The problem of finding all four-bar linkages whose coupler curve passes through nine prescribed points has been a longstanding unsolved problem in kinematics. Using a combination of classical elimination, multihomogeneous variables, and numerical polynomial continuation, we show that there are generically 1442 nondegenerate solution along with their Roberts cognates, for a total of 4326 distinct solutions. Moreover, a computer algorithm that computes all solutions for any given nine points has been developed.
Complete Solution of the Nine-Point Path Synthesis Problem for Four-Bar Linkages
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Wampler, C. W., Morgan, A. P., and Sommese, A. J. (March 1, 1992). "Complete Solution of the Nine-Point Path Synthesis Problem for Four-Bar Linkages." ASME. J. Mech. Des. March 1992; 114(1): 153–159. https://doi.org/10.1115/1.2916909
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