The Hyperbola depiction of how the GPS equations are solved:
From several psuedoranges measured at the same time, we can calculate the (absolute) difference in the distances between us and these GPS satellites. 2 satellites gives us 1 known difference in distance, and we can plot our position along the surface of a hyperboloid where all points at the surface have the same (absolute) diff in distance to the 2 satellites. From there we can add an other or several other additional satellites to make more hyperboloids which will create an intersection point at our (receiver) position.
We will need 3 hyperboloids to make one point of intersection.
Why will 3 satellites give us only 2 hyperboloids as opposed to 3?
(hyperboloids between satellites 1 + 2, 2 + 3, AND 1 + 3?)
And why will 4 satellites give us only 3 hyperboloids as opposed to 6?
(hyperboloids between satellites 1 + 2, 2 + 3, AND 3 + 4
AND 1 + 3, 2 + 4, 1 + 4 ?)
Formula taken from one of the answers under:
k satellites in general position do determine k(k-1)/2 hyperboloids
So, why do we need the 4th satellite, when it is not needed to make 3 hyperboloids?
Best Answer
EDIT: I answered this question originally with the assumption that there was only one way of understanding the way GPS works. There's a good few paragraphs on Wikipedia: https://en.wikipedia.org/wiki/Global_Positioning_System#Geometric_interpretation
I don't understand why you can't have more hyperboloids. I wonder if there is no value at adding more hyperboloids, i.e., including more than 3 'differences' when you have 4 satellites doesn't increase the amount of information?
--old post follows
I thought having two satellites to measure your distance from would give you a circle (the intersection of two spheres. And that three satellites gives you two points, one near the Earth's surface and one up in space somewhere. I don't think that you do get hyperboloids so I am keen to see a reference.
Also, note that in order to measure distances, your receiver needs to solve for time too, so you need at least four satellites. (You also need the almanac that tells you where the satellite is at a particular time, and that's transmitted in the signal from the satellites, or used to be, which is why it took ages to get a first fix on old hardware).
This seems like a pretty good resource for reading up: https://www.maptoaster.com/maptoaster-topo-nz/articles/how-gps-works/how-gps-works.html
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