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Transformable polyhedral surfaces with rigid facets, ie, rigid origami, are useful for designing kinetic and deployable structures. In order to apply rigid origami to various architectural and other engineering design purposes, it is essential to consider the geometry of origami in kinetic motion and provide sufficiently generalized methods to produce controlled variations of shapes that suit the given design conditions. In this paper, we introduce the author’s recent studies and their extensions on the geometry of rigid origami for designing transformable and deployable structures.