There are six distinct input-output (IO) equations for any given four-bar 4R linkage that relate distinct pairs of the four exterior angles between the edges of the deformable quadrilateral. The algebraic \(v_i\) - \(v_j\) IO equation expresses one joint angle tangent half-angle parameter, \(v_i\) , in terms of another, \(v_j\) , as an implicit biquadratic function scaled by the link lengths. In this paper a novel method is presented using the \(v_1\) - \(v_2\) IO equation for deriving the parametric coupler point curve equation for both assembly modes in terms of the input angle parameter \(v_1\) , expressed in the non-moving linkage reference coordinate system. To do this, the \(v_1\) - \(v_2\) equation is solved for \(v_2\) and the result is substituted into the forward kinematics \(4\times 4\) homogeneous transformation matrix for the open 2R kinematic chain of the first two moving links. Finally, the coupler point coordinates in the \(x_2, y_2\) coordinate system that moves with the coupler are transformed into the relatively non-moving \(x_0\) , \(y_0\) linkage coordinate system, yielding the parametric coupler point curve equation.