## 3 The Method for Describing the Positional Relation of Two Parallel Surfaces  
### The Description Method of Positional Relation between Two Horizontal Parallel Surfaces  
On the basis of the measurement mechanism in Section 2.1, when \(A\), \(B\), and \(C\) are in an absolute smooth surface, the spring is in the initial compression state, this moment, the data recorded by laser displacement sensor is \(X_{0}\), and \(X_{0}\) is defined as the initial value.  
Figure 7 shows the measurement model of two horizontal parallel surfaces. With \(PA\) as the datum surface, \(PB\) as the measured surface. \(L_{\mathrm{s}}\) is defined as the starting position, \(L_{\mathrm{e}}\) is defined as the terminal position. By coinciding contact line \(A\) with \(L_{\mathrm{s}}\), the measurement mechanism is driven to move in the direction indicated in Figure 7, and the measurement mechanism is clinging to the auxiliary surface to ensure that there will be no sliding during the measurement process. When contact line \(A\) reaches to \(L_{\mathrm{e}}\), the measurement is completed, and the data obtained in this process represent effective measurements.  
Figure 7(a) shows the geometric model of \(PA\) and \(PB\) cross section direction. The shape contour of \(PB\) is determined by the shape contour of \(PA\), the relative height difference of \(PA\) surface shape contour relative to \(PB\) surface shape contour and the positional relation of the two surfaces. According to the mathematical relationship between the effective measurements and \(X_{0}\), the extraction algorithm for combined projection waveform is developed and the combined projection waveform can be obtained. Next, since the measurements of the laser displacement sensor change with time, a time-space conversion algorithm is developed. This algorithm is used to map the time series waveform to the coordinate position of \(PB\) surface, which is called the spatial sequence waveform. Then the trend curve of this waveform is found by the method of neural network fitting (a fitting method in the toolbox of MATLAB), which is named as the combined projection curve. This curve is the height difference of \(PB\) surface shape contour relative to \(PA\) surface shape contour. Due to the shape contour of each surface have been measured before, so they are known quantities. According to the combined projection curve of the \(PB\) and the surface shape contour of \(PA\), the separation algorithm for the combined projection curve is studied. The shape contour of \(PB\) surface can be obtained by this algorithm, which is called the calculated surface shape contour. When the two surfaces \(PA\) and \(PB\) are coplanar, as shown in  
Figure 6: The positional relation between contact line \(A\) and detection line \(C\): (**a**) The distance between the contact line \(A\) and the detection line \(C\) is unchanged (**b**) Contact line \(A\) and detection line \(C\) are located in two surfaces with shape contours.  
Figure 8b, the positional relation between the two surfaces is zero, and the calculated surface shape contour is the same as the known shape contour of _PB_ surface. If the two results are different, it proves there exists positional relation between two surfaces. Therefore, a description algorithm for positional relation curve between two surfaces is studied. According to the difference between the calculated surface shape contour and the known shape contour of the measured surface, the positional relation between two surfaces can be determined.