The Riemann sphere

We can map the unit disk $\{u^2+v^2<1\}$ onto the lower hemisphere of the unit sphere simply by appending the appropriate $z$-coordinate, namely,

$$(u,v)\mapsto (u,v,-\sqrt{1-u^2-v^2})$$

This maps line segments in the disk to semicircles in the lower hemisphere meeting the equatorial circle at right angles.