My question pertains to the Lamé curve. We have the equation: $$ \left|\frac{x}{a}\right|^n+\left|\frac{y}{b}\right|^n = 1. $$
If we have a circle of radius 1, turned "inside" like a Lamé curve with $n < 1$, we have something like: $$ \left|x\right|^n + \left|y\right|^n = 1. $$ I am trying to find the exact solution to n where the Lamé curve lines up with the quarter sections of the circles closes to the origin. In other words, the Lamé curve that approximates: $$ (x \pm 1)^2 + (y \pm 1)^2 = 1, \quad -1 \leq x \leq 1, -1 \leq y \leq 1. $$ The closest approximation I have is: $n = 0.564168562$, a number that cannot be neatly described in radians as far as I know, nor a constant I recognize off the top of my head.
In general, a Lamé curve cannot line up with an arc of a circle. Take the points $(1,0),(0,1)$ and $(2^{-1/n},2^{-1/n})$ on the Lamé curve $|x|^n + |y|^n = 1.$ Let us assume that we are working in the first quadrant (so $x\geq 0$ and $y \geq 0$) the other cases follow by symmetry.
The circle passing through these points is $(x-a)^2 +(y-b)^2 +c=0.$ Where $$ a=b = \frac{ -2^{-1/n} + 2^{(1-n)/n)}}{-2+2^{1/n}}. $$
Solving $a=1$ for $n$ gives $n=-\frac{\ln(2)}{\ln\left(1-\frac{\sqrt{2}}{2}\right)} \approx 0.5644763827.$
You can check that the point $(1/2, (1-(1/2)^n)^{1/n})$ lies on the Lamé curve, but not on the circle.