I'm trying to expand a Haversine-based spherical navigation library I've created for a computer game I'm developing. Currently, missiles "chase" their targets by simply computing the initial heading for the shortest great circle arc to the target, per frame; I'm trying to find a way to get them to "intercept" their target along the shortest possible arc instead.
The nearest I can get to this with Google is the intersection of two great circle arcs. My problem extends that somewhat with speed and time.
I'm trying to work out; given an object, at a given set of spherical coordinates, moving along a great circle arc with a current given compass bearing; compute, from another set of spherical coordinates, the initial bearing such that another object setting off from here with a given different speed will intercept the first object along the shortest possible great circle arc.
Layman's description: An aircraft is cruising around the world with a constant speed. You are a SAM site. Calculate the initial bearing your interceptor missile has to take in order to shoot it down by following the shortest possible path.
My own working so far is has been rubbish. The best I've done is come up with the obvious stipulation "They must end up in the same place, after travelling for the same amount of time", which applies to trivial 2D planar geometry.