Timeline for What is the covering density of a very thin annulus? Is it $\frac{\pi\sqrt{51\sqrt{17}-107}}{16}$?
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| Jul 27, 2021 at 7:56 | comment | added | user44143 | I would center the first annulus at $(1-\epsilon, \epsilon)$ or the like, which probably leads to a unique choice at each step for the minimal point $p$ on the boundary of the union of the annuli. Then I would choose the next annulus to be tangent at $p$ to one of the previous annuli and so that the new union covers a neighborhood of $p$; there will be just a few options and the greedy choice is the one maximizing the newly covered area. | |
| Jul 27, 2021 at 6:12 | comment | added | RavenclawPrefect | @MattF.: I tried this out using a discrete approximation to the plane with a fairly simple greedy algorithm: an annulus is lined up to the current closest uncovered point $P$, with its tangent line through $P$ altered by up to $0.1$ radians from the default setup (where it's perpendicular to the vector from the origin to $P$) to add some stochastic behavior. After placing $350$ annuli with thickness $0.1$ times the radius, the image looks like this, and its average density up to the radius covered so far is around $3.88$. | |
| Jul 27, 2021 at 3:35 | comment | added | RavenclawPrefect | @MattF.: "appropriate annulus" is hiding a fair bit of complexity there, I think. For instance, suppose that I try to minimize overlap with existing annuli while covering a point of minimum distance. Then if my currently-covered region is a disc centered at the origin, it will take infinitely many placements to cover the points just beyond the disc, since each new annulus gets placed tangent to the disc. Is there a particular algorithm for placing new annuli you had in mind? | |
| Jul 26, 2021 at 22:20 | comment | added | user44143 | You could try a greedy algorithm: Start with one annulus. Find the point closest to the origin which is outside all the annuli so far. Cover that point with an appropriate annulus and repeat. Then at each stage, what is the covering density for the largest fully covered circle? Does that density seem to have a limit? | |
| Jul 26, 2021 at 18:59 | comment | added | RavenclawPrefect | Originally posted at Math StackExchange here, with no progress on the question despite some interest. | |
| Jul 26, 2021 at 18:58 | history | asked | RavenclawPrefect | CC BY-SA 4.0 |
