I'll have a look at it. This is a small adaptation on the algorithm matplotlib uses for generating contours (link). Generating contours is more complicated than it seems at first so I needed this external help. Solving problems there is not as easy as solving problems in my own code, so this review might take a while.

BOUNDARY_S(quad) does return a bool, so it doesn't really make sense to compare it with BOUNDARY_S(quad) > 0.
This is just a std::cout and the functions write_cache and write_cache_quad are not really being used anyway (it's just something that came with the original code) so that's not a problem. I could safely change (BOUNDARY_S(quad) > 0) for BOUNDARY_S(quad) but I'll try to understand the logic behind that to fix everything at once.