I'd like to see this answered by someone knowledgeable too. From my limited knowledge I would guess the function being C2 gives the curvature a "global" domain

I'd like to see this answered by someone knowledgeable too. From my limited knowledge I would guess the function being C2 gives the curvature a "global" domain; while if the function is only C1 the curvatures will have limited influence, and you can use that to enforce other curvatures "prohibited" under C2. Anyway, I might be completely wrong, just the idea I got from the article :)