Consider the following diagram. I am given the values of $h$, $a$, and $\theta$, along with the positions at both endpoints of chord $a$. Using these values, I would like to consider a traversal from one endpoint to the other at a constant rate along arc $s$, and be able to calculate the current position at any given time.
The trick is that with $a$ being held constant, $h$ and $\theta$ can go to zero while the center of the circle heads off to infinity. As a result, if my method of computing the point on the arc depends on computing the center point of the circle or its radius then it will break down as $h$ approaches zero. This, despite the fact that the arc segment to be traversed is still nice and finite.
I would like to avoid introducing a special case for small values of $h$ and instead find a way of expressing the point purely as a function of $\theta$ and a linear combination of $h$ and $a$, all of which are consistently defined even if they happen to go to zero. Is such a parameterization possible?