Goal: To gain a better understanding of Euclidean space, $\Bbb R^3,$ conformally compactified into a unit sphere.
Question: How can I visualise and mathematically describe Euclidean space, $\Bbb R^3,$ conformally compactified into a unit sphere? Is my model of it correct?
My attempt:
Start with the conformally compactified disk model of Euclidean geometry (the arcs are the geodesics):
and extend it to three dimensional space. So rotate the disk model (rotate the arcs) to get a collection of surfaces one of which looks like this, (of varying volume, s.t. there will be disjoint "layers" of these surfaces with different volume, with the largest volume corresponding to the surface with the volume of the unit sphere):
Then take the space of the collection of all these surfaces, with endpoints equally spaced across the unit sphere (subspace of four structures represented below):
I think if you eliminated the arcs, you'd be left with the space of lines going through the center of the sphere, which would be the real projective plane, $\Bbb R P^2.$
Could you consider not just the manifold created by identify these antipodal points,$\Bbb R P^2$, immersed in $\Bbb R^3,$ but also include the arcs I've described to make a manifold and immerse it into $\Bbb R^3?$ How would it be different than let's say Boy's surface?