The Magical Quarter-Circle

In my last post I presented a new model for the connection of the arced paths of reflective motion around a sphere. In the model I drew secondary vectors all over a central ‘star of vectors’, allowing me to identify the points around which the particles in motion curve. Granted, saying it like that, probably very few people understand what I mean.

In today’s post I understand why my last post was still hard to understand and present a small glimpse into a way to make the concept somewhat more understandable. And this new Eureka moment I had was in seeing that the reflection model indicates that there are quarter-circles of reflection.

So, what are these quarter-circles? I’m still trying to figure it out myself, so I might not be super-clear yet, but I’ll give it a try. And note, what I say is more of an intuitive take on what I see in the model and not some precisely constructed theory from mathematical first principles. So, I might be wrong. But at least at the moment I think the model indicates that there are two types of motion depicted in the model: there is 1° arced motion around a fixed point, as depicted in the four quarter-circles in the below model, but also 2° arced motion with a constantly moving center of arcing.

This means that beyond the quarter-circles of reflection, the vectors of arcing form evenly spaced spokes along their path. However what you see above is a rough idea of the sketched manually with Blender, so the idea might not be as convincing compared to me finding actual equations to describe the motion.

I can’t say with any certainty whether this idea holds water, but at least I’ve managed to convince myself, at least to a point.

Actually, what I fear the most is that even if the model was correct, I’m not 100 % certain that I’ll be able to prove it or disprove it. But at least, I’ll take a shot at it. And even if I get stuck with this as well, I’m quite happy to have drawn this.