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Joseph O'Rourke
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Not an answer. Just attempting to track the specified geometry. I used $\epsilon=0.1$ radians to delimit the subset $B \subset T$.

My question is, what does $C$ look like (as a set of points)?

It looks like as shown below:


  [![SphereArcQ][1]][1]
Pardon that I did not

(Assume we choose the orthogonal vector with positive x-value)

but instead showed both $\pm$.

In any case, have a nice day! :-)


[![Smiley][2]][2]
Joseph O'Rourke
  • 150.8k
  • 36
  • 358
  • 958