parametric equations of great circle on sphere

Hello all,

I'm new to the forums and I hope this is the appropriate subforum for this question.

I'm programming a sphere that rolls around in straight paths. for the particular library i'm using, i essentially indicate phi/theta and it essentially returns 2D graphics of a sphere rotated to the given angles.

My goal is to rotate the sphere starting at a given starting angle and every frame i would supply the new spherical coordinates. thus i would need to have phi and theta as a function of time for a regular rolling ball.

so the question essentially:

what are the equations for phi and theta of a great circle in parametric form, specifically, in terms of time__ given a starting phi/theta and a velocity__?

I've been juggling a few ideas in my head but this is really not my forte. any help would be appreciated :)

__edit underlined__