Investigating the geometric similarity of trajectory data to extract movement patterns in urban environments is an emerging area of research that has attracted several efforts over the past few years. This paper uses a convex-hull algorithm whose objective is first to identify curvatures and turning points in a given trajectory and secondly to provide a computable solution to identify the similarities of trajectories. Moreover, the present paper tries to detect additional capabilities that will support the exploration of regular patterns efficiently. This approach is supported by a series of geometrical definitions and algorithms that reduce the complexity of primary trajectories significantly and identify a trajectory geometrical decomposition modeled by an abstract trajectory descriptor (ATD). The main novelty of this paper is to find out the similarity between the row trajectory's geometry and the results of the ATD method using the known geometric measures as distance, orientation, complexity, and shape. Based on this decomposition principle, trajectory similarities can be studied using physical, geometrical, or both descriptors as considered in the ATD method. The proposed method has been evaluated using Geolife benchmark trajectory database. the results show that the proposed algorithm not only successfully identify curvatures and turning points at different scales, but also proved to provide relevant trajectory similarities with efficient computation times as the overall similarity difference value equals 0.002 between the two row trajectories. The resulted trajectories using the ATD method applying less than 5% of the primary points. In addition, the computation time of about 93% is reduced using the detected critical geometric points using the ATD method.