I want to know how many points ($n$) can be placed in a circle of radius $r$, with a minimum spacing $s$ between points. I find postings for several similar problems -- smallest circle around a set of points, packing circles, etc -- but not for this case.
Clearly it's not simple. I'm pretty sure the following are correct but have not attempted proofs, as I'll be happy with a reasonable degree of certainty:
For $r<\frac{s}{2}$, $n=1$. The one point can be placed anywhere in the circle.
For $r\geq \frac{s}{2}$ and $r<\frac{s}{\sqrt{3}}$, $n=2$. When $r=\frac{s}{2}$, the points must be at opposite ends of a diameter.
For $r\geq \frac{s}{\sqrt{3}}$ and $r<\frac{s}{\sqrt{2}}$, $n=3$. When $r=\frac{s}{\sqrt{2}}$, the points must be equally spaced on the circumference of the circle.
For $r\geq \frac{s}{\sqrt{2}}$ and $r<\frac{s}{2}\times \tan(54^\circ)$, $n=4$. Again, the smallest value of $r$ requires equal spacing on the circumference. Perhaps there's a better expression for the upper bound.
For $r\geq \frac{s}{2}\times \tan(54^\circ)$ and $r<s$, $n=5$.
Now it gets fun, because when $r=s$, $n=7$. That's one point in the center and six points equally spaced around the circumference. There is no value of $r$ for which n=6. This means there is no continuous function which when rounded gives n as a function of $r$ for all positive values of $r$.
Then of course it gets even more complicated. I'm pretty sure that the next step is $n=8$, with a point in the center and seven points around the circumference. It's tempting to think that at some point one could turn alternate points in, making sort of a star and possibly reducing the diameter. I don't have an argument, but my inclination is to think this does not happen: that you keep adding points one at a time, through $n=12$ points -- one in the center and $1$ around the circumference -- and then, suddenly, it jumps to $n=19$, because $12$ points on the circumference allows both the center point and the first ring of $6$.
Perhaps this would lead to a system: divide the range ($r$) into $ds\leq r< (d+1\times s)$, with a continuous formula (rounded) within each division. But I'm getting a headache. It's been over $40$ years since I did this sort of stuff to get my degree.
Oh, the application is geocaching. Geocaches must be placed a minimum distance apart: $s = 0.1$ mile. I'm writing a program (a GSAK macro to be specific) which needs to download all the geocaches within a specific distance ($r$) of a given latitude and longitude. The only reasonable way of invoking the download requests a number of geocaches, not a radius. But I don't want to make an overkill request, because the user running the macro has an allocation of downloads/day. Therefore I'd like to make the request for the maximum number, but no more. (There's another entry in the API which might enable a continuation call, but I'm not sure and it's vastly more complicated to use.)
Of course geocaches are placed on the surface of a sphere, the earth, so technically the analysis should be done in spherical geometry. (The minimum spacing requirement in geocaching ignores elevation.) But $r$ will be vary small (typically less than a mile) compared to the size of the earth, and this combined with the errors inherent in measurements in the game make Euclidean geometry sufficient for the purpose.
Edward
