Projection to the sphere

Question

I want to map the set $\{(x,y)\in \mathbb{R}^2 | 0\leq x\leq 1, 0\leq y\leq x^2\}$ to the unit sphere $S^{2}\subset \mathbb{R}^3$ in a way so that the image of the point $(0,0)$ is the north pole and the cusp of the 2d set is still a cusp on the sphere. I wasn't capable to find the correct function term of the projection. Can someone help?

Answer 1

One can just use the map $(x,y) \mapsto (x,y,1)$ and then normalize the resulting 3d element. This should do the trick.