I am currently on the hunt for a way to approximate a half-circle function $$t \to f(t) = \sqrt{1-t^2}$$ By means of truncated power series (polynomials). I found several issues with a previous polynomial / power series approach to approximate a circle here. In that question, I tried approximating on the set $t\in [-1,1]$. I suspect one of the main issues arise from the fact that $$\frac{\partial f}{\partial t}(\mp 1) = \pm\infty$$
Something which polynomials cannot achieve at any point. So then the next idea would be to limit the definition set for our polynomial so that these critical points $(t,f(t) = (\pm 1, 0)$ are not included in the approximation. In order to do this we might do better to try and utilize several symmetries which exist for this function.
Two symmetries that can be used are:
$f(t) = f(-t)$, this lets us restrict to fit only for $t\in [0,1]$ or $|t| \to f(t)$
The point ($t,f(t)$) reflected in line $t-f(t)=0$, we can reduce the interval to $t\in [0,1/\sqrt{2}]$
Now to the question, how can we (assuming we can approximate well with a polynomial on this shrunken, restricted set) handle the points $t\in[1/\sqrt{2},1]$ ?
We have the identity $t^2+f(t)^2-1 = 0$. How - if at all - could this be useful?
Or is it maybe better to use some function solver algorithm? Like Newton's?