Looking at the stereographic projection, there is a longer distance between points the father you get from the center of the map. Although the latitude lines remain circular in a polar projection, the map scales to avoid distortion father from the constant growth of the map once you leave the projected hemisphere. The northern hemisphere in an artic projection still must distort, making geometry a mess.
Goode homolosine projection is closer to keeping that distortion down, but all maps are an estimate due to the way a 3d curve is translated to a flat surface.
All that said, and I know I’m being pedantic, you could come really close by calculating the center of the circle in a sphere, then projecting the map stereographically from the center. That specific projection would come the closest, given the irregular shape of the Earth.
All map projections are arbitrary. The only way to do this is on a globe.
Different projections preserve different properties. From memory there are ones that leave circles circular, so would allow this.
Edit: It’s stereographic projection that maps circles to circles.
Looking at the stereographic projection, there is a longer distance between points the father you get from the center of the map. Although the latitude lines remain circular in a polar projection, the map scales to avoid distortion father from the constant growth of the map once you leave the projected hemisphere. The northern hemisphere in an artic projection still must distort, making geometry a mess.
Goode homolosine projection is closer to keeping that distortion down, but all maps are an estimate due to the way a 3d curve is translated to a flat surface.
All that said, and I know I’m being pedantic, you could come really close by calculating the center of the circle in a sphere, then projecting the map stereographically from the center. That specific projection would come the closest, given the irregular shape of the Earth.