The kinematic synthesis equations of fairly simple planar linkage topologies are vastly nonlinear. This indicates that a large number of solutions exist, and hence a large number of design candidates might be present. Recent algorithms based in polynomial homotopy continuation have enabled the computation of entire solution sets that were previously not possible. These algorithms are based on a technique that stochastically accumulates finite roots and guarantees the exclusion of infinite roots. Here we apply the Cyclic Coefficient Parameter Continuation (CCPC) method to obtain for the first time the complete solution of a Stephenson III six-bar that traces a path and coordinates the angle of its input link along that path. Linkages of this type, called timed curve generators, are particularly useful for controlling the motion of an end effector point and influencing its transmission properties from a rotary input. For a numerically general version of the synthesis equations, we computed an approximately complete set of 1,017,708 solutions that divides into subsets of four according to the Stephenson III cognate structure. This numerically generic solution set essentially represents a design tool. It can be used in conjunction with a parameter homotopy to efficiently obtain all isolated roots of other systems of this same structure that correspond to a specific synthesis task. This is demonstrated with two example synthesis tasks.