## 5 Error Analyses of the New Method
### The Effect of the Zero-Adjustment Error  
For multi-probe methods using three or more probes, the zero-adjustment error always has a significant impact on the measurement and reconstruction results [18]. Due to the cumulative effect, using a high precision reference surface to calibrate the zero points of probes is not good enough for eliminating this type of error. Therefore, the influence of the zero-adjustment error in the new method proposed in this paper is analyzed.  
Figure 8: Reconstruction profiles of the existing method: (**a**) the profile \(f_{1}\); and (**b**) the profile \(g_{1}\).  
Figure 9: Reconstruction profiles of the new method: (**a**) the profile \(f_{2}\); and (**b**) the profile \(g_{2}\).  
Assume that the zero-adjustment errors of probes are \(ez_{j},j=1,2,3,4\); the outputs of probes in Equations (1)-(4) become:  
\[m_{1}(x_{i})=f(x_{i})-e_{y}(x_{i})+ez_{1}, \tag{15}\]  
\[m_{2}(x_{i})=f(x_{i}+d_{1})-e_{y}(x_{i})+ez_{2}, \tag{16}\]  
\[m_{3}(x_{i})=-[g(x_{i})-e_{y}(x_{i})]+ez_{3}, \tag{17}\]  
\[m_{4}(x_{i})=-[g(x_{i}+d_{2})-e_{y}(x_{i})]+ez_{4}. \tag{18}\]  
Correspondingly, the data processing process in Equations (5) and (6) becomes:  
\[\begin{array}{l}mf(x_{i}+d_{2})=\frac{m_{2}(x_{i})+m_{4}(x_{i})-[m_{1}(x_{i} +d_{2})+m_{3}(x_{i}+d_{2})]}{\begin{array}{l}=\frac{1}{s}[f(x_{i}+d_{1})-f(x _{i}+d_{2})]+\frac{1}{s}(ez_{2}+ez_{4}-ez_{1}-ez_{3})\\ =f_{s}^{\prime}(x_{i}+d_{2})+\alpha\end{array}} \tag{19}\]  
\[\begin{array}{l}mg(x_{i}+d_{2})=\frac{m_{2}(x_{i})+m_{4}(x_{i})-[m_{1}(x_{i} +d_{1})+m_{3}(x_{i}+d_{1})]}{s}\\ =\frac{1}{s}[g(x_{i}+d_{1})-g(x_{i}+d_{2})]+\frac{1}{s}(ez_{2}+ez_{4}-ez_{1}- ez_{3})\\ =g_{s}^{\prime}(x_{i}+d_{2})+\alpha\end{array}} \tag{20}\]  
The differential values in Equations (19) and (20) are also integrated by Equations (7) and (8). It can be seen that there is one more component \(\alpha\) due to the zero-adjustment error in \(mf\) and \(mg\) in Equations (19) and (20), which cause a rotation of the reconstruction profiles compared with the actual profiles and the tangent value of the rotation angle is \(\alpha\).  
When we evaluate the geometric parameters of a pair of parallel profiles, it is more essential and difficult to evaluate the straightness and the parallelism. When the straightness is evaluated using the reconstruction profiles, the rotation of the reconstruction profiles compared with their original profiles will not affect the evaluation result. When evaluating the parallelism of a pair of parallel profiles, if the rotation angles of the two reconstruction profiles are equal, the rotation has no effects on the evaluation result. It is known that the reconstruction result of this new method belongs to the situation that two profiles rotate by the same angle, and the reconstruction errors shown in Figure 6b are also consistent with the above analyses. Therefore, it can be concluded that the zero-adjustment error will not affect the evaluation results of the straightness and parallelism. As a result, in most cases, it is not necessary to calibrate the zero points of probes to obtain prior knowledge before measurements, which makes the new method proposed in this paper simply to operate.