Rectangle Filled with equal Circles

2018 3 25
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更新时间：2018 3 25
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Dear All, I am wondering how to generate a rectangle with (W X L dimensions) filled with equal circles (with Radius = R)? Can you help me ? Kind regards,

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回答（1 个）

Walter Roberson
Walter Roberson 2018-3-25

0 个投票

This is more difficult that it sounds.
https://en.wikipedia.org/wiki/Circle_packing_in_a_square
https://www2.stetson.edu/~efriedma/mathmagic/0610.html
That is, if the task is to fit the maximum number of possible circles with given radius into the rectangle, then the best fit can be fairly sensitive to the ratios involved, and the formulas for the coordinates can be fairly complicated.

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Ahmad Hani
Ahmad Hani 2018-3-25
Thanks, But I'm Looking for Matlab function or code for this problem. Please, where can I find that? Thanks
Walter Roberson
Walter Roberson 2018-3-25
Could you confirm that the task is to pack the maximum number of circles of given radius within the given rectangle size? That would probably be a fair bit of work to do well, and I do not know that you would be able to find MATLAB source for it.
One genetic algorithm implementation for packing in a square is at http://staff.itee.uq.edu.au/marcusg/cias.html
Note that the question would be quite different if the requirement were only to "do well" rather than the maximum. Calculations for a hexagonal packing can be done quite efficiently -- the problem is just that for ranges of ratios that you can do a little better than hexagonal if you work at it long enough.
Like if the task at hand is really only "placing nodes for a wireless sensor network" or something like that, then absolute best packing is probably not relevant.
Ahmad Hani
Ahmad Hani 2018-3-25
Yes,I have a rectangle with fixed dimensions L X W (L=6,W=2), and n equal circles with radius = R, How can I packed max number of non-overlapped circles inside this rectangle. Thanks
Stephen23
Stephen23 2018-3-25
编辑：Stephen23 2018-3-25
@Ahmad Hani: it does not matter how many times you ask this question, mathematics is not going to find a general solution for you in the next five minutes. Please do not repeatedly ask the same question, just because you did not like the first (correct) answer.
https://www.mathworks.com/matlabcentral/answers/389947-how-can-i-packed-equal-circles-with-raduis-r-inside-a-rectangle-l-x-w
Walter Roberson
Walter Roberson 2018-3-25
There is no known deterministic algorithm for this.
https://arxiv.org/abs/1008.1224 " Circle Packing for Origami Design Is Hard", Erik D. Demaine, Sandor P. Fekete, Robert J. Lang
"We show that deciding whether a given set of circles can be packed into a rectangle, an equilateral triangle, or a unit square are NP-hard problems"

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