I am tyring to cover a simple concave polygon with a minimum rectangles. My rectangles can be any length, but they have maximum widths, and the polygon will never have an acute angle.
I thought about trying to decompose my concave polygon into triangles that produce a set of minimumally overlapping rectangles minimally bounding each triangle and then merging those rectangles into larger ones. However, I don't think this will work for small notches in the edges of the polygon. The triangles created by the reflex vertices on those notches will create the wrong rectangles. I am looking for rectangles that will span/ignore notches.
I don't really know anything about computational geometry, so I'm not really sure on how to begin asking the question.
I found other posts that were similar, but not what I need:
Some examples: Black is the input. Red is the acceptable output.
Another exmaple: The second output is prefered. However, generating both outputs and using another factor to determine preference is probably necessary and not the responsibility of this algorithm.
Polygons that mimic curves are extremely rare. In this scenario much of the area of the rectangles is wasted. However, this is acceptable because each rectangle obeys the max width constraint.
Also, I found this article to be close to what I need:
Maybe a better question is "How can I identify rectangular-like portions of a concave polygon?"
Here is an image showing the desired implementation:
The green is the actual material usage. The red rectangles are the layouts. The blue is the MBR of the entire polygon. I am thinking I should try to get little MBRs and fill them in. The 2-3 green rectangles in the upper left corner that terminate into the middle of the polygon are expensive. That is what I want to minimize. The green rectangles have a min and max width and height, but I can use as many rows and columns necessary to cover a region. Again, I must minimize the number of rectangles that do not span across the input. I can also modify the shape of the green rectangle to fit in small places that is also very expensive. In other words, getting as many rectangles as possible to span as much as possible is ideal.
Maybe I should simply be trying to identify rectangular regions like this:
Or, perhaps a better approach would be using largest-inscribed rectangles instead of MBRs. I could continually cut my polygon down using rectangles until I am left with regions were the largest-inscribed rectangle is not sharing an edge with the original polygon. The remaining regions would have to be handled with a heuristic approach.
Again
I've been working with the engineering and manufacturing departments at my company to bring more clarificaiton to this problem. I am still waiting to confirm, I'm but I am now thinking an algorithm that would return sets of largest inscribed rectangles would work. While it does not sure if completely cover the shape, it would give preference to the orthognal regions while leaving the non-orthogonal regions to some heuristics. The only trick is necessarily getting me closer to my goalmaxamize those orthogonal regions.
