It sounds like you want a point with argument theta such that a bunch of constraints are satisfied. I don't think you'll easily be able to find a general polar "r(theta)=...", since your curve is defined by a "sweep" of other non-trivial function outputs (i.e. the intersection of two circles), which is another way of saying "solve for x in f(x)=0 where f(x) is complicated".

I would be inclined to use a non-linear solver, which will will almost certainly converage within 2 or 3 iterations especially if you're starting with a good initial estimate from time=t-1. 40khz should not be a problem.

You can start with scipy.optimize, and look into ceres if you need something faster.