Design of Infinitesimally and Finitely Flexible Origami Based on Reciprocal Figures

This paper investigates novel computational design method for infinitesimally and finitely foldable rigid origami based on solving a first-order folding mode, which can be represented by a reciprocal figure. We derive these graphical conditions from a matrix representation of rigid origami, and extend the conditions to cases when the surface includes holes. We propose an algorithm to obtain forms that satisfy the conditions and an interactive system to freely design infinitesimally foldable forms. We show design examples of shaky polyhedron and origami, and finitely foldable quadrivalent mesh origami.