Randomly placing circles into square equally apart I am setting up some wireless microphones. The transmitter will transmit 360 degrees for 10 meters. I would like to use an algorithm to work out how many more transmitters I need.
I have the area of the hall, and each transmitter will need to be placed 10 meters to the right,left,up and down.
So by using the algorithm I can find how many transmitters I need and where to put them.
I'm sure there are algorithms out there that might fit with slight adjustments, if anyone can point me in the correct place I will appreciate it. My maths isn't too good :)
 A: If you want to completely cover a hall, the circles must be overlapped, since placing them at regular distances of $10$ meters would leave many regions uncovered. In particular, considering the positions of the centers as lattice points in a plane, placing centers at vertical and horizontal regular distances of $\displaystyle \frac{10}{\sqrt{2}}$ meters would allow complete covering of the area (in this pattern, adjacent circles in the $45^o$ and $-45^o$ diagonal directions are tangent each other, and the distance between consecutive centers along these diagonal lines is twice the radius, i.e. $20$ meters). For a sufficiently large area, the resulting asymptotic density of centers is $\displaystyle \frac{1}{(\frac{10}{\sqrt{2}})^2}=0.02$ centers per square meters. For a hall of area $A$ (measured in square meters), the number of circles would then be estimated by $0.02\cdot A$. Note that, particularly for relatively small areas, this estimate must be adjusted in excess to take into account the borders of the hall, where a higher density is necessary to ensure complete covering.
