This paper proposes a fast and unsupervised scheme for the polygonal approximation of a closed digital curve. It is demonstrated that the approximation scheme is faster than state-of-the-art approximation and is competitive with Rosin's measure and aesthetic aspects. The scheme comprises of three phases: initial segmentation, iterative vertex insertion, iterative merging, and vertex adjustment. The initial segmentation is used to detect sharp turns, that is, vertices that seemingly have high curvature. It is likely that some of the important vertices with low curvature might have been missed in the first phase; therefore, iterative vertex insertion is used to add vertices in a region where the curvature changes slowly but steadily. The initial phase may pick up some undesirable vertices, and thus merging is used to eliminate redundant vertices. Finally, vertex adjustment was used to enhance the aesthetic appearance of the approximation. The quality of the approximations was measured using the Rosin's method. The robustness of the proposed scheme with respect to geometric transformation was observed.