[Math] Find, inside a large circle, the maximum number of small circles placed 60 degrees to each other and

circlespacking-problem

… starts with a small circle in the center of the large circle.

enter image description here

The above picture shows a program I wrote to actually draw the circles out. But you can see that this method does not yield maximum number of blue circles. There are still spaces around the red circle.

The method i used is to draw blue circle "rings" starting from the center outwards. i.e move out in the blue arrow direction for one circle diameter, then go around in the red arrow direction, then repeat next ring in the blue arrow direction.

Anyone can share a smarter method? Thank you all. I need only to calculate the number, but if there is a systematic way to draw will be better.

Best Answer

If your goal is to make a program, then considering you already have that done, it seems like the easiest strategy would just be to: 1. Add on more rings until you know that every circle in a new ring will be outside the main circle. 2. Iterate through the small circles, removing all of the circles who's centers are further than R-r from the main circle, where R and r are the radii of the large and small circles respectively.

That however assumes that you can't fit more circles in by translating your entire set of small circles to the side a bit more. If you want to make sure you have the maximum, you might have to do some more fudging.

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[Math] Diameter of nth circle between two circles problem

Translate everything towards the negative $x$-direction for unit distance. Under this translation,

the contact point of the red and blue circle become the new origin.
the old $x$-axis becomes the line $y = -1$.

Invert with respect to the unit circle in the new coordinate system. Under this circle inversion,

The red circle become the line $x = \frac12$.
The blue circle become the line $x = -\frac12$.
The contact points $(\pm 1, -1)$ of the red/blue circles and the line $y = -1$ get mapped to $(\pm \frac12, -\frac12)$. Since any straight line will mapped to a circle passing through origin, we find the line $y = -1$ get mapped to a circle of radius $\frac12$ centered at $(0,-\frac12)$.
Since the green circle are tangent to the red/blue circles and the line $y = -1$. It get mapped to a circle radius $\frac12$ centered at $(0,-\frac32)$.
By a similar argument, one find the orangle circle get mapped to a circle of radius $\frac12$ centered at $(0,-\frac52)$.

Combine these observations, it is not hard to see in general, the $n^{th}$ circles we wish to construct get mapped to a circle of radius $\frac12$ centered at $(0, -(n + \frac12))$. Notice the nearest and farest distance of the inverted $n^{th}$ circle to the origin is $n$ and $n+1$. When one one invert it back, one find the diameter of the original $n^{th}$ circle is simply $$C_n = \left(\frac1n - \frac{1}{n+1}\right) = \frac1{n(n+1)}$$

Draw a line from point to circle circumference, but pointing at the center

The asin function requires you to do various tricks to deal with lines that can come from any direction (upper right, upper left, lower right, lower left) because it only gives you half a circle's worth of angles, $-\frac\pi2$ to $\frac\pi2,$ whereas you need angles all around the circle.

The atan2 function, if your software library has it, is usually much better for applications like this. You call it like this:

    atan2(b, a)

and it gives you a full range of angles from $-\pi$ to $\pi,$ which will be sufficient for lines coming from any direction. Moreover, you don't even need to compute $c.$

Best Answer

Related Solutions

[Math] Diameter of nth circle between two circles problem

Draw a line from point to circle circumference, but pointing at the center

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