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]