Given a single degree of freedom mechanism, a moving reference frame attached to any link has a motion that can be described with a single parameter. A point relative to this moving frame is sought such that it either continually increases or decreases in distance from a point in the fixed frame over the entire motion. These points can be used to define a revolute–prismatic–revolute (RPR) chain for a planar mechanism or a spherical–prismatic–spherical (SPS) chain for a spherical mechanism capable of actuating the device over its entire range of motion. Moreover, the singularities relative to the joints in the original mechanism are not a concern and the dimensional synthesis can focus on creating the set of circuit-defect free solutions. From this analysis, a unique fixed point is determined in the planar case relative to two positions and their velocities with the following characteristic. All points in the moving reference frame that are moving away from it in the first position are approaching it in the second position, and vice versa. This point is as critical to the identification of singularity-free driving chains as the centrodes or the poles.