A note on the univariate nonic derived from the coupler curve of four-bar linkages

The coupler curve is the curve traced by one of the points on the coupler link of a four-bar linkage, which is algebraically a sextic bivariate polynomial. A system of eighth-order polynomial equations of nine unknowns can be established from the algebraic coupler curve in order to determine linkage parameters. However, such a system of higher order multivariate polynomial equations poses a big challenge to find solutions efficiently. In this paper, a univariate 9th order polynomial is derived from the system of higher order polynomials. The derivation of the univariate polynomial is presented in details. Examples of coupler curve synthesis are included to show the efficient solutions with the univariate polynomial.