GPS - Part 2

After reading yesterday's post, a friend asked why the handheld GPS device needs to get a distance from 4 satellites with known locations? Why won't three known points suffice?

Here is one way to think about this. Imagine a train traveling on a North-South railway line, say Mumbai's suburban local train. If I told you that the train is 4 km away from Andheri stations, that is not sufficient information to precisely locate the train for it could be either North or South of Andheri Station. But if I added another piece of information, that the train was also 10 km from Dadar, then you could locate the train at the point of intersection of these two distances. Alternately, I could tell you the train was 4 km South of Andheri - giving you again, two pieces of information, the distance and the direction. Long story short, to locate a point along one dimension, you need 2 pieces of information.

Lets try this in 2 dimensions. Let us say, you wanted to know the precise location of a certain village near Pune. If I told you the village was 50 km from Pune, it could be anywhere on the circumference of a circle with a radius of 50km centered on Pune. If we also added that it was 130 km from Mumbai, you are now better off, but these two circles could intersect each other at 2 points. If we add a third piece of information, that it is 1250 km from Delhi, then you have pinpointed the location.

Thus, the number of measurements needed is one more than the number of dimensions you wish to locate the object in. To get a location in 3 dimensions, i.e. on the surface of the earth, I explained patiently, we would need distances from 4 satellites.

My friend was not quite so naive. He was setting me up. After I had laboured through this explanation, he insisted that three satellites might suffice. His argument: the fourth measurement is known without reaching for a satellite. You know you are on the surface of a sphere with a radius of 6371 km.

"True," I said, in my best smart-ass tone, "Your solution would work if you were on the surface of the earth at mean sea level. But handheld GPS devices give you Latitude, Longitude and Altitude. And since the designers of the GPS system wanted the system to be usable by aircraft too, they figured you would need 4 pieces of information.

That leaves one question still open. If the satellites are buzzing about the earth in orbits along multiple planes, how do we know where the satellites are at any given point?

What say, Heisenberg?