In solving a problem in physics, I have arrived at the solution set
$x(\theta)=\frac{1}{2}k^2[(\theta - \sin\theta)+\mu(1-\cos\theta)] + x_0$
$y(\theta) = \frac{1}{2}k^2[(1-\cos \theta) + \mu(\theta+\sin\theta)] + y_0$
In order to fit this parametric curve to connect two points $a$,$b$ the best method I have found in to numerically solve an equation which eliminates $k^2$ to find a theta that fits and then substitute back in to find k. Computationally this is hugely taxing and inefficient, but I can't for the life of me isolate the equations in a way that yields either the upper bound on theta or the $k^2$ term. Is there a way to compute $k^2$ and the bounds on $k$ without using computational estimation methods?