Carbonaceous mesophases are discotic nematic liquid crystals that are spun into high performance carbon fibers using the melt spinning process. The spinning process produces a wide range of different fiber textures. Planar polar (PP) and planar radial (PR) textures are two ubiquitous ones. This paper presents theory and simulation of the texture formation process using the Landau-de Gennes mesoscopic theory for discotic liquid crystals. The computed PP and PR textures phase diagram, given in terms of temperature and fiber radius, is presented to establish the processing conditions and geometric factors that lead to the selection of these textures. Thin fibers adopt the PR texture, while thicker fibers and higher temperatures adopt the PP texture. The influence of elastic anisotropy to the formation of textures and structure is thoroughly characterized.