Let's suppose we have a line with a known length $L$ that we transform from a straight line to a curved one matching a semi-circle.

Is there a way to find a formula for the height of the smallest rectangle containing that line? (the width of that rectangle will be the variable)

We can consider the thickness of the line to be equal to $1$.

We have $f(L) = 1$ and $f(2*L/\pi) = L/\pi$

What about $f(x)$ where $(2*L)/pi\le x \le L$ ?

It will be something for the web, so if it's not possible to find the exact formula and we can approximate it, then it's fine. Having a few pixels off won't be a big deal in my case.

Let's suppose we have a line with a known length $L$ that we transform from a straight line to a curved one matching a semi-circle.

Is there a way to find a formula for the height of the smallest rectangle containing that line? (the width of that rectangle will be the variable)

We can consider the thickness of the line to be equal to $1$.

We have $f(L) = 1$ and $f\left(\frac{2L}{\pi}\right) = \frac{L}{\pi}$

What about $f(x)$ where $\frac{2L}{\pi}\le x \le L$ ?

It will be something for the web, so if it's not possible to find the exact formula and we can approximate it, then it's fine. Having a few pixels off won't be a big deal in my case.