Determining GPS Receiver Position with Satellite Distance Differences

[Maruchan]

Homework Statement

The global positioning system GPS uses the fact that a receiver can get the difference of distances to two satellites. Each GPS satellite sends periodically signals which are triggered by an atomic clock. While the distance to each satellite is not known, the difference from the distances to two satellites can be determined from the time delay of the two signals. This clever trick has the consequence that the receiver does not need to contain an atomic clock itself.

a) (7 points) Given two satellites P=(2,0,0), Q=(0,0,0) in space. Identify the quadric of all points X, such that the distance d(X,P) to P is by 1 larger than the distance d(X,Q) to Q. b) (1 point) Assume we have three satellites P,Q and R in space and that the receiver at X knows the distances d(X,P) - d(X,Q) and d(X,P) - d(X,R). Why do we know the distance d(X,Q) - d(X,R) also? Conclude that 3 satellites are not enough to determine the location of the receiver. c) (2 points) Assume we have 4 satellites P,Q,R,S in space and that the receiver knows all the distance differences from X to any pair of satellites from the 4. What is in general the set of points for which these distances match? Conclude that with some additional rough location information we can determine the GPS receiver position when 4 satellites are visiable.

The Attempt at a Solution

I solved part a fairly easily; I set the distance formulas equations equal and squared both sides, manipulated then squared again.
I ended up with an answer of 12(x-1)^2-4y^2-4z^2=3 which is a hyperboloid of two sheets.

Part b I'm not entirely certain, but I believe that an application of the vector addition triangle law might be able to solve this. Can someone elaborate/clarify for me?

Part c is the portion I need the most help with. I'm not certain what the question is asking or what form my solution needs to take. If anyone can help, I would greatly appreciate it.

Thank you very much.

 

[Maruchan]

Can anyone help?

I have somewhat of an idea about part c...if we have 2 satellites it goes to a plane, 3 satellites goes to a curve, 4 satellites goes to a point...but how would I show this mathematically?