In my post from yesterday I told you that I had probably figured out the equations for refracting the vector connecting two elementary particles of energy (dots). However, I also told that I hadn’t been able to check these equations yet and that for simplicity I would just show the equations for tilted coordinates, as these would be significantly simpler.
However, to do actual vector manipulation, the equations for tilted coordinates must be tilted back to the original coordinates. So, this is what I did for today’s post. As for the logic, this shift back is pretty simple. In the above image the shifted coordinates are depicted with two orthogonal red lines, XSR and ZSR. These are connected with a hypothenuse LXZ, whose length is easy to determine. The angle γ between ZSR and LXZ is easy to determine as well.
Next, to determine the coordinates shifted back, we can keep the hypothenuse LXZ, but we need to increase the angle γ by φ to get to the new shifted-back triangle.
The mathematics here isn’t too difficult, but it is quite tedious. However, with clever use of copy-paste, it isn’t even that much of an ordeal in writing. Not showing any of the intermediate steps, here are the new tilted-back coordinates:
The astute observer might notice that the Y-coordinate remains unchanged and that I used the subscript R for refraction.
Even though these equations look intimidating, I’ve checked that when φ = 0 these give
While the above equations might look intimidating, they only have one variable φ, and deal with very basic trigonometry. The only 'scary' thing is that when dealing with a three-dimensional world, the equations get quite long.
The next thing for me to do is to check whether the equations make any sense.
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