This is me responding to my own post again haha!
Anyways, I beleive I have solved the problem.  I will have rendered images
for you tommorrow that will demonstrate what the "banding" looks like.  It
seems I needed to use a homogenous axis system everywhere, except for two
singularities at the poles of the unit sphere, which are handled as rare
cases in the code.   Your suggestion about singularities really pointed me
in the right direction.

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