ in the parametrization given in that link, you get \begin{align*} x &= 3\cos\theta + \cos 3\theta \\ y &= 3\sin\theta - \sin 3\theta. \end{align*} Now the triple angle identities (use deMoivre's formula if you know it, addition formulas in trigonometry otherwise) come to your rescue: \begin{align*} \cos 3\theta &= 4\cos^3\theta - 3\cos\theta \\ \sin 3\theta &= 3\sin\theta - 4\sin^3\theta. \end{align*} Inserting these, we find \begin{align*} x &= 4\cos^3\theta \\ y &= 4\sin^3\theta. \end{align*} Taking the $2/3$ power gives \begin{align*} |x|^{2/3} &= 4^{2/3}\cos^2\theta \\ |y|^{2/3} &= 4^{2/3}\sin^2\theta, \end{align*} from which we deduce that $|x|^{2/3} + |y|^{2/3} = 4^{2/3}$.