In my last post I introduced the concept of elliptical refraction and showed that it explains refraction better than using two non-connected circles (or tori). At the end of the post the concept was very preliminary and couldn’t yet be used to make equations out of.
So, this is what I’m up to now. I’m trying to understand the geometry at a level where I can write equations. To begin with, I need to add circles to each sphere of connection (or whatever I call the larger, more transparent sphere) that are orthogonal to the ellipse of refraction. Then I add small green spheres along the circles for the non-refractive states and small red spheres along the circles for the refracted states. Then I can draw vectors to both. Like this:
From this angle, the orthogonal circles of refraction look like they are centered at the centers of dots (yellow and blue), but because of the ellipse of refraction, the circles are slightly offset from the center, and thus have radii just a bit smaller then length of the vector connecting the dots.
If viewed from a different angle, both the orthogonality is more pronounced (a red square is added to highlight the orthogonality), as is the offset of the orthogonal circle of refraction:
The next question is: “How does one convert this image into equations?” The answer for the moment is that I don’t know. Three-dimensional geometry can be a right old hassle. However, the figure looks completely solvable, and I already have some ideas that might, or might not, pan out.
As a teaser, here are just the relevant parts of the image, needed to crack the code, at an angle where the ellipse of refraction and orthogonal circle of refraction are visible as lines:
And the same as a 2D PowerPoint image:
Except the above angle isn't the best possible. When viewed directly from above it is possible to see the non-refracted green circle, the red ellipse of refraction and the blue orthogonal circle of refraction that also looks like an ellipse at this angle.
While still not exactly easy, this shape should produce the easiest equations to work with.
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