In this section, the results of the data analysis described above are presented for the three freeform specimens investigated. All measurements shown here were recorded at room temperatures with variations of approximately 1.5 K between the different sites. These temperature differences mostly affected the radius components of the measurements, which were eliminated for the bilateral comparisons. Since the material of the specimens has good thermal properties, even the effect on the radius is small. For each specimen, we first present the processed measurements after subtracting the design form and an individual best-fit sphere which had been interpolated on a common grid. These measurements are shown in Fig. 3 for the two-radii specimen, in Fig. 7 for the convex toroidal surface and in Fig. 11 for the concave toroidal surface.
The x−z− profiles (at y=0) and the y−z− profiles (at x=0) after subtracting the design form and an individual best-fit sphere are shown in Fig. 4 for the two-radii specimen, in Fig. 8 for the convex toroidal surface and in Fig. 12 for the concave toroidal surface.
The processed measurements after subtracting the design form and an individual best-fit sphere formed the input data for the bilateral differences shown in Fig. 5 for the two-radii specimen, in Fig. 9 for the convex toroidal surface and in Fig. 13 for the concave toroidal surface.
Two-radii specimen
For the two-radii specimen, the data were evaluated for a common aperture with a radius of 16.5 mm. For this measurement aperture, five data sets were available: 1: USTUTT TWI (Nraw= 777229), 2: PTB TWI (Nraw= 470629), 3: Mahr TWI (Nraw= 109225), 4: Mahr MFU (Nraw= 28410) and 5: IPP MFU (Nraw= 29829). The curvatures of the subtracted best-fit spheres were 0.0056/m (USTUTT TWI), 0.0029/m (PTB TWI), 0.0035/m (Mahr TWI), 0.0027/m (Mahr MFU) and 0.0007/m (IPP MFU).
The processed measurements after subtracting the design form and an individual best-fit sphere had peak-to-valley values below 1 μm (see Fig. 3), while for the bulk of the data sets, the processed measurements had peak-to-valley values below 500 nm. The results of the MFU at IPP have some large deviations at the cosine transition zones between the different spherical segments of the specimen located at the diagonals between the x- and y- axes (cf. design deviation from a best-fit sphere shown in Fig. 2, lower row, left). Some minor deviations in this area can also be seen in the results of the MFU at Mahr. The processed measurements of the three different TWI setups seem to be in good agreement. This is an excellent result when one considers that these three setups of the same measurement principle are very different. The good agreement between the three TWIs is also reflected in their profiles (see Fig. 4). Of course, this does not guarantee that these measurements are correct, since all TWI measurements might suffer from the same systematic errors.
The pointwise differences between the processed measurements are shown in Fig. 5 for the two-radii specimen and reinforce the similarities between the three TWI measurements. The largest deviations occur in the transition zones of the spherical segments of the specimen for the MFU measurement at IPP. Furthermore, several small high-frequency characteristic deviations between the MFU and the TWI measurements are visible. However, because these deviations are also visible in the bilateral comparison between the two MFU measurements, this seems to be more of a systematic influence of the MFU measurements than a feature of the surface which was not measured by the TWI measurements.
To evaluate the bilateral deviations in a quantitative manner, the RMS values of the pointwise difference plots are illustrated in Fig. 6. Each element (row i, column j) represents the RMS value of the difference plot between partners i and j (1: USTUTT TWI, 2: PTB TWI, 3: Mahr TWI, 4: Mahr MFU, 5: IPP MFU). While the RMS values between the three TWI measurements are in the range below 20 nm, the RMS values between the Mahr MFU measurement and all three TWI measurements are below 50 nm. The RMS values between the IPP MFU measurement and all other measurements are around 80 nm.
Convex toroid
For the specimen which has a convex toroidal surface form, the data were evaluated for a common aperture with a radius of 21.8 mm. For this measurement aperture, five data sets were available: 1: USTUTT TWI (Nraw= 783260), 2: PTB TWI (Nraw= 427119), 3: Mahr TWI (Nraw= 124296), 4: Mahr MFU (Nraw= 40641) and 5: IPP MFU (Nraw= 30062). Since the centres of the four Gaussian peak markers were located at (x=0 mm, y=±22 mm) and (x=±22 mm, y=0 mm), the markers were not completely visible in the measurement data sets. Nevertheless, since several partners measured a larger area of the specimen, and since the Gaussian markers had a full width at half maximum of 0.5 mm, at least some parts of the markers were visible for all partners, allowing the markers (or parts of them) to be used to align the measurement data sets.
