A constructive geometric approach to the foldability

This paper introduces a unified geometric formulation, providing a comprehensive, Euclideanfoundation for planar four-bar linkages. Moving away from traditional algebraic frameworks thatclassify mechanisms through link length inequalities, this work establishes a complementary geometricframework by utilizing a rigid generating triangle as the fundamental evolutionary structure.By introducing a fourth hinge joint along the perimeter of the triangle, the system transforms intoa mobile kinematic linkage. The proposed framework proves that total collinear foldability is bothconstructively generated and universally characterized through a two-fold geometric law, supportedby rigorous synthetic proofs for both genesis and structural configuration. This formulation offersa coordinate-free approach that enriches the morphological analysis of linkage configurations.