I recently learnt about how GPS works and how it uses the intersection of spheres to locate a person which got me thinking whether 4 spheres can always guarantee a position fix. My understanding is that greater than 3 spheres can always give 2 points of intersection provided all the centres of the spheres are in the same plane. I imagine that it rarely occurs that all the satellites are coplanar but also I imagine its possible that we can interpret which of the two intersections give a position on Earth rather than in space. Still I am curious to know whether I'm correct in thinking that no number of satellites can guarantee in every case a singular intersection since it is possible that the find themselves all to be coplanar.
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1$\begingroup$ Considering that satellites by definition are in eliptical orbits around the earth " all the centres of the spheres are in the same plane", there are more than centers of rotation to take into account. It seems to me that the probability of overlay you imagine is zero "Satellites undergo an elliptical orbit around the Earth. The shape and size of ellipse can be defined by i) eccentricity (e): degree of perturbation from the perfect circle, and ii) the semi-major angle (a): the half distance between the perigee and apogee in the ellipse. $\endgroup$– anna vCommented Oct 14, 2023 at 11:58
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$\begingroup$ Finally, the angle between the perigee and the satellite with respect to the Earth’s center can be defined by true anomaly (f)." quote from "12.1.1 Satellite Orbit"sciencedirect.com/topics/earth-and-planetary-sciences/… $\endgroup$– anna vCommented Oct 14, 2023 at 12:00
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I am curious to know whether I'm correct in thinking that no number of satellites can guarantee in every case a singular intersection since it is possible that the find themselves all to be coplanar.
Yes. You can think of each satellite defining a sphere, where the center of the sphere is the satellite and the receiver is some point on the sphere. The spheres all intersect at the receiver, but they may intersect in other points also.
Two spheres intersect in a circle. Any number of additional colinear spheres, spheres centered on the line defined by the center of the two, will intersect in the same circle.
Three non-colinear spheres intersect in two points. Any number additional coplanar spheres, spheres centered on the plane defined by the centers of the three, will intersect in the same two points.
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$\begingroup$ So I imagine that despite this being theoretically possible it is extremely unlikely / maybe impossible depending on their orbits and hence not a real world problem? $\endgroup$– NullityCommented Oct 14, 2023 at 13:52
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$\begingroup$ Yes, quite unlikely. There are 24 satellites in 6 different orbital planes. So you really should never see more than two from the same orbital plane. $\endgroup$– DaleCommented Oct 14, 2023 at 14:12
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1$\begingroup$ @Nullity, unlikely is correct. In the days when we had fewer satellites, there was software you could use to predict how good/bad it was going to be at a particular location by predicting the en.wikipedia.org/wiki/Dilution_of_precision_(navigation) (DOP) based on the available vehicles. You might plan an operation or measurement for when the DOP was within certain parameters. That's less necessary with today's constellations. $\endgroup$ Commented Oct 14, 2023 at 16:34