The efforts for measuring the mirror’s 3-D shape and calculating the mirror’s manufacturing errors could be traced back to the time when the telescope was invented. In 1858, Jean Foucault invented a method to measure the telescope mirrors with the knife-edge test. Unfortunately, it could only measure spherical mirrors while most telescope mirrors are aspheric instead of spherical. In addition, the Foucault method could only provide qualitative results instead of quantitative results. To yield quantitative results, it must be combined with other methods and requires great effort and considerable skills to make accurate judgments. In 1922, Vasco Ronchi invented a different technique to measure telescope mirrors and it could not provide quantitative results either. Based on the previous methods, several new methods were proposed later, e.g., the star test, the Ross null test, and the autocollimation test. However, none of them are satisfactory. In the late 1960s, the laser unequal path interferometer was invented to test spherical concave surfaces.11 In the early 1970s, Karl Bath invented an interferometer to test telescope mirrors with quantitative results and it was recognized as the most informative method of that time. The Ceravolo interferometer is an alternative method with similar performances. Unfortunately, these three interferometers are only suited for testing spherical surfaces. Later on, an interferometer made by ZYGO used the peak-to-valley (PV) and root mean squares (RMS) to evaluate the quality of the mirror. However, the complexity of the optics makes PV/RMS incapable of adequately describing the mirror quality. Hence, power spectral density, slope RMS, inverse Hartmann test, and structure function (SF) are adopted widely in mirror quality evaluations.12–18 In Ref. 16, an inverse Hartmann test was proposed for surface form measurement in the spherical coordinates with increased dynamic range and resolution. However, its accuracy was decreased compared to that in the rectangular coordinates. In Ref. 17, a tutorial about the SF analysis is presented and its advantages over Fourier-based methods were proved. In Refs. 19 and 20, researchers at the University of Arizona used the laser tracker to obtain the direct shape measurement for the GMT mirror and they achieved a measurement accuracy of . In Ref. 21, a large deformable aspherical mirror is measured with sub- accuracy by the software configurable optical test system. Its measurement principle is the same as that of deflectometry and it is based on the integration of the surface slope. In Ref. 22, ray tracing was used to measure the optical aberrations of aspherical lenses. All the above methods except for Ref. 22 could only indirectly give quantitative results of the surface errors for the mirror. The paraxial radius, geometry dimensions, and eccentric errors of the mirror, are outside of their capabilities.