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        riddles >> medium >> How many towers to cover the whole world?
        
(Message started by: MacNCheese on Apr 17^th, 2012, 2:01pm)

Title: How many towers to cover the whole world?
Post by MacNCheese on Apr 17^th, 2012, 2:01pm

Hi :D

This is my first question (and post), so dont be too hard on me :P
I haven't solved it yet, but I think I've a good approximation, for a particular case.

Consider the problem of constructing towers for transmitting radio waves all over the world.

At frequencies above 40 MHz, communication between antennas is essentially limited to line-of-sight (LOS) paths. Because of line-of-sight nature of propagation, direct waves get blocked at some point by the curvature of earth (a tangent to the surface of earth connecting the top of the communicating towers being the maximum displacement between tops of the towers).

[I'd upload a pic to demonstrate but it won't let me]

Your problem is to find the minimum number of towers so that all the teenagers in the world can share their soulless pop music, assuming the following:

1. The earth is a perfect sphere with radius R.
2. The height of each tower is a constant H.
3. All towers can transmit signals to and from other towers (and to every point) within LOS. Hence if A is within LOS of B, and B is within LOS of C, A is said to be connected to C as well.
4. The whole world is said to be connected if every point is connected to every other point on the surface.

Since I haven't studied non-euclidean geometry at all, I'm not sure if this belongs in this or the easy section.

Establishing a lower bound should be fairly easy. But there will obviously be overlap, so it'll require a few more towers :P

Let me know if anything needs clarification.

Sidenote: Alternatively, try minimizing cost where the
cost for a tower of variable height H = AH² + B
where A, B are constants, and assuming towers can be any height.

Title: Re: How many towers to cover the whole world?
Post by SMQ on Apr 17^th, 2012, 6:34pm

Sounds like you want towers to be [hide]approximately equally spaced on the surface of the sphere[/hide] so that all the LOS between nearest neighbors just graze the horizon. In that case I would start by considering [hide]the vertices of geodesic domes[/hide] (http://en.wikipedia.org/wiki/Geodesic_dome).

--SMQ

Title: Re: How many towers to cover the whole world?
Post by rmsgrey on Apr 18^th, 2012, 2:25pm

I'd model it as placing (overlapping) discs on a sphere to cover the surface - obviously that's necessary in order to connect to every point on the surface, and it's almost as obviously sufficient since if two internally connected regions touch or overlap at any point, the two towers directly involved in the contact will have LOS on each other, so, for any two towers not to be connected, there would have to be an uncovered strip separating them...

Title: Re: How many towers to cover the whole world?
Post by MacNCheese on Apr 18^th, 2012, 3:25pm

That's exactly how I went about trying to do it, finding the area of coverage for any one tower. Since the positions would be symmetric, we would place it on the vertices of geodesic domes, as SMQ said. But I can't quite figure out how much area of overlap per tower there will be, so I can't calculate the minimum number of towers. The lower bound was simple: divide the area of sphere with area of coverage of a tower, but after that, I could only approximate overlap for H << R.

Any help with that?

Title: Re: How many towers to cover the whole world?
Post by arun gupta on Apr 25^th, 2012, 5:39am

Interesting question :o but i don't know answer ???

Title: Re: How many towers to cover the whole world?
Post by prachisharma on May 27^th, 2012, 11:54pm

at list 1 is there don't no about maximum