Solving a planar four-bar linkages design problem

The problem of synthesizing a planar four-bar linkages whose curve passes through five prescribed precision points based on elimination methods has been a difficult problem in kinematics because it is of high polynomial degree and highly nonlinear. A new elimination method to solve the problem is presented. Based on the presented method, a 36th degree univariate polynomial of the problem is derived from the determinant of a Sylvester's matrix, which is relatively small in size, without factoring out or deriving the greatest common divisor. Conclusion shows that there are generically 36 non-degenerate solutions of the problem. To verify the obtained conclusion, the completely same result can be obtained using continuation method.