  Geometric stress focusing, e.g. in a crumpled sheet, creates point-like
vertices that terminate in a characteristic local crescent shape. The observed
scaling of the size of this crescent is an open question in the stress focusing
of elastic thin sheets. According to experiments and simulations, this size
depends on the outer dimension of the sheet, but intuition and rudimentary
energy balance indicate it should only depend on the sheet thickness. We
address this discrepancy by modeling the observed crescent with a more
geometric approach, where we treat the crescent as a curved crease in an
isometric sheet. Although curved creases have already been studied extensively,
the crescent in a crumpled sheet has its own unique features: the material
crescent terminates within the material, and the material extent is
indefinitely larger than the extent of the crescent. These features together
with the general constraints of isometry lead to constraints linking the
surface profile to the crease-line geometry. We construct several examples
obeying these constraints, showing finite curved creases are fully realizable.
This approach has some particular advantages over previous analyses, as we are
able to describe the entire material without having to exclude the region
around the sharp crescent. Finally, we deduce testable relations between the
crease and the surrounding sheet, and discuss some of the implications of our
approach with regards to the scaling of the crescent size.
