Angles between 2 points along a varied path

[Ralajer]

I am writing a program that uses Snell's Law for refraction of light through two interfaces and I've encountered a problem representing the geometry symbolically. I could determine the values numerically but I don't want the overhead as it would have to be called up to 1E7 times.

The generalized diagram below shows the known values in green and the unknown in red. These are the equations that I am working with:

[*]B=tan(d)*D + tan(e)*E + tan(f)*F
and from Snell's Law the relationship between angles d, e, and f.

• sin(f)*n_F = sin(e)*n_E
• sin(e)*n_E = sin(f)*n_D
  
  
  where all the refraction indices n_x's are known
  ​

In an attempt to solve for one of the angles f in this case I get the following:
f=-atan(1/F*(E*tan(asin(n_F*sin(f)/n_E))-A+D*tan(asin(n_F*sin(f)/n_D))))
I cannot solve for f explicitly. Is there something that I am missing in my analysis of the problem? Any help would be appreciated.

Thanks
Rob

[PLAIN]http://www.wtrresources.com/img/Refraction_Problem_Diagram.jpg
http://www.wtrresources.com/img/Diagram.png"

 

[pat_connell]

One possible way to solve your problem is to use a numerical method such as Newton's Method. This method involves iteratively solving for the unknown angle f until it converges to the true solution. The idea is that you start off with an initial guess for the value of f, and then use the equation to calculate a new guess for the value based on the previous guess. You repeat this process until the difference between successive guesses is small enough that you can consider them to be equal. This technique is often used to solve equations that cannot be solved analytically.