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In general, a quadrilateral-mesh surface does not enable a continuous rigid motion because an overconstrained system, in which the number of constraints around degree-4 vertices (three for each vertex) exceeds the number of variables (the number of hinges), is constructed. However, it is known that the developable double corrugation surface, known as Miura-ori, produces a rigid deployment mechanism. The rigid-foldability of Miura-ori is due to the singularity in its pattern, where a single vertex is repeated. We generalize the geometric condition for enabling rigid motion in general quadrilateral-mesh origami without simple repeating symmetry. To ensure the existence of a finite motion, we derive the identity of functions from the formula for degree-4 single-vertex origami. This yields a variety of unexplored generalized shapes of quadrilateral-mesh origami that preserve one-DOF finite rigid-foldability in addition to …