Stitching together piece of flat space If I start with an infinite flat sheet of graph paper, and in polar coordinates cut out a piece according to:
$r>0, \ \ -f(r) < \theta < f(r)$
Now I want to stitch the remaining graph paper together, by associating each point $(r,f(r))$ to $(r,-f(r))$ on the seam.
How do I calculate what the curvature is of the stitched up paper along the seam?
 A: Perhaps your description of what you wanted to do doesn't match with what you intended. If we go backwards, i.e., start with a right circular cone, and make some squiggly cut in it, from base to apex, and then flatten that out we will get a disk with a wedge with squiggly sides cut out. The two sguiggly sides match up. 
Assume that if we did the cut with a straight line, then we'd get a wedge described by the lines $\pm \tilde{\theta}$. Now, we can imagine replacing those straight lines by the squiggly one described by your function. To keep from making a mess, we'd need constraints on the function, corresponding to your squiggly line on the cone not wrapping around and intersecting itself. Say $f'(0) = 0$, and $|f(r)| < \pi r$, but we might have to think about that more. Now we have a wedge cut out by curves described by $\pm \tilde{\theta} + f(r)$. In other words, we're removing the piece according to 
$$
-\tilde{\theta} + f(r) \leq \theta \leq \tilde{\theta} + f(r),
$$
which is not the same as you have in your question. If we get it right, and we have a seam with no curvature, then it will be represented by a curve on the cone we get, so it makes sense to go backwards like this. 
