This paper introduces a 3D object tracking method for an augmented reality (AR) assembly assistance application. The tracking method relies on point clouds; it uses 3D feature descriptors and point cloud matching with the iterative closest points (ICP) algorithm. The feature descriptors identify an object in a point cloud; ICP align a reference object with this point cloud. The challenge is to achieve high fidelity while maintaining camera frame rates. The point cloud and reference object sampling density are one of the key factors to meet this challenge. In this research, three-point sampling methods and two-point cloud search algorithms were compared to assess their fidelity when tracking typical products of mechanical engineering. The results indicate that a uniform sampling maintains the best fidelity at camera frame rates.