Try wrapping the curves to a similarly sized cylinder then project the wrapped curves onto the hemispherical surface (use the "along face normal" direction option). It may require significant tweaking of the input curves to get what you want.
If you have access to NX 11, you may be able to make use of the new "flatten surface" command. Supposedly it allows you to flatten a surface, create curves on it, and map the curves back onto the surface.
For the simple example you show it can be done analytically (program/journal) since on a sphere or torus section the lines for the simple planar lines would produce arcs in the latitude and longitude direction on the sphere or torus.