## 2 Materials and Methods
### Measurement Data Processing  
The data obtained from the tool position measurement while  
to the sample cross-section. In order to compare the results, it was necessary to unify their structure. The first step was to convert the coordinates stored in the measurement files from cartesian into polar coordinates. In the case of the CMM measurements, the angular interval in the polar coordinates was not constant. Therefore, the next step was to oversample the measurements to obtain the same number of measurement points in both data sets. The spline interpolation algorithm was used for this purpose. Then, in order to keep only the harmonics describing the surface waviness in the signal, a digital low-pass filter was applied to the limit frequency equal to fifty times the base harmonic (of the nominal shape). The examples of deviation as a function of angular orientation prior to and after applying the filter for the as-measured contour are presented in Figure 6.  
Finally, the results were compared using the best fit algorithm to align the measurement data to the reference considering the measurement contour center displacement by angle \(\psi\) and radius \(r\) and contour rotation by angle \(\beta\). The algorithm assumes obtaining the minimum width of the deviation from the nominal profile zone defined as the difference between the largest and smallest deviations. The examples of data prior to and after alignment are presented in Figure 7.  
Figure 6: Shape deviation in mm as a function of angular orientation of the machined part: (**a**) unfiltered; (**b**) filtered at cut-off equal to 50 upr (undulations per revolution). Figure 7: Adaptation of real data to the theoretical nominal outline: (**a**) prior to adaptation; (**b**) after adaptation. The discrepancies in the deviation measurement result from the assumed measurement technique and its accuracy. It must be noted that the tool position measurement while processing is influenced by additional errors that are caused by phenomena occurring during the processing, e.g., vibration in the machine tool, tool holder, shaft and lathe, build-up edge effect, etc. Table 3 and Figure 11 present the statistical measures for the geometric deviations for both measurement methods \(e_{\text{CMM}}\) (from off-line coordinate measurement machine) and \(e_{\text{NCDT}}\) (from on-line laser transducer): arithmetic mean \(m\), standard deviation \(s\), minimum \(min\), first quartile \(Q1\), median \(Me/Q2\), third quartile \(Q3\), and the maximum value. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Sample \(f\) [Hz]**} & \multicolumn{8}{c}{_e_CMM-eNCDT [\(\mu\)m]**} \\ \cline{2-9}  & \multicolumn{2}{c}{**Mean**} & \multicolumn{2}{c}{**Standard**} & \multirow{2}{*}{_Min_} & \multicolumn{2}{c}{**Quartile**} & \multirow{2}{*}{_Median_} & \multirow{2}{*}{_Quartile_} & \multirow{2}{*}{_Max_} \\  & & \(m\) & & & & & & & \\ \hline S1.1 & 1.42 & 9.91 & 133.48 & \(-10.00\) & \(-4.86\) & 9.99 & 22.74 & 31.39 \\ S1.2 & 2.81 & 4  
For sample no. S1.4, the tool position recorded by the optical laser NCDT sensor was influenced by a gross error due to the contamination of the sensor face with dirt. These results were excluded from further considerations in this work. The maximum mean shape deviation recorded during the tests amounted to 34.29 \(\upmu\)m, its standard deviation amounted to 35.10 \(\upmu\)m. The maximum and minimum deviations were 82.86 \(\upmu\)m and \(-60.61\)\(\upmu\)m, respectively. It can be noticed that variation in differences between both measurement techniques grow with FTS operating frequency when it exceeds 9 Hz, which is the limit operational frequency of the piezoelectric drive. The obtained profile deviation from an ideal shape and the standard deviation from the mean were designated for each measurement series. The data are visualized in the form of two groups. The first, tagged with green triangle markers, includes test results for which the FTS operating frequency did not exceed the limit frequency \(f_{\text{g}}\) equal to 9 Hz.