Rectangular to circle

[Hai-Etlik]

Are you familiar with projections?

If you have a map in a known cylindrical projection (One which produces a rectangular map), you just need to reproject each hemisphere into an Equatorial Azimuthal projection. In particular I'd recommend Stereographic Azimuthal. There is plenty of software that can do this including GDAL, QuantumGIS, and Hugin.

If you just drew an arbitrary map on a rectangular canvas with no regard to it being a representation of a sphere, you are pretty much screwed in terms of getting results that look good and make sense without reworking the map significantly.

[Naeddyr]

A good place to start is http://en.wikipedia.org/wiki/Cartographic_projection Wikipedia. Read up on how projections work. It's also fun to check out various galleries online which showcase different kinds of projections.

Most computer programs that deal with projections (that I've used: g-projector, Flex Projector, Hugin has it as one of its projections) use as their base map a map in the Equirectangular, plate carree projection, which is very simple to compute (on computers). G-projector is the easiest to use, and now that it's upped its maximum output to 3000x1500 pixels it's much more useful.