  An efficient way to introduce elastic energy that can bias an origami
structure toward desired shapes is to allow curved tiles between the creases.
The bending of the tiles supplies the energy and the tiles themselves may have
additional functionality. In this paper, we present the theorem and systematic
design methods for quite general curved origami structures that can be folded
from a flat sheet, and we present methods to accurately find the stored elastic
energy. Here the tiles are allowed to undergo curved isometric mappings, and
the associated creases necessarily undergo isometric mappings as curves. These
assumptions are consistent with a variety of practical methods for crease
design. The h^3 scaling of the energy of thin sheets (h = thickness) spans a
broad energy range. Different tiles in an origami design can have different
values of h, and individual tiles can also have varying h. Following
developments for piecewise rigid origami, we develop further the Lagrangian
approach and the group orbit procedure in this context. We notice that some of
the simplest designs that arise from the group orbit procedure for certain
helical and conformal groups provide better matches to the buckling patterns
observed in compressed cylinders and cones than known patterns.
