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The singularities' problem.
On a sphere, if you choose any system of coordinates,
you cannot avoid singularities (for example two polar singularities):
(50)
Notice you can map a sphere with a single "polar singularity".
Cut the sphere by this first family of planes, passing by the same straight
line:
(51)
Then introduce the second family of planes, which intersects
the sphere too.
(52)
If we except this local mess, elsewhere there is no problem.
Looking at the sphere from the other side, you get this:
(53)
But, at the S points, the values of a
and b are simply not
defined!
However a sphere is basically a regular surface. Turn
an egg in your hand: you will not discover any peculiar point, any intrinsic
singularities.
(54)
Conclusion: these singularities are an artefact due
to the choice of coordinates.
These polar singularities are not "real", are not intrinsic
singularity. You choose a system of coordinates. Then an arbitrary point,
or two, become singular. The two singularities of a mapped sphere: the northern
pole and the southern pole, are a pure creation of the choice of coordinates
system, an artifact.
