We consider the problem of computing a minimum-weight polygonal path between two points in a weighted polygonal subdivision, subject to the constraint that the path have few segments (links). We give an algorithm that generates paths of weighted length at most (1 + ε) times the weight of a minimum-cost k-link path, for any fixed ε > 0, while using at most 2k - 1 links. This is an improvement over the previous (1 + ε)-approximation algorithm, which used at most 5k - 2 links. Further, we have implemented our new algorithm and we have conducted a performance study of these algorithms (old and new) on a variety of real-world and synthetic data, comparing not only the efficiency but also the quality of paths generated using these algorithms. We also consider the implications of these results on the practical usage of these algorithms.