Several efforts have recently been made to relate the displacement of swimming three-link systems over strokes to geometric quantities of the strokes. In doing so, they provide powerful, intuitive representations of the bounds on a system's locomotion capabilities and the forms of its optimal strokes or gaits. While this approach has been successful for finding net rotations, noncommutativity concerns have prevented it from working for net translations. Our recent results on other locomoting systems have shown that the degree of this noncommutativity is dependent on the coordinates used to describe the problem and that it can be greatly mitigated by an optimal choice of coordinates. Here, we extend the benefits of this optimal-coordinate approach to the analysis of swimming at the extremes of low and high Reynolds numbers.