Kedaman: The general problem you have described looks overwhelming. I do not think you are going to be able to solve it.

Even if you work on a particular case with more than 4 intercepts, I do not think you are going to find a solution if you do not allow human interaction with computational programs. I am not sure you could solve a particular case with as many as four intercepts.

If I really had to solve this problem, I would first try solving a simple case requiring one intercept. Then see if I could use the technique in some fashion to solve a problem with two intercepts. Then try three. Then see if I could generalize the approach.

If I discovered that I could not find a general method applicable to any problem involving two intercepts, I would just give up.

Why not try solving a simple example of this type of problem? We should learn to walk before we try running.