## Theories and Methods
### Analysis of Planeness Measurement Uncertainty Based on GUM  
For the sake of judging the flatness results objectively and reasonably, the most widely used uncertainty evaluation method at present [28; 29], namely the GUM (guide to the expression of uncertainty in measurement) is adopted to analyze the measurement uncertainty. Firstly, the measurement method and measurement device parameters need to be defined. Then, a mathematical model would be established and the uncertainty components will be introduced separately, calculated, and synthesized together. And, finally, the uncertainty results could be acquired [30]. According to the planeness calculated using the equation described above, the transfer coefficients of each parameter are listed as follows:  
\[\left\{\begin{array}{c}\frac{\partial f}{\partial x_{M}}=\frac{a}{\sqrt{a^{ 2}+b^{2}+1}}\\ \frac{\partial f}{\partial x_{L}}=\frac{a}{\sqrt{a^{2}+b^{2}+1}}\end{array}\right. \tag{8}\]  
\[\left\{\begin{array}{c}\frac{\partial f}{\partial y_{M}}=\frac{-b}{\sqrt{a^{ 2}+b^{2}+1}}\\ \frac{\partial f}{\partial y_{L}}=\frac{b}{\sqrt{a^{2}+b^{2}+1}}\end{array}\right. \tag{9}\]  
\[\left\{\begin{array}{c}\frac{\partial f}{\partial z_{M}}=\frac{1}{\sqrt{a^{ 2}+b^{2}+1}}\\ \frac{\partial f}{\partial z_{L}}=\frac{-1}{\sqrt{a^{2}+b^{2}+1}}\end{array}\right. \tag{10}\]  
\[\left\{\begin{array}{c}\frac{\partial f}{\partial a}=\frac{x_{L}-x_{M}}{ \sqrt{a^{2}+b^{2}+1}}-\frac{a[x_{M}-x_{L}-a(x_{M}-x_{L})-b(y_{M}-y_{L})]}{(a^{2 }+b^{2}+1)^{\frac{3}{2}}}\\ \frac{\partial f}{\partial b}=\frac{y_{L}-y_{M}}{\sqrt{a^{2}+b^{2}+1}}-\frac{ b[x_{M}-x_{L}-a(x_{M}-x_{L})-b(y_{M}-y_{L})]}{(a^{2}+b^{2}+1)^{\frac{3}{2}}}\end{array}\right. \tag{11}\]  
Analyze the uncertainty of each parameter. Assume that the uncertainty of a single measuring point is \(u_{0}\), and the measured plane is parallel to the \(xoy\) plane. Therefore, the uncertainty of \(z\) is the main uncertainty factor of a single measuring point, and the uncertainty is as follows:  
\[\mu(Z_{M})=\mu(Z_{L})=\mu_{0} \tag{12}\]The uncertainty of coefficients \(a\) and \(b\) are calculated as follows:  
\[S=\begin{bmatrix}n&\sum x_{i}&\sum x_{i}\\ \sum x_{i}&\sum x_{i}^{2}&\sum x_{i}y_{i}\\ \sum y_{i}&\sum x_{i}y_{i}&\sum y_{i}^{2}\end{bmatrix} \tag{13}\]  
To simplify the equation representation, make the setups \(p=\sum x_{i}\), \(q=\sum y_{i}\), \(\mu=\sum x_{i}^{2}\), \(v=\sum y_{i}^{2}\), \(w=\sum x_{i}y_{i}\). Then, \(S\) is expressed as follows: \(S=n\mu v+2 pqw-q^{2}\mu-p^{2}v-nw^{2}\). The expressions of \(a\) and \(b\) are given, respectively, by  
\[a=\frac{(wq-pv)\sum z_{i}}{S}+\frac{(nv-q^{2})\sum x_{i}z_{i}-(pq-nw)\sum y_{i} z_{i}}{S} \tag{14}\]  
\[b=\frac{(pv-qu)\sum z_{i}}{S}+\frac{(pq-nw)\sum x_{i}z_{i}-(nu-p^{2})\sum y_{i }z_{i}}{S} \tag{15}\]  
Then, the measurement uncertainties of \(a\) and \(b\) are, respectively, followed by  
\[u(a)=\sqrt{u_{0}\sum_{i=1}^{n}(\frac{\partial a}{\partial z_{i}})^{2}}=\frac {(wq-pv)+(nu-q^{2})p+(pq-nw)q}{S}\cdot u_{0} \tag{16}\]  
\[u(b)=\sqrt{u_{0}\sum_{i=1}^{n}(\frac{\partial b}{\partial z_{i}})^{2}}=\frac {(pw-qu)+(pq-nw)p+(nu-p^{2})q}{S}\cdot u_{0} \tag{17}\]  
Finally, the synthetic measurement uncertainty of the flatness could be described by  
\[u_{f}=\sqrt{[\frac{\partial f}{\partial Z_{M}}u(Z_{M})]^{2}+[\frac{\partial f }{\partial Z_{L}}u(Z_{L})]^{2}+[\frac{\partial f}{\partial a}u(a)]^{2}+[ \frac{\partial f}{\partial b}u(b)]^{2}} \tag{18}\]