  A recent result shows that a general smooth plane quartic can be recovered
from its 24 inflection lines and a single inflection point. Nevertheless, the
question whether or not a smooth plane curve of degree at least 4 is determined
by its inflection lines is still open. Over a field of characteristic 0, we
show that it is possible to reconstruct any smooth plane quartic with at least
8 hyperinflection points by its inflection lines. Our methods apply also in
positive characteristic, where we show a similar result, with two exceptions in
characteristic 13.
