I'm wordering about parametric equations in the form $(x,y)(t)=(\cos^n(t),\sin^n(t))$ for $t\in[0,\frac{\pi}{2}]$ for various $n\in\mathbb{N}.$
$(\cos(t),\sin(t))$ is a (quarter of a) circle. For points on the curve, the Euclidian distance to the origin is $1$. This is obvious since $\sin^2(t)+\cos^2(t)=1$.
$(\cos^2(t),\sin^2(t))$ is a (quarter of a) square. For points on the curve, the Manhattan distance to the origin is $1$. Again, this is obivious since $|\sin^2(t)|+|\cos^2(t)|=\sin^2(t)+\cos^2(t)=1$.
$(\cos^3(t),\sin^3(t))$ is a (quarter of an) astroid. For lines tangent to the curve, the Euclidian distance between the $x$ and $y$ intercepts is $1$. Proof.
$(\cos^4(t),\sin^4(t))$ is (part of) a parabola. For lines tangent to the curve, the Manhattan distance between the $x$ and $y$ intercepts is $1$. The proof of this is along the same lines as $n=3$ case.
What is $(\cos^5(t),\sin^5(t))$, or the curve for any $n>5$? The $n=5$ curve looks pretty close to a hyperbola (but isn't one). Is it just a coincidence that the Euclidian vs Manhattan distance properties I found arise in pairs? Or can something similar be said for $n=5$ and $6$?