# Parametric path of great circle on sphere

hello, i've posted this question on a math forum, but they weren't much help - this really is more appropriate in physics ;)

i'm working on a computer program and i'm using a library that generates a 2D graphics of a circle given a theta and phi. if i want to make the ball rotate around its normal axis, i would call the library's rendering function and pass the same theta every time, but with phi being slightly different. this, of course, is only the case for for rotation around an axis through the poles.

my question
given a starting phi and theta and a velocity phi and theta, what are the parametric expressions for phi and theta in terms of time.

things to keep in mind
1. i'm working purely with spherical coordinates (no cartesian) and i would prefer a solution that solved in such a coordinate system.
2. it needs to be time dependent in such a way that dl/dt is constant (ie: constant angular speed)

what i've tried
i figured i would first need an equation of the path and then parameterize it later. unfortunately i don't know spherical that well :/

any help will be greatly appreciated!