Packing arbitrary polygons within an arbitrary boundary

Question

I was wondering if anybody could point me to the best algorithm/heuristic which will fit my particular polygon packing problem. I am given a single polygon as a boundary (convex or concave may also contain holes) and a single "fill" polygon (may also be convex or concave, does not contain holes) and I need to fill the boundary polygon with a specified number of fill polygons. (I'm working in 2D).

Many of the polygon packing heuristics I've found assume that the boundary and/or filling polygons will be rectangular and also that the filling polygons will be of different sizes. In my case, the filling polygons may be non-rectangular, but all will be exactly the same.

Maybe this is a particular type of packing problem? If somebody has a definition for this type of polygon packing I'll gladly google away, but so far I've not found anything which is similar enough to be of great use.

Thanks.

Answer 1

The question you ask is very hard. To put this in perspective, the (much) simpler case where you're packing the interior of your bounded polygon with non-overlapping disks is already hard, and disks are the simplest possible "packing shape" (with any other shape you have to consider orientation as well as size and center location).

In fact, I think it's an open problem in computational geometry to determine for an arbitrary integer N and arbitrary bounded polygonal region (in the Euclidean plane), what is the "optimal" (in the sense of covering the greatest percentage of the polygon interior) packing of N inscribed non-overlapping disks, where you are free to choose the radius and center location of each disk. I'm sure the "best" answer is known for certain special polygonal shapes (like rectangles, circles, and triangles), but for arbitrary shapes your best "heuristic" is probably:

1. Start your shape counter at N.
2. Add the largest "packing shape" you can fit completely inside the polygonal boundary without overlapping any other packing shapes.
3. Decrement your shape counter.
4. If your shape counter is > 0, go to step 2.

I say "probably" because "largest first" isn't always the best way to pack things into a confined space. You can dig into that particular flavor of craziness by reading about the bin packing problem and knapsack problem.

EDIT: Step 2 by itself is hard. A reasonable strategy would be to pick an arbitrary point on the interior of the polygon as the center and "inflate" the disk until it touches either the boundary or another disk (or both), and then "slide" the disk while continuing to inflate it so that it remains inside the boundary without overlapping any other disks until it is "trapped" - with at least 2 points of contact with the boundary and/or other disks. But it isn't easy to formalize this "sliding process". And even if you get the sliding process right, this strategy doesn't guarantee that you'll find the biggest "inscribable disk" - your "locally maximal" disk could be trapped in a "lobe" of the interior which is connected by a narrow "neck" of free space to a larger "lobe" where a larger disk would fit.