# Configuration space-based stitching model

The stitching model in Eq. (30) implies that the overlapping correspondence must be known. When testing a flat, we can easily determine the overlapping point pairs according to the nominal positions of subapertures. However, when testing curved surfaces, different subapertures have different positions and orientations. It is hard to determine the overlapping correspondence simply from the subaperture measurements on imaging pixels.

On the other hand, the stitching model in Eq. (30) is based on a linear approximation of the six-dofs rigid body transformation plus defocus. Such an approximation may fail to stitch with high accuracy in the case of large misalignments or testing aspheres with null optics or near-null optics. For example, this model uses slopes to correct the surface error change due to lateral shift. It works well for stitching flat subapertures. However, it may be ineffective for the stitching of aspheric subapertures measured with null optics, because the radial shift of the test mirror or the null optics results in a change from a null test to a nonnull test condition. The deviation of the shifted nominal surface from the aspheric reference wavefront generated through the null optics is now included in the subaperture measurement. It cannot be simply derived from the slope of the subaperture surface error. We can well understand the subaperture aberrations, such as astigmatism and coma induced by lateral shift of the test mirror with regard to the null optics as shown in Fig. 18. It is not easy to imagine that such induced aberrations cannot be simply related to the nominally aligned subaperture surface error (basically zero heights due to null test) by the slope.

Aiming to develop a unified stitching model and algorithm for general surfaces including aspheres tested in null or near-null conditions, we proposed the configuration of a space-based stitching model. The subaperture configurations in 3D space as well as the defocus coefficients are related to the overlapping deviations and then are best estimated. Moreover, the overlapping correspondence is naturally determined because the Cartesian coordinates of the measuring points are obtained with rigid body transformation.

**Figure 18 **Surface error slope can relate flat subapertures but not aspheric subapertues to the misalignment due to the change of the null test condition.

**Figure 19 **Ray tracing the object-image mapping.