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I have two circles. Both origin at San francisco $(37.77493,-122.419415)$, The larger circle has Radius $R_1$, the smaller circle has radius of $R_2$.

What's the fewest number of additional overlapping smaller circles (/w Radius of $R_2$) that fit inside the larger one, that creates zero empty space (except near the edge of the larger circle), and the corresponding co-ordinates of each circle's origin.

I've bounced this problem across my peers, and we've decided that the circles are optimally fitted to how the Flower of Life is laid (http://en.wikipedia.org/wiki/Flower_of_Life).

Thus one unit of $R$ from the centre would produce $7$ circles ($1$ in the middle, and $6$ surrounding).

Origin = $(x,y)$

Surrounding circles: (x+cos(60x)R,y+sin(60x)R) // Guess enter image description here

From here I'm stuck on how to continue to fill the circle, and how to give real-world coordinates of each of the circle's origins. With input like $R_1 = 5,000$ meters, and $R_2 = 1000$ meters.

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It would help if you were more specific about what you want. It's not clear to me exactly how much space you want to cover with your overlapping small circles considering that the flower of life has a good bit of empty space around the edges. –  Jim Feb 19 '13 at 20:38
    
Sorry if I wasn't clear, empty space around the edges are fine. –  Mr. Demetrius Michael Feb 19 '13 at 20:51
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If empty space is fine then I would say the fewest number of additional circles that meet your criteria is $0$. But I'm certain that's not what you intend, hence I still think you need to be more specific. –  Jim Feb 19 '13 at 21:30

1 Answer 1

up vote 0 down vote accepted

The coordinates of all the centers are: $$ (\frac{\sqrt{3}j}{2},k + \frac j 2) R $$ for $k,j$ integers.

To check if the small circle (radius R) is inside the larger one (radius R_1) compute the distance between the centers: $$ R \sqrt{\frac 3 4 j^2 + (k+\frac j 2)^2} \le R_1 - R. $$

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This is true in a flat plane -- but the OP apparently wants the inside of a circle on a sphere to be filled. –  Henning Makholm Feb 19 '13 at 22:48
    
@Henning, with the largest radius just $5$ kilometers, the Earth is pretty nearly flat. But perhaps OP could clarify. –  Gerry Myerson Feb 20 '13 at 0:33
    
@GerryMyerson: Huh. Were there concrete values for the radii in the question when I commented? Yes, it looks so. Okay then. –  Henning Makholm Feb 20 '13 at 0:57
    
It was an example, but it's a good estimation. Wouldn't be too hard to incorporate haversine, en.wikipedia.org/wiki/Haversine_formula thank you. –  Mr. Demetrius Michael Feb 21 '13 at 18:08

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