The curvatures of the subtracted best-fit spheres were 0.0005/m (USTUTT TWI), 0.0041/m (PTB TWI), 0.0038/m (Mahr TWI), 0.0013/m (Mahr MFU) and 0.0097/m (IPP MFU).
The processed measurements, after subtracting the design form and an individual best-fit sphere, have peak-to-valley values below 1 μm (see Fig. 7), while the bulk of the data sets have peak-to-valley values below 400 nm. The processed measurements show characteristic features between the different measurement instruments. The results from the MFU at IPP contain some larger deviations. A similar structure can be seen in the results from the MFU at Mahr, but to a much smaller extent. For this specimen as well, the processed measurements of the three different TWI setups seem to be in good agreement. For all processed measurements, apart from the measurement results of the MFU at IPP, there was a larger peak in the centre of the specimen. This was also the case for the two-radii specimen and may be an artifact from the manufacturing process in which SPDT was used.
The profiles of the processed measurements (see Fig. 8) show very good agreement between the results from the TWI at PTB and those from the TWI at Mahr. The measurement of the TWI at USTUTT is also close to these results. While the measurement results of the MFU at Mahr mainly differ from the TWI profiles in the centre, the MFU results from IPP also contain larger deviations on the outer part of the specimen. Note that the two MFU measurements do not contain the same deviations from the three TWI measurements. Instead, they show deviations with opposite signs, especially in the centre of the specimen.
The pointwise differences between the processed measurements are shown in Fig. 9 for the convex toroidal surface; once again these bilateral comparisons emphasize the similarities between the three TWI measurements. Furthermore, the pointwise differences between the MFU measurement at Mahr and the three TWI measurements are much smaller than the pointwise differences between the MFU measurement at IPP and all other measurements. To evaluate this in a quantitative manner, the RMS values of the pointwise difference plots are illustrated in Fig. 10. Each element (row i, column j) represents the RMS value of the difference plot between partners i and j (1: USTUTT TWI, 2: PTB TWI, 3: Mahr TWI, 4: Mahr MFU, 5: IPP MFU). While the RMS values between the three TWI measurements are around 30 nm, the RMS values between the Mahr MFU measurement and all three TWI measurements are around 50 nm. The RMS values between the IPP MFU measurement and all other measurements are around 110 nm.
Concave toroid
For the concave toroidal specimen, the data were evaluated for a common aperture with a radius of 34.9 mm. For this measurement aperture, four data sets were available: 1: Mahr TWI (Nraw= 64117), 2: Mahr MFU (Nraw= 137438), 3: IPP MFU (Nraw= 567709) and 4: NMIJ UA3P (Nraw= 313634).
The curvatures of the subtracted best-fit spheres were 0.0020/m (Mahr TWI), 0.0023/m (Mahr MFU), 0.0016/m (IPP MFU) and 0.0020/m (NMIJ UA3P).
The processed measurements after subtracting the design form and an individual best-fit sphere have peak-to-valley values around 500 nm (see Fig. 11) and show characteristic features between the different measurement instruments. The results of the MFU measurements at IPP and Mahr show some structures which were also slightly visible in the measurements of the other two specimens. These structures may be calibration artifacts of the MFU which occur mainly in regions where the gradient in the azimuthal direction is high (cf. design deviation to a best-fit sphere shown in Fig. 2, lower row). For the measurement of the UA3P at NMIJ, several data points had to be masked out, since a large deviation from the design was visible in this area which was not present in the other/previous measurements. This deviation from the design may be due to damage to the specimen during transport. Therefore, we decided to mask out these data points to prevent them from contributing to the following analysis. The four processed measurements show good agreement in the basic form measured. This is also reflected in the profiles shown in Fig. 12, which also show high-frequency ripples detected by all measurement devices. These ripples may be caused by vibrations of the tool during manufacturing. In the centre of the specimen, a likely explanation for why the MFUs are closer to the other instruments is that the form of this concave toroidal surface is much flatter (see radii of curvature in Table 1) compared to the other two specimens.
The pointwise differences between the processed measurements are shown in Fig. 13 for the concave toroidal surface. All pointwise differences are in the same order of magnitude. To evaluate this in a quantitative manner, the RMS values of the pointwise difference plots are illustrated in Fig. 14. Each element (row i, column j) represents the RMS value of the difference plot between partners i and j (1: Mahr TWI, 2: Mahr MFU, 3: IPP MFU and 4: NMIJ UA3P). The RMS values between all bilateral difference measurements range from 43 nm to 53 nm. The smallest RMS value has the pointwise difference between the measurement of the TWI at Mahr and the MFU at IPP.