I have a simple globe view in my app. Previously we were using WhirlyGlobe, but have opted to use SceneKit instead. Everything works great and we are happy with it EXCEPT
If you already have images and need to reproject them (say from Mercator to Plate Carree/Equirectangular), take a look at GDAL (http://www.gdal.org). In addition to the format conversions, it contains a reprojection library. GDAL runs on most systems, and can be used from the command line or bundled into other programs.