In my last post I talked about the elegance with which I was able to use laws of sines and cosines to figure out the first two angles relating to refraction.
However, in the post I already identified a small problem in progressing with the projections that I had chosen. To cut a long story short, the whole concept relied on me being able to sum up two angles: 60° and θ, which is the angle of refraction. But I already had an idea that there was something a bit off in this concept.
The problem is that you can’t add θ to the original 60° angle of the non-refracted triplet of dots. So, while the shape wasn’t too far off, it wasn’t accurate either.
So, what I had to do was not exactly change the shape itself, but I had to tilt it so that refraction would be along a single plane. Below, you can see two of the new projections. On the left is the x-z projection, where the refraction is visible as vertical blue line connecting a green (unrefracted) sphere with a red (refracted) sphere. The red horizontal line is a projection of the ellipse of refraction. Almost surprisingly this ellipse isn’t an active component of refraction, but just a tool to align the shapes. On the right is the y-z projection, where the refraction is visible as blue circular arc, with a radius almost (but not quite) 2r.
I’m still trying to figure out the X-Y projection, but this is roughly what it looks like:
I’ve already had to tweak the picture a few times in writing this post for it to make sense, so it might not be ready.
However, I think this solves the inaccuracy problem. If I choose the projections like this, I should have no problem figuring each angle out. I’m still in the early stages, but I seem to be on the right tracks.
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