Singularities on a sphere

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This has absolutely nothing to do with game development, but I hope someone here feels like answering this question for me. I'm reading a book where some process is pictured as the propagation of a latitude parallel along a sphere. The parallel will start at the north pole with 0 radius, expand to some maximum at the equator and contract back to 0 radius at the south pole:

The main point being made using this metaphor, is that there are no singularities at the north and south poles. I find this a bit confusing, since the first thing that occured to me when I saw this picture was the Hairy Ball Theorem. By definition it'd follow that there are no continuous tangents for the north and south poles in this particular picture. Doesn't this mean that the north and south poles are in fact singularities since they're not differentiable?

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Of course the poles are singularities in that flow. The only possible value for the vector field at the poles is 0, since anything else would mean there is a discontinuity. What you say your book says is wrong.

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Thank you for your reply, good to know my hunch made sense. Maybe I'm reading too much into this metaphor, but it just rather strange the author would pick just this example to prove a point about these singularities. Would it make any difference if complex numbers are involved?

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You can have the vector field on the sphere with only one discontinuity.

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