## 3 The Method for Describing the Positional Relation of Two Parallel Surfaces
### The Description Method of Positional Relation between Two Vertical Parallel Surfaces  
Figure 9 shows the detection model of two vertical parallel surfaces. The detection process and algorithm flow are the same as the detection of two horizontal parallel surfaces  
Figure 8: Geometric model of _PA_ and _PB_ cross section direction: (**a**) Positional relation exists between _PA_ and _PB_ surfaces. (**b**)Positional relation does not exists between _PA_ and _PB_ surfaces.  
Figure 7: Detection model of two horizontal parallel surfaces.  
Figure 10 shows the geometric model of _PC_ and _PD_ cross section direction. When _PD_ is the measured surface with _PC_ as the datum surface. If there is no positional relation between the _PC_ and _PD_ surfaces, as shown in Figure 10a. The existed geometric relationship is the surface shape contour and relative displacement of the two surfaces. Since both are known. Therefore, the calculated surface shape contour is the same as the known surface shape contour. If there is an unknown positional relation between the _PC_ and _PD_ surfaces, the relative distance between the shape contours of the two surfaces will change, as shown in Figure 10b. At this time, the shape contour of _PD_ surface cannot be obtained only from the known shape contour of _PC_ surface and relative displacements. Therefore, the calculated shape contour of _PD_ surface is different from the known shape contour of _PD_ surface. Their shape difference is the positional relation curve.  
Due to the existing methods can only perform the parallelism evaluation, but the spatial position curve of the two parallel surfaces cannot be depicted. Therefore, it is difficult to carry out a comparative test to verify the test results in this paper.  
Figure 10: Geometric model of _PC_ and _PD_ cross section direction: (**a**) Positional relation does not exists between _PC_ and _PD_ surfaces. (**b**)Positional relation exists between _PC_ and _PD_ surfaces.  
Figure 9: Detection model of two vertical parallel surfaces.  
By definition, parallelism is different to the positional relation of this article. Parallelism requires a datum direction as a datum, which is an orientation error. The positional relation has no datum direction. In this paper, the method of mutual detection by the two surfaces is used to verify the correctness of this detection method. The significance of mutual detection is that, ideally, the positional relation curve obtained by _PA_, _PB_ mutual detection must be the same shape and opposite direction. The positional relation curve obtained by _PC_, _PD_ mutual detection must be the same shape. When different measuring mechanisms are used to measure two horizontal parallel surfaces and two vertical parallel surfaces, the same positional relation description method is used to obtain the positional relation curves. However, the datum surface and the measured surface have changed, that is, the movement trajectories of the detection lines \(A\), \(B\), and the contact line \(C\) have changed. Under this condition, if the results of the mutual detection are all consistent with the objective law, the correctness of the description method of the positional relation can also be proved.