Fully cover a rectangle with minimum amount of fixed radius circles

Question

I\'ve had this problem for a few years. It was on an informatics contest in my town a while back. I failed to solve it, and my teacher failed to solve it. I haven\'t met anyone

Answer 1

For X and Y large compared to R, a hexagonal (honeycomb) pattern is near optimal. The distance between the centers of the circles in the X-direction is sqrt(3)*R. The distance between rows in the Y-direction is 3*R/2, so you need roughly X*Y/R^2 * 2*/(3*sqrt(3)) circles.

If you use a square pattern, the horizontal distance is larger (2*R), but the vertical distance is much smaller (R), so you'd need about X*Y/R^2 * 1/2 circles. Since 2/(3*sqrt(3) < 1/2, the hexagonal pattern is the better deal.

Note that this is only an approximation. It is usually possible to jiggle the regular pattern a bit to make something fit where the standard pattern wouldn't. This is especially true if X and Y are small compared to R.

In terms of your specific questions:

1. The hexagonal pattern is an optimal covering of the entire plane. With X and Y finite, I would think it is often possible to get a better result. The trivial example is when the height is less than the radius. In that case you can move the circles in the one row further apart until the distance between the intersecting points of every pair of circles equals Y.

2. Having a regular pattern imposes additional restrictions on the solution, and so the optimal solution under those restrictions may not be optimal with those restrictions removed. In general, somewhat irregular patterns may be better (see the page linked to by mbeckish).

3. The examples on that same page are all specific solutions. The solutions with more circles resemble the hexagonal pattern somewhat. Still, there does not appear to be a closed-form solution.