## 4 Case Study and Discussion
### Discussion on the Approach Efficiency
The simplest strategy is the traversal strategy, in which all patches on the target plane should be compared with the current point such that the target patch can be determined. The traversal strategy is feasible, but has low efficiency because the patches far from the current point are also traversed, even though they cannot be the target patch. To avoid unnecessary traversal computations, the adjacent-area strategy is employed in this study, and its principle is displayed in Figure 16. The mating area should be determined first such that the discrete points that cannot be in contact will never be traversed. Next, two steps are performed twice. Namely, the mating area is divided into four sub-areas, and the sub-area that corresponds to the current point is determined. As a result, the sub-area with only \(1/16\) of the mating area is determined and is the adjacent area of the current point. Finally, the target patch can be acquired by performing horizontal and vertical bisection iterations  
Figure 15: Principle of computing the distances between the corresponding points by projecting [26]. of the adjacent area. The adjacent-area strategy is more complex than the traversal strategy, which questions its efficiency if the density of the point clouds is not high. However, the efficiency advantage of the adjacent-area strategy will prevail with the increasing density of the point clouds. **Effect of the mesh size of point clouds. **  
Point clouds are employed to describe the mating planes. Thus, the surface deviations of each mating plane are collections of discrete points. The mesh size refers to the distance between two discrete points in the length or width direction of a mating plane, and is a key parameter of point clouds. If small mesh size is given, the morphologies of the mating plane will be fully described, but the time required to compute the distances between the corresponding points will increase. As a result, the efficiency of determining contact points is low. On the contrary, if mesh size increases, some geometric fluctuations of the mating plane will be ignored. However, the efficiency of determining contact points will increase because the time required to compute the distances between the corresponding points is directly reduced. To verify the impacts of the factors mentioned above on the approach efficiency, the PODs were computed under different sizes of point clouds. The computation time was recorded in the case of assembly state 1 (Section 4.1) as an example. The software environment of these simulations was **MATLAB(r) R2015b**, and the hardware environment was an ordinary desktop computer (Windows 7 OS, 3.50 GHz Intel Core i5-4690 CPU, and 16 GB RAM). The results are listed in Table 6.  
The computation time is only 14.1s when the size of the point clouds is 2.5 mm. Thus, the efficiency of the proposed approach is verified. Moreover, the computation time increases with decreasing point cloud size; for example, it increases by about 40 times when the point cloud size is reduced to 0.5 mm. Consequently, the appropriate size of the point clouds should be determined by weighing the accuracy and the efficiency when the proposed approach is employed. The size of the point clouds should be as large as possible to achieve the highest efficiency when the accuracy is acceptable. Otherwise, a small size of the point clouds should be given if the required accuracy is high, even though the computing cost are inevitably affected.