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This paper describes a method of approximating a shape contour with a polygon. The polygon vertices are extracted from the curvature extremes, through a scale-space description of the contour, via linear diffusion. These vertices are located on the contour points where the sharper changes of the contour directions occur. Using a proper strategy, a set of extremes that result in a given approximation level is chosen. By adding new vertices, the approximation level can be improved, and a scalable representation of the contour is identified. This method results in an approximation that discriminates local from global geometric features and provides a good visual representation of the original contour. This polygonal approximation method is used for scalable encoding of the shape contours. In this regard, an encoding technique suitable for scalable polygonal approximation has been developed. We show that encoding the approximated polygons result in a good relation between the distortion and the bitrate. Finally, we show that in addition to coding this method can be efficiently used for shape comparison and shape retrieval.