Mapping a Toroid 'Planet'?

[samkatz]

Hello party people. I'm toying with the idea of making the setting for a TTRPG game a hoop/donut, but I'm having trouble thinking about how to map it. I want to 1) represent it in 2D, and 2) project it to the 3D shape. What I have so far for each:

1) So I'm thinking the best map projection will be essentially two trapezoids joined at the wide edge. That joint wide edge will be the larger equator (on the outside of the planet), and the top and bottom will represent the smaller equator (inside the donut hole). That way each latitude is the right relative length/circumference to the others. (Unless there's some reason the circumference would decrease non-linearly that I can't think of?)

2) This I'm totally ill-equipped for. There are tools that map equirectangular projections to globes for me, so I've never bothered to learn the actual math involved, or how to invoke such math on an image. I think wrapping up the above-described image into a cylinder (kind of a cone?) and then folded around into a donut will do it without much distortion. Am I wrong? How would I do it?

3) I might be back asking for guidance with weird coriolis forces and seasons on such a world. This is the best treatment I can find, but there some details I haven't figured out how to fill in.

[samkatz]

Okay, now I'm thinking something like this might be better / easier to fold into a torus, but I'm still not sure how to approach the mathematics of switching between this and A) the 3D torus view itself and B) an uninterrupted plane so I can draw landmasses.

[Falconius]

I wouldn't try and draw it on a plane, it's a losing proposition. Your best bet is to draw it on a 3d model. Doing so has the added of advantage of having UV maps which will unwrap to a flat plane in the 'projection' that you set them up with. I think that the edge on projection you have there is probably your best bet, because of the donut hole in the middle, which can also unwrap nicely that way.

[Tiana]

If I wanted to do that, I'd probably make a 3D toroidal shape (easily achieved, you will just have to drop in a model someone else made), unwrap it, save that UV map, and then draw on that, which can theoretically be then rewrapped around the 3D object in Blender.