I’ll start today’s post with a confession. I’d been writing this post for a while without saving and I managed to lose the first draft in my computer forcing a restart. But sometimes an apparent disaster can be a good thing. I still have my thoughts on the post that I can rewrite, so I can start from a blank slate, with the new ideas fresh in my mind.
So, what was I talking about? Long story short, once again I had made some errors to the image that I presented in my last post. The error wasn’t big in terms of geometry, but it was quite big in terms of logic. You see, the shape of dot reflection involved drawing two identical tori (toruses) in the middle of a sphere and stretching them to allow for two dots drawn to one torus to fly to the other torus. When I connected two pairs of bent cones of reflection, I managed to mangle half of the reflections. I was about to explain how, but that’s not very important. If you wish to, you can read the post and try to figure out what the error is.
I then started working on fixing the error and ended up on a major detour, thinking that the idea of bent cones of reflection itself was faulty. However, when played around with the shape enough, I realized that my biggest error in duplicating the image to show several reflections was that I didn’t align the two vertical dots along a single circular plane of reflection at exactly 90-degree angle to the average of the of the two tilted semi-horizontal circular planes of reflection. This is what it looks like as a side projection:
And here is what it looks like from the front:
And as always, here is what it looks like when rotating:
Well, actually from the above video you can see that the shape is still quite rough, and the vectors of reflection don’t align perfectly. However, I think getting them to align is just a matter of finessing the shape. The model is still rough around the edges. I think I should be able to apply proper mathematics to it to fix the last kinks.
So, what is this cone-to-cone connection in the title of this post? Well, reflections can be described with connecting the cones of reflection in a way where two neighboring cones merge to form bicones, which in turn are connected to other bicones from their tips. What is omitted from the above video and pictures is that to describe the reflection of dots, you need to draw double the number of bicones, where the new cones have a reverse shape to the first cones, with the circular face and the sharp tip reversing places.
Hopefully I’ll be able to show this shape accurately already in my next post.
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