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Null subaperture layout designThe subaperture layout design for null subaperture stitching interferometry is quite straightforward because we do not need to consider the dynamic range of measurement. When testing a larger flat or planar wavefront, we simply put the subapertures one by one with given centertocenter distances in two lateral directions. As shown in Fig. 5, the overlapping ratio is completely determined by the centertocenter distance /. A simple geometric calculation gives the ratio ro as follows:
where d is the diameter of the subaperture. For an instance, the overlapping ratio is about 39.1% for / = 0.5 d and 28.5% for / = 0.6 d. When testing spherical surfaces with a Fizeau interferometer, the TS is first selected according to the radius of curvature of the test surface. The halfangle 0 of the test beam cone is then calculated from the f/number of the TS. After selection of the TS, subapertures are roughly arranged according to the test geometry shown in Fig. 1. The optical axis of the surface is located at the geometric center of the full aperture. Without loss of generality, we consider the offaxis subaperture centered on the x axis. The offaxis distance x_{0} of the subaperture is given by the offaxis angle p:
Exact calculation of the overlapping ratio in this case is subtle and complicated, typically in the form of a surface integral.^{42} However, that is not necessary because the small uncertainty of the ratio has little effect on the measurement uncertainty. A simple approximation is applying Eq. (1) for the rough arrangement of subapertures. The overlapping ratio can further be checked numerically by applying the convex hull algorithm on the OXY plane, with the determined Figure 6 Coordinate frames for an offaxis subaperture. offaxis distance x_{0} or offaxis angle p. The problem of determining the overlapping point pairs as required in the subaperture stitching algorithm will be discussed in Section 5.3. 
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