Intersection of 3 Spheres
Problem
MathCityMap (MCM, https://mathcitymap.eu/) allows to solve tasks which are bound to coordinates. The MCM app on today’s smartphones uses a global navigation satellite system (GNSS) to determine the current position. GNSS uses the intersection of spheres in the 3D space. Determine the algebraic solution of the intersection of three spheres assuming that it exists. Feel free to use a Dynamic Mathematics Software (DMS, i.e., GeoGebra) to represent the problem visually.
Solution
We will consider the following three spheres, where and can be freely chosen (slider in GeoGebra) while is dependent and will be determined later, as they can be easily constructed in a DMS and an affine change of basis can transform any three spheres into the chosen ones.
(1)
(2)
(3)
Let us start with the spheres and by combining (1) and (2) using (2) – (1).
(4)
We define to this value
and conclude that the intersection of the spheres and lies in the plane parallel to through , i.e., having the equation . Putting (4) into (1) gives
which is a circle with centre and radius
i.e., having the equation
(5)
In the plane the equation (3) simplifies to
(6)
which is a circle with centre and radius .
The intersection of the three spheres has thus been reduced to the intersection of two circles which is solved in a similar way by combining (5) and (6) using (6) – (5).
(7)
Putting (7) into (5) gives
which solves to
Regarding GNSS, one of the two solutions can be eliminated as it is geometrically incoherent. Nevertheless, a fourth satellite is necessary, as it provides solutions in the measurement of signal propagation time, due to desynchronization of the receivers’ time compared to the satellites’ atomic clocks.