Homeomorphism from the interior of a unit disk to the punctured unit sphere I need help constructing a homeomorphism from the interior of the unit disk,  $\{(x,y)\mid x^2+y^2<1\}$, to the punctured unit sphere, $\{(x,y,z)\mid x^2+y^2+z^2 = 1\} - \{(0,0,1)\}$. I was thinking you could take a line passing through $(0,0,1)$ and a point in the disk and send that point to the part of the sphere the line passes through, but this function wouldn't cover the top half of the sphere.
 A: Your idea is fine if you use the whole plane (stereographic projection). Then compose with a homeomorphism between the plane and the unit disk.
A: Stereographic projection, as mentioned in the other answers, usually comes up in this context. Indeed, it is a beautiful way of identifying the punctured sphere with $\mathbb{R}^2$, in a conformal manner (preserving angles).
However, if you only care about finding a homeomorphism to the disk (and there is no way to make this conformal anyway),  then perhaps it is easier to just define the homeomorphism from the punctured sphere to the unit disk by
$$ \phi(x,y,t) := \frac{t+1}{2} e^{i\theta}, $$
where $\theta$ is the angle of the point in the $x-y$-plane; i.e. $\theta=\arg(x+iy)$.
If you prefer real coordinates, this means
$$ \phi(x,y,t) = \frac{t+1}{2\sqrt{x^2+y^2}} \cdot (x,y).$$
A: You can make that homeomorphism by coposing two homeomorphisms $T_1$ and $T_2$, which $T_1:S^3\setminus\{(0,0,1)\}\rightarrow\mathbb{R}^2$ is the  stereographic projection given by:$$T_1(x,y,z) = (\frac{x}{1-z},\frac{y}{1-z})$$
and $T_2$ is any homeomorphism between $\mathbb{R}^2$ and unit ball, $B^2$, writing in the polar coordinate for example:$$T_2(r,\theta) = (\frac{r}{1+r}, \theta)$$
and note that composition of two homeomorphism is a homeomorphism.
