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# Approximate Circle Packing

I’d like to be able to have a region that is densely packed with non-overlapping circles, whose sizes vary spatially in a pre-defined way. There are lots of resources for Poisson-disk sampling, but that doesn’t result in snugly packed circles. Time to do some tinkering!

## Random search

The laziest way to do this: try to add a circle at a random point, and see if it fits. Repeat until you fail for way too long. This will give non-overlapping circles, but they are not densely packed.

This found 264 valid circles. I had the algorithm try 50,000 times, and it only added 7 circles during its second half. Super inefficient at the end!

One of the main problems here is that it doesn’t know which regions are covered, so it picks useless points way too often, and increasingly so at the end.

## Pruning

Let’s start with a set of candidate locations instead, rather than a continuous space. Then whenever we add a circle, we can remove the too-close points from the list of candidates, and continue.

First, we remove all the candidates that are too close to a wall. Then the first candidate will be a valid circle, so we add that. Then we check the distance to all the other candidates, and remove the ones that are too close to be valid. And repeat!

The order of the candidates becomes important, then. For example, if they are randomly ordered, this will quickly reduce the number of candidates, but will not be densely packed. Here, the circles are in black, and the candidate positions are in blue.

This was much faster than the entirely random search, but still gave loosely packed circles. Instead, we can sort by their location, say their y-position:

Which increases the number of circles by 20%! Excellent.

Now how expensive is this approach? Let’s say there are $n_p$ candidate points, $r$ is the uniform radius, and the dimensions are width and height $w,h$. The number of points should be proportional to the area of the region, $n_p=\alpha w h$. The resulting number of circles is approximately $n_c \approx \frac{wh}{\pi r^2}$.

If we weren’t discarding the bad points, we would have to check every point at each iteration. That means that the number of computations would be $N=\sum_{i=1}^{n_c} n_p = h^2 w^2 \alpha/(\pi r^2)$.

If we are discarding points, they’ll be discarded at a roughly uniform rate. So at iteration $i$, we should have $n_p(1-i/n_c)$ points remaining. Then $N=\sum_{i=1}^{n_c}n_p (1-i/n_c) = n_p(n_c-1)/2 =\frac{h^2w^2 \alpha}{2 \pi r^2}-\frac{h w \alpha}{2}$, which means that we’ve halved the necessary computations by pruning.

A further optimization would be to only check points that are within a distance of twice the maximum radius. Since the points are sorted by their y-position, this works well. In this example case, this means that a swath of approximately $\beta n_p 2 r/h$ points are checked in each iteration. By experimentation, $\beta\approx 0.6$. This reduces the number of computations to $N=\sum_{i=1}^{n_c} (\beta n_p 2 r/h) = \frac{h w^2 \alpha \beta}{\pi r}$. Removing that factor of $h$ from the expense is very helpful!

On a similar note, if you could also limit the x-width of each check, that would be even better. Then we could have $N=n_c (\gamma n_p (2r)^2/(wh)) = \frac{4 h w \alpha \gamma}{\pi}$, with $\gamma\approx 0.4$, which is much, much faster. That’s now linear with the area of the region!

Unfortunately, our 1D list of points isn’t great for checking squares. So let’s use a 2D array instead! This gives the points a useful property: it’s easy to know how far they are from any other point, just based on their storage structure. Now we can easily achieve that area-proportional method!

Unfortunately, this will result in circles that are on a grid, and the circles won’t be touching perfectly. That can be fixed with a little shift based on gradients. If you know the gradient of the distance functions and local radii, you can shift the circle down until it’s touching, and life will be OK again.

## Halftoning

Of course, I have to use this for halftoning with dots and circles. Time for some quick math. Define the min and max radii of the circles to be $r_{min},r_{max}$, and a packing efficiency of $\phi$. The radius of a given dot is $r$. The image intensity (compared to darkness) is $I_0=1-k_0$.

First, dots. Let’s say that the radius of the drawn dot is $r_{dot}$. The average darkness of a region is $k=\phi \pi r_{dot}^2/(\pi r^2)$. If scale the input to be between the minimum and maximum achievable values with these radii, and solve for r, we get

$r=r_{max}/\sqrt{ 1+k_0 \left( \left(\frac{r_{max}}{r_{min}}\right)^2-1 \right) }$.

If we do the same sort of thing for an outlined circle, we have $k=\phi 2\pi r w/(\pi r^2)$, where $w$ is the width of the line. Then the output radius is

$r=r_{max}/\left( 1+k_0 \left(\frac{r_{max}}{r_{min}}-1 \right) \right)$.