It would be worthwhile to look into the way flat sections are fitted to a sphere. If a flat plane is warped to the spherical surface the linear segments between the corner vertices can diverge, producing gaps. If the flat sections are kept flat, so that the global surface is polygonal instead of perfectly spherical, then such gaps cannot occur.
This is somewhat analogous to the demand that the game screen be drawn on the gnomonic projection. Such a projection has straight lines everywhere drawn as straight, thus ensuring no gaps at the joining sides of polygons. No other projection can do this, because anywhere off axis (screen center) straight lines would be curved, with a failure to correctly fit polygonal sides; gaps or overlap would result.
To further impress the characteristics of the gnomonic projection, as a means to divine what might be causing the gaps we see thus far in this graphics extender...
In the real world, the straight line between any two points seen by the observer is a segment of a great circle. That's because the viewpoint and the two endpoints define a plane, and the viewpoint is always the center of the environment. This plane is thus a section of the larger plane that bisects a sphere centered on the observer.
The only projection which retains great circles as straight lines no matter where crossing the FoV is the gnomonic. As an indicator of its real world usefulness, note that this is the projection used for plotting the paths of meteors on the sky (no matter where they appear, the brief flight of a shower meteor is essentially straight, and thus represents a great circle segment.) Any other projection preserves the straightness of a great circle when passing through the center of projection (view center) only; everywhere else a straight line becomes curved.