## 3 Numerical Simulations and Experiments
### Experimental Results  
An investigation of off-axis PMD is carried out as well, and Figure 8 shows the experimental setup, which includes a CCD camera with an external diaphragm, a flat surface to be measured, and a screen to display sinusoidal fringe patterns with a period of 21 pixels. The optical axis of the camera lens is approximately \(13.1^{\circ}\), \(15.4^{\circ}\), \(16.4^{\circ}\), \(17.8^{\circ}\), and \(19.1^{\circ}\) away from the normal of the surface being measured. The size of the test flat glass element in diameter is 200 mm, and the measured area of it is a square with side lengths of roughly 142 mm. Based on the setup in Figure 8, the distorted fringe pattern captured by the CCD camera at different angles, as shown in Figure 9, and the calculated area, which is boxed by the dotted red lines, includes \(355\times 355\) pixels. Figure 6: Height distribution of a simulated free-form surface. Figure 7: RMS of residual error of a free-form surface using Southwell and RBF with different angles. Decreasing the sample points can reduce computation time and save computer memory because the flat primarily contains a large amount of low-frequency information, as shown in Figure 9. As a consequence, the data points are sampled to create a 121 \(\times\) 121-pixel matrix. The camera records the phase-shifted sinusoidal fringe patterns in two directions. Their deformation depends on the surface figure, the posture of the camera, and the surface shape of the element inducing. Next, a sixteen-step phase shifting method is used to determine the correspondence between the camera pixel and the screen pixel, Starting with a surface assumption, \((x_{m},y_{m},z_{m})\) are determined with ray tracing as described in Section 2.3. Then, the slope calculated from Equation (1) is integrated. The integrated surface is put back to update the sampling point by ray tracing to iteratively calculate the slope. Finally, the improved Southwell method and RBF interpolation algorithm are applied to reconstruct the surface with the slopes in the x and y directions. Then, the tilt of the acquired surface shape is removed using the Chebyshev polynomial fitting algorithm. The experimental result is given in Figure 10 for the maximum angle of 19.1\({}^{\circ}\), and experimental results for other angles are given in the Appendix from Figures A1-A4. Meanwhile, the RMS values of the residual error of the two methods are plotted as Figure 11.  
Figure 8: Experimental setup. (**a**) Experimental components and general placement; (**b**) The sinusoidal fringe patterns to be displayed on a screen. Figure 9: Distorted fringe pattern captured by the camera at different angles in the x and y direction, respectively. Taking Figure 10 as an example, the first three terms of the Chebyshev polynomial are removed from the result of the interferometer, as shown in Figure 10a. Figure 10b,c are the reconstructed surface shape without the first three terms using the Southwell and RBF, respectively. Figure 10d is the surface shape difference between the interferometer result and the reconstruction result of Southwell, and Figure 10e is the surface shape difference between the interferometer result and the reconstruction result of RBF. For the five different angles, by comparing with the result of the interferometer, it shows that all of the RMS obtained by the RBF interpolation algorithm are smaller than those of the improved Southwell algorithm. Noticeably, the RMS of the improved Southwell algorithm reconstruction residual error is 131.6 nm, while the RMS of RBF is 97.16 nm when the angle is the maximum of 19.1\({}^{\circ}\) in our experiment. In addition, we conducted repeatability experiments for a flat element, and the results of our repeatability experiments are depicted in Figure 12; the numbers 1, 2, and 3 in the label in the figure's upper right corner indicate that there are three experiments at five angles. The results verify that the reconstruction results of the RBF interpolation algorithm have higher accuracy.