## 3 Results and Discussions
The next step is to calculate the distance of all points from the plane described by the equation, that is, the location of each point, constituting the partial flatness deviation of the surface under examination \(d_{i}\). We use Formula (3):  
\[d_{i}=\frac{z_{i}-ax_{i}-by_{i}-c}{\sqrt{a^{2}+b^{2}+1}} \tag{3}\]  
From the obtained partial flatness deviations, we determine the maximum and minimum value, and the sum between their absolute values defines the flatness error (\(f\)) of the actual element in accordance with Formula (4):  
\[f=|d_{max}|+|d_{min}| \tag{4}\]  
The value \(f\) (Table 2) is thus a flatness parameter defined by the distance of the extreme points (minimum and maximum) from the reference surface. The analysis of the accuracy and time required for measuring flatness deviations depends on several factors. The key factors include the number of measurement points, the repeatability of results, and the characteristics of individual measurement methods, such as the coordinate measuring machine (CMM), a manual machine, and a measuring arm. The choice of the appropriate method and the determination of the optimal number of measurement points have a significant impact on the reliability of results and the efficiency of the measurement process. The following section discusses how these elements influence the differences between the methods. * CMM (coordinate measuring machine): The most accurate method (flatness error f = 61.9 \(\upmu\)m) with the highest measurement repeatability. However, a disadvantage is the long measurement time as the number of points increases (e.g., 60 points require 23 min). This method allows for process automation and only requires basic operator training. * Manual Machine: Good accuracy (flatness error f = 63.6 \(\upmu\)m), but weaker measurement repeatability. The measurement time is shorter compared to the CMM machine but \begin{table}
\begin{tabular}{c c c} \hline \hline
**Measuring Device** & **Testing Time** & **Obtained Flatness Error** \\  & **t, min** & **f, \(\upmu\)m** \\ \hline Automatic Measuring Machine & 32.0 & 61.9 \\ Manual Measuring Machine & 19.5 & 63.6 \\ Measuring Arm & 19.0 & 79.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of obtained results [own elaboration]. requires a skilled operator. At 100 points, it achieves a similar accuracy to the CMM within the range of 60-65 \(\upmu\)m.
* Measuring Arm: The least accurate method (flatness error f = 79.7 \(\upmu\)m) with greater variability in results for a small number of measurement points. At 130 points, it achieves an acceptable repeatability range (77.5-82.5 \(\upmu\)m). It is a fast method in practical use but demands physical effort and operator training. In summary, the difference in results between the most precise method (CMM) and the least precise method (measuring arm) is 17.8 \(\upmu\)m. The value of 17.8 \(\upmu\)m is the difference between the \"Obtained Flatness Error f, mm\" for the measuring arm (79.7 \(\upmu\)m) and the automatic measuring machine (61.9 \(\upmu\)m) from Table 2. The results show that the greater the number of measurement points, the closer the results become. For devices with lower accuracy, systematic collection of a greater number of measurement points positively impacts the results, reduces the dispersion of the obtained values, and brings the results obtained using different methods closer together. Consequently, a properly planned collection of measurement points allows the use of a mobile device to achieve approximate results. However, achieving more accurate results requires the use of stationary machines. Because in the case of measuring shape deviations of elements, the commonly used strategy of even distribution and the quantity of collected points do not always yield the desired results, an analysis of the obtained results for random and systematic sampling was performed [21]. In two devices (manual machine and automatic machine), the same measuring tip was used, while in the measuring arm, a ball of the same diameter was used.