A fundamental discovery in segmented-mirror active optics is described: symmetrizing the geometry of thesensor–actuator array provides a computationally effective symmetrization of the mathematical description for the figure control of a Keck-type telescope mirror.

The author establishes a universal mathematical control model and provides an efficient algorithm to solve this model equation set. This model can be applied to multifarious Keck-type mirror configurations with a similar sensor–actuator geometry design, no matter what kind of outline shapes and how many segments they have. With the underlying symmetry, a further extension of this algorithm is possible without increasing the number of parameters to be estimated for the recently proposed extremely large telescopes, such as the 30-m California Extremely Large Telescope (CELT) and the 100-m OverWhelmingly Large (OWL) telescope.

Moreover, careful choice of boundary conditions in conjunction with the proper choice of minimization algorithm yields results that exceed the performance of the current existing techniques given by Nelson and Mast [Appl. Opt. 21, 2631 (1982)]. This method allows noise performance analysis. Several computer simulation models for application of this algorithm are given for the Keck 10-m Telescope and the Large Sky Area Multi-Object Fiber Spectroscopic Telescope’s (LAMOST’s) MA (Reflecting Schmidt plate) and MB (spherical primary mirror).