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I am looking for an equation of intersection of 3 circles or 3 spheres, on the surface of the fourth (central) sphere, in a spherical coordinate circle. This should really be just a simple trilateration problem.

I know this is usually done by transforming the spherical coordinate system to a Cartesian coordinate system, then calculating the Intersection point, and then transforming it back to the spherical coordinate system. But due to large input errors my final error is then far too big to be useful.

Currently I am obtaining the result with a numerical calculation, but it takes too long (approx. 1 second for a single calculation), thus I would like to find an analytic solution. I have tried to derive it myself many times, but all I get is an unsolvable mix of sin and cosin functions (or tand and cotand)…

In short: I have the spherical coordinates of 3 points on the fourth ‘’center’’ sphere. On the center sphere I have a fourth point of which I do not know the coordinates, but I know all 3 angular distances between the first 3 points and the forth point.

How do I find the coordinates of the forth point, without having to transform everything into the Cartesian coordinate system?

I have scoured the internet and books for months, but failed to find a solution. Is it even possible?

I know it should be…

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# Spherical trilateration

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