The way the dome works out where to point to match a given mount position relies on a whole load of mathematics. The actual implementation of that maths is a bit fiddly but the principals behind it are fairly straightforward. It goes something like this:
This can be expressed mathematically by the vector equation of a line
r = m + td (where t is a number that can take any value from 0 upwards)
(Mathematically that is what value of t do we require such that |m + td| = Rdome)
The problem with this is that it assumes that the dome is very thin. It assumes that, if the line of sight goes through the centre of the dome gap, that the dome will be centred within the image, however the outer extremity of the dome may not be centred even if the innermost part is.
We had the idea that if we changed the dome radius value in the algorithm to be half-way between the inside and the outside, that it would give a better position for the dome. I ran simulations for the whole sky with different radii. (Each simulation run takes my work laptop 6 hours to complete, for various reasons, so this took a while).
The image below shows what the visibility looked like to start with, with green being image 100% sky and red being image 2/3 sky or less. Hover over it to see how this changed with an increased radius parameter.
|Hover to see the improvement with larger dome radius|
I still wasn't satisfied though. I had invested a lot of time in this endeavour so far and the resulting solution, though effective, was not very elegant. We were still assuming that the dome was a thin sphere, just a larger one. And we had some thoughts that perspective effects due to the wide angle of the camera would also be skewing things. But this I will talk about in my next and final post...
Click here for part 4