I think the ATLA world map is in a type of azimuthal projection by default. The following is the product of that observation. (OC)
(Repost due to weird UI shenanigans)
I noticed that the ATLA world map is unique in the fact that the poles are both uninterrupted, which is a quality that is prominent in azimuthal projections. While the map isn't explicitly stated to be in any specific projection, or really any projection at all, I felt that I would only be satisfied with a more detailed rendition of that map if I knew what projection the original could be in.
After scrolling the G.Projector for a while, just to see what options they had, I selected three azimuthal projections that I thought would best preserve the shape and artistic direction of the landmasses: stereographic, Berusing harmonic, and Lambertian equidistant. When the original map is overlaid on most other azimuthal projections, the longitudes further from 0 became increasingly distorted. The three I selected weren't without issue - Berusing harmonic and Lambertian equidistant still distorted shapes at the edges, just to a lesser degree; and stereographic being borderless was difficult to work with.
My first selection was the Lambertian equidistant. I obtained a gridmap from G.Projector and overlaid the ATLA world map over it. I then morphed grid cells one by one so they would fit the grid of an equirectangular projection, then drew over the coastlines to create a finished map. After about 5 hours of work, I became unsatisfied with the distortion and switched to the Berusing harmonic projection, to which I worked and additional 5 or 6 hours. I was again unsatisfied with the distortion, and then discovered my entire process was invalidated by the existence of GIS software, which I subsequently learned how to use for this purpose. Annoyingly, searching through the coordinate reference systems of qGIS, I didn't find Berusing harmonic or a sufficient stereographic CRS, so I had to settle with my least favorite, but most accessible Lambertian equidistant.
TL;DR: I wanted the ATLA world map to be more immersive, and the process of making it immersive was very time consuming.
For the final map, I must say I'm not entirely unsatisfied. It definitely makes the setting of ATLA feel much more grounded, which is the goal of this whole project. However, you can tell why the Lambertian equidistant azimuthal projection isn't ideal for this situation. The Fire Nation and far East of the Earth Nation are suuuper stretched. If there's no GIS software that allow conversion between stereographic/Berusing harmonic and equirectangular, I'm either going to have to manually stretch some landmasses which is very much not ideal, or I'm going to have to settle with these unusual shapes.
The map was projected into the following images above:
1: Equirectangular
2: Two-hemisphere orthographic
3: The original Lambertian equidistant
Edit: 1 spelling fix, slight change in wording, slight change in formatting (why does Reddit automatically cancel out your italics, and what happened to image captions?)