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February 2nd, 2018, 10:20 AM   #1
Joined: Mar 2016
From: UK

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Overlapping Circles & Lattice Efficiencies


I was wondering if anyone could help me with an overlapping circle problem... I'm trying to prove that the efficiency of a triangular lattice is higher than that for a square lattice.

Given the figure below:

The blue dots represent LEDs and the black lines show the lattice. The red circles represent the radiance of the LEDs. To simplify the problem, I'm assuming the radiance of each LED is the same regardless of distance from the source, for a given radius.

The left hand side of the figure shows a triangular lattice and the right hand side shows a square lattice. It can be seen that there are 4 unique levels of radiance. Let 1 be the weakest and 4 be the strongest. This gives the triangular lattice 2 radiance levels of 3 and 4 and the square lattice 3 radiance levels of 2, 3 and 4.

Finally, assume the radius of each circle is 1 and therefore the length of a side for either the triangle or square in each respective lattice will be 1.

So the question is, how do I calculate the percentage area for each radiance level for either the square or the triangular lattice?

I've successfully answered this for the triangular lattice...

The area of a triangular cell is simply: $\displaystyle \frac{\sqrt{3}}{4}a^2$

As $\displaystyle a = 1$ we get: $\displaystyle \frac{\sqrt{3}}{4}\approx 0.433$ (Eqn. 1)

The area of a circle is: $\displaystyle \pi r^2$

As $\displaystyle r = 1$ we get: $\displaystyle \pi$

Given the arc is $\displaystyle \frac{1}{6}$ of the circle, we get the area of the arc as: $\displaystyle \frac{\pi}{6}$ (Eqn. 2)

Subtracting (1) from (2) we get: $\displaystyle \frac{\pi}{6}-\frac{\sqrt{3}}{4}$

As there are three such segments in one cell, this gives: $\displaystyle 3\left ( \frac{\pi}{6}-\frac{\sqrt{3}}{4} \right )\approx 0.272$ (Eqn. 3)

To find the percentage of the arc segments in the triangular cell, simply divide (3) by (1) which gives $\displaystyle \approx 62.8\%$

So, this solves the triangular lattice, but I really can't get my head around the square lattice?

I know that each arc segment in the square lattice is $\displaystyle \frac{\pi}{4}$ and that the area of a cell is $\displaystyle 1^2=1$ but don't know how to separate each individual segment.

I look forward to hearing from you. If you have any questions, or I've missed anything out, please do let me know and I'll do my best to answer them or fill in the gaps.


Kind Regards

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Last edited by edwardholmes91; February 2nd, 2018 at 10:24 AM.
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