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Wavefront sensing (WFS) is one of the key elements for active alignment of the new Wide-Field Corrector (WFC), as it
tracks sidereal motion, with respect to the fixed Hobby-Eberly Telescope (HET) primary mirror. During a track, part of
the 10m-pupil of the WFC can lie outside the primary periphery and be clipped off. An