Not a precisely quantitative 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 indicated below:
<hr />
&nbsp;
[![SphereArcQ][1]][1]
<hr />
Pardon that I did not

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

but instead showed both $\pm$.

$C$ appears to be bounded by a circular arc connected to a `V`
whose angle is determined by $\epsilon$,
in particular, the straight-line boundaries of $C$ are orthogonal
to the $\epsilon$ extremes of $B$,
as is the circular arc boundary.
<hr />
[![Smiley][2]][2]
<hr />
In any case, have a nice day! :-)


  [1]: https://i.sstatic.net/aM4kx.jpg
  [2]: https://i.sstatic.net/SRv7m.jpg