  Folding paper along curves leads to spatial structures that have curved
surfaces meeting at spatial creases, defined as curve-fold origami. In this
work, we provide an Eulerian framework focusing on the mechanics of arbitrary
curve-fold origami, especially for multi-curve-fold origami with vertices. We
start with single-curve-fold origami that has wide panels. Wide panel leads to
different domains of mechanical responses induced by various generator
distributions of the curved surface. The theories are then extended to
multi-curve-fold origami, involving additional geometric correlations between
creases. As an illustrative example, the deformation and equilibrium
configuration of origami with annular creases are studied both theoretically
and numerically. Afterward, single-vertex curved origami theory is studied as a
special type of multi-curve-fold origami. We find that the extra periodicity at
the vertex strongly constrains the configuration space, leading to a region
near the vertex that has a striking universal equilibrium configuration
regardless of the mechanical properties. Both theories and numerics confirm the
existence of the universality in the near-field region. In addition, the
far-field deformation is obtained via energy minimization and validated by
finite element analysis. Our generalized multi-curve-fold origami theory,
including the vertex-contained universality, is anticipated to provide a new
understanding and framework for the shape programming of the curved fold
origami system.
