I am looking for some help to figure out this math problem. I want to know an optimal scenario for the problem I am trying to solve. The details are as follows:
1. there is an inner and outer ring with openings in them.
2. only one ring will spin, though instead of being as shown, in some cases it would be the inside ring, but I don't see this having any relevance to the solution for this problem
3. when I say "optimal" what I mean is, having as few openings (empty space), as possible on both rings, but always having at least one opening in both rings being aligned as possible at all times, so that the opening of the inside ring always allows for a path from inside the inner ring to the space outside the outer ring (as shown in the upper right corner, when it pauses at 0:06) . The inner ring opening can be slightly blocked by the outer ring, but the closer to fully open, the better.
4. it's a bonus if there are a greater number of times during rotation when there is more than one clear path through the openings, but this is a very distant second in criteria, and only worth considering if it has no impact (or almost no impact) on criteria #3.
I can figure this out by playing around with a pen and paper, cutting out a circle for the inner ring, and rotating it, but I assume that really figuring out the absolute (or close to it) most optimal configuration for this requires some sort of math equation/algorithm/whatever.
From google, I'm guessing the particular type of math to figure this out is either "combinatorial geometry" or "combinatorial trigonometry", but I really don't know.
Important for finding the best solution:
1. there is no strict guidelines on how many openings are in the inner or outer rings, or if the two rings have the same number of openings. Fewer openings is better though.
2. there is no need for the size of the openings to be consistent, though I wouldn't want any one solid part of either ring to be any smaller than the smallest piece shown, and having fewer larger solid parts is a plus.
3. there is no need for the spacing of the openings to be consistent
4. despite the above three points, one configuration is better than another for various reasons, which can't be explained in words, so I'd like to get a few of the "most optimal" configurations that have some recognizable differences between them, in terms of the different variables involved.
The ideal thing to have, would be a "plug and play" program, where I can enter specific dimensions for the depth for inner and outer rings, varying diameters, minimum and maximum width of openings, number of openings, etc. and have a computer generated outcome, working with the variables that are inputed.
Taking the next step with really figuring this out involves detailing exact dimensions of the 2 rings, and what the minimum width of what is considered "a clear path" must be determined, but I wanted to get this video up here to get this going in the meantime. Obviously, this is in the preliminary stages of even investigating this problem.