injective map between the semi-unit circle and a parabola How can we injectively map the semi-unit circle to a parabola bounded between two points?
I earnestly think this can be done, as the curves are homeomorphic (one can be "bent" to the other), but the explicit map is out of my imagination. Any hints? Thanks beforehand.

Thanks to the hint provided by @AdamRubinson, supposing for the sake of representation, we wish to find a map between the upper unit semicircle and the parabolic segment $x=y^2$ bounded between $(0,0)$ and $(0,1)$. Then, the map $f(x,y)=\left(\frac{(x+1)^2}{4},\frac{x+1}{2}\right)$ does the job. For other cases, appropriate scaling can be done.
