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This paper presents a computational design method to obtain collapsible variations of rigid-foldable surfaces, ie, continuously and finitely transformable polyhedral surfaces, homeomorphic to disks and cylinders. Two novel techniques are proposed to design such surfaces: a technique for obtaining a freeform variation of a rigid-foldable and bidirectionally flat-foldable disk surface, which is a hybrid of generalized Miura-ori and eggbox patterns, and a technique to generalize the geometry of cylindrical surface using bidirectionally flat-foldable planar quadrilateral mesh by introducing additional constraints to keep the topology maintained throughout the continuous transformation. Proposed methods produce freeform variations of rigid-foldable structures that have not been realized thus far. Such a structure forms a one-DOF mechanism with two possible flat states. This enables the designs of deployable structures …