This paper introduces a novel technique for analyzing digital curves by identifying turns and twists. A turn or twist, being a local feature, is well-defined for any three consecutive runs. A twist indicates an inflection, while a turn signifies no inflection. Determining the presence of inflections in a digital curve involves identifying twists interspersed among turns. We construct a definite grammar tree within the domain of regular language using a word-theoretic interpretation of pixel runs. We demonstrate how the pre-images of these runs facilitate a combinatorial classification of digital curves in discrete geometry, correlating with the grammar tree classification. This approach results in efficient vectorization of digital curves through a well-defined segmentation that adheres to certain invariant properties . Experimental results are provided to demonstrate the merit and efficacy of our method.