How can I transform an open disc to a sphere minus a point

Question

If I have an open disc $D$ centered at the origin in $\mathbb R^2$ with radius 1, how can I continuously transform $D$ to the sphere $S^\times$ with radius $1/2$ centered at $(0,0,1/2)$ excluding the point $(0,0,0)$. In other words, I'm looking for a continuous function $$f: D = \{x^2+y^2 < 1\} \to S^\times = \{x^2+y^2+(z-1/2)^2=1/4 \text{ : } (x,y,z) \neq (0,0,0)\}.$$

For example, the image of the line $x=0$ on the disc $D$ might map to the circle on $S^\times$ with $y^2+(z-1/2)^2 = 1/4$ and $x=0$ excluding the point $(0,0,0)$. I would think the circle in $D$ of radius $1/2$ centered at the origin might map to the "equator" of $S^\times$, given by $x^2+y^2=1/4$ and $z=1/2$. Neither of these are constraints, but examples of how such a function might behave.

Could somebody help me with one such example of a function fitting my description? Thanks.

Answer 1

This is a heuristic description of one such mapping. You can figure out the details.

Make the disc lie on the equatorial plane of the sphere. Since the disk radius is twice the sphere radius, it will stick out on all sides. For the portion of the disk inside sphere, project each point straight down to the sphere. Now you have an "ashtray": a halfsphere with a wide lip all around.

Next, bend the lip up all around: Consider the cylinder of radius $1/2$ about the $z$-axis. This cylinder circumscribes the sphere. For each point on the disk outside the cylinder, take the line from the point to the $z$-axis which makes a 45 degree angle with that axis. Project the point on disk to the intersection of this line and the cylinder. Now you have a cylindrical segment whose height matches its radius, capped off at its lower end by a hemisphere.

Finally, do an "anti-Mercatur" projection of the cylinder onto the upper half of the sphere. That is, draw a horizontal line segment from each point on the cylinder to the $z$-axis. This line segment will intersect the sphere at some point. The cylinder point maps to the intersection of the line segment and the sphere.