Coords of frustum intersection with a sphere

I need to determine the nearest area formed on the surface of a sphere by the intersection of the view frustum.
Ideally the coordinates would be in longitude and latitude.
I’ve been pondering this one for a while now, and can’t see the wood for the trees, so a second perspective would be really helpful.
If the frustum falls full on the surface of the sphere, all would be ok - I could simply find the intersection of the 4 rays projected down the corners of the frustum, but what if only the bottom (& left & right) planes of the frustum intersect the sphere surface? How do I get any useful info from that?
I basically want to draw just that patch of the surface of the sphere (using a kind of quadrilateral rasterizer).
Is this easy?

I don’t have an idea what you’re trying to achieve and what a nearest area is supposed to be. Maybe a better explanation or a picture could help.