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The problem of singularities.
On a sphere, no matter what coordinate system you choose, you cannot avoid singularities (for example, two polar singularities):
(50)
Note that it is possible to map a sphere using a single "polar singularity". Cut the sphere along this first family of planes, all passing through the same straight line:
(51)
Then introduce the second family of planes, which also intersect the sphere.
(52)
If we set aside this problematic local area, there is no problem elsewhere. Looking at the sphere from the other side, you get this:
(53)
However, at the S points, the values of a and b are simply not defined!
However, a sphere is fundamentally a regular surface. Turn an egg in your hands: you will not discover any special point, any intrinsic singularity.
(54)
Conclusion: these singularities are an artifact due to the choice of coordinates.
These polar singularities are not "real", they are not intrinsic singularities. You choose a coordinate system, and then an arbitrary point, or two, becomes singular. The two singularities of a mapped sphere—the north pole and the south pole—are a purely artificial creation due to the choice of the coordinate system.