Best Projection for a Ring World

[Hai-Etlik]

Map projections are used to translate from a sphere to a plane. A ringworld (in the sense of a ribbon around a star) is already planar, just very long and thin

Projections are mappings from spaces onto lower dimension spaces: Not necessarily just from a planetary surface in ℝ³ onto ℝ². And the surface of a ringworld is not planar, it's cylindrical.

The obvious projection of that cylindrical surface onto a plane (Technically its a form of "Equidistant Cylindrical") is as much a projection as any geographic one. This projection does produce distortion for vertical features: the tops of two mountains on a ringworld are closer together than the points directly beneath them at the vertical datum. There's also a discontinuity where you 'break' the ring in order to 'unroll' it.

The actual amount of distortion is extremely small, but it is there and 'unrolling' it is a projection even if it's much simpler than any we have to deal with on our approximately spherical world.