0
$\begingroup$

Given a circle area with radius $r$. There are $n$ points which are to be distributed evenly within that area.

For each point, what is the distance $d$ to the next neighboring point?

Am I right that this distance $d$ is equal for all points? So all points have the same distance to their next neighbor?

Is there an algorithm which helps me putting the points into the circle area in the first place?

Thanks a lot for some hints :)

  • 0
    How can you distribute $n$ points evenly in a disk? Do you mean you want to use a uniform _random_ distribution, each point having identical distribution and being stochastically independent of the other points? – 2017-02-18
  • 0
    Imagine a 4x4 m square. I want to place 16 points evenly into the square. So I place them in a 1x1 m grid, resulting in a distance of 1 meter to the nearest neighbor for each point. Now I want to do the same thing, but this time the area is not a square, but a circle. – 2017-02-18
  • 0
    So you want a square lattice to lie inside the disk, and you ask what the maximal edge length $d$ in that square lattice can be when there must be at least $n$ lattice points inside the disk? Or could it be another lattice, like one made up of equilateral triangles? – 2017-02-18
  • 0
    Exactly. The kind of lattice is arbitrary. – 2017-02-18
  • 0
    Wheras it would be intersting to find the lattice where `d` is maximized. – 2017-02-18
  • 1
    I changed some `\`` into `$` in your question code to get the usual math typography. Will it need to be a "lattice" with the kind of symmetry one associates with that word, or is some non-symmetric configuration of the $n$ points OK if that gives a greater isolation of the points? For each choice of $n\ge 2$ points, given one of the points $x$, we can denote by $d_x$ the distance from $x$ to the nearest other point. Then we can take $D=\min_x d_x$. So fixing the radius at $r=1$ (no loss of generality), for each $n$ we could ask what the maximal value of $D$ is. Now it is a precise question. – 2017-02-18

1 Answers 1

0

Disclaimer: I am not a mathematician.

The way I would solve this is assuming that I can sub-divide the area of the circle into $n$ regular hexagons. Points are places as Centroid of the hexagons. Connecting the centroids will also yield regular hexagons. This could be used to compute the distance between the points. Let s be the side of each hexagon. the formula for finding the area of a hexagon is Area = $\frac {3\sqrt3}{2} s^2$, where $s$ is the length of a side of the regular hexagon. The total area of these hexagons should be less than the area of the circle due to some loss. Given the radius of the circle and the number of points to be fit in, we can calculate the maximum ‘s’. This will be the distance between the points.

Related useful links: https://mathoverflow.net/questions/109515/the-gauss-circle-problem-on-a-hexagonal-lattice/110143#110143