For some time, I’ve been attempting to come up with the equations for the ellipse of refraction. However, now that I finally found the equations, with the help of ChatGPT, I realized that I had been looking for the wrong thing.
You see, when I finally got the exact equation for tilting an ellipse that encircles two spheres, I could also view the tilted circle-ellipse pair. And doing so, I realized something that doesn’t make sense. If you zoom into the edge of a tilted circle-ellipse pair (with the circle depicting the edge of the sphere along which the ellipse is located), the ellipse doesn’t touch the circle. Like this:
But by definition, this shouldn’t be possible. Regardless of the tilt, the ellipse should always touch the circle. Except that as it is clearly possible, this means that the ellipse doesn’t touch the edge of the sphere. To confirm this realization, I first made cuts from a sphere and realized that while viewed from an angle the cut might appear elliptical, such a cut is always circular. Like this:
The only way to cut an ellipse from a three-dimensional shape is if there’s ellipticity in the shape. In Wikipedia, an ellipse is cut from a cone.
So the only option left for me was to once again kill my darlings The ellipse of refraction was definitely a necessary step to figuring out the solution to refraction, but once I found a flaw in the logic, I had to revisit my original logic. and see where I went wrong.
While I cannot be a 100 % certain yet, I think I know what the problem is. When viewed as an x-z projection, it appears that the refraction relating to the blue sphere is connected to the refraction relating to the yellow sphere, because the green circle of no refraction is tilted by the same angle for both refractions.
However, I had already known that the refraction for these two weren’t the same, so I had conveniently split the ellipse of refraction to half-ellipses: one for the blue sphere and one for the yellow. However, in retrospect, this idea seems silly. What I now realize is that the circle of no refraction is expanded into two wider circles of refraction that occupy the same plane. The radius of the circle of refraction is:
In the case of no refraction, φ = 0 and L = √3r.
And this is what it looks like in three dimensions.
And almost as a teaser, this is what the projections look like in Blender.
X-Y projection:
X-Z projection:
And Y-Z projection:
At the moment, I don’t have any of the equations to figure out what this means for the vectors. However, I’m quite confident that these equations will be easier than anything relating to ellipses.
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