I am working on a microscopy project and keep running into the same roadblock.
Some context: The microscope has a detector base with a variable distance from the source beam (you can move it up and down). There are 56 [0-55] distinct positions that the base can take. I know the angle of the beam across the base for the top and bottom positions but need to find a way to figure it out for all positions (this is dark field electron microscopy if you want to look into it more).
The question: Let me start with a picture
How can I determine what $\theta_3$ and $\theta_4$ are?
My trigonometry is not very strong so I am not sure if there is a relationship between the two which I am missing.
Progress so far: In addition to googling around, I have tried writing down equations for the situation but haven't gotten something I could work with yet. Here is a list of them.
$$tan(100) = (x+y)/a$$ $$tan(50) = x/a$$ $$tan(\theta_3) = (x+y)/(a+b)$$ $$tan(\theta_4) = x/(a+b)$$ $$tan(700) = (x+y)/(a+b+c)$$ $$tan(350) = x/(a+b+c)$$
As you can see this provides me with 7 variables and 6 equations, not enough to solve for anything.
I also looked into seeing if there was a relationship between the angle and step position (e.g. if linear: if you went to the middle, step 27, $\theta_4$ = 200rad and $\theta_3$ = 400rad) but I wasn't able to find anything online or on paper to convince me this would work.