In what the parametrisation $(x,\sqrt{1-x^2})$ of the circle is interesting? We all know the parametrisation $\gamma (t)=(\cos(t),\sin(t))$ of the circle that has the advantage to be smooth and is easy to use. Sometime, my teacher use the parametrisation $$\left\{\varphi(t)=\left(t,\sqrt{1-t^2}\right)\mid t\in [-1,1]\right\}\cup\left\{\tilde \varphi(t)=\left(t,-\sqrt{1-t^2}\right)\mid t\in [-1,1]\right\},$$
and I would be curious to know in what this parametrization can be interesting. Moreover, the speed goes to infinity when $x\to \pm 1$, so it's not very good, is it ? I would be curious to know is what this paramtrisation is more interesting rathe $t\longmapsto (\cos t,\sin t)$.

The other parametrisation comes from pythagorean version of circle:
 $ x^2+y^2=1 $
This version of circle is of form $f(x,y)=1$.
To get the parametrisation, you need to use substitution $y=f(x)$, i.e.
$f(x,f(x))=1$ and thus $x^2+f(x)^2=1$, which means $f(x)^2 = 1-x^2$ which gives $f(x)=\pm \sqrt{1 - x^2}$.
The interesting part is that this same process can be done to any equation of the form $f(x,y)=c$.
