  Rigidly and flat-foldable quadrilateral mesh origami is the class of
quadrilateral mesh crease patterns with one fundamental property: the patterns
can be folded from flat to fully-folded flat by a continuous one-parameter
family of piecewise affine deformations that do not stretch or bend the
mesh-panels. In this work, we explicitly characterize the designs and
deformations of all possible rigidly and flat-foldable quadrilateral mesh
origami. Our key idea is a rigidity theorem (Theorem 3.1) that characterizes
compatible crease patterns surrounding a single panel and enables us to march
from panel to panel to compute the pattern and its corresponding deformations
explicitly. The marching procedure is computationally efficient. So we use it
to formulate the inverse problem: to design a crease pattern to achieve a
targeted shape along the path of its rigidly and flat-foldable motion. The
initial results on the inverse problem are promising and suggest a broadly
useful engineering design strategy for shape-morphing with origami.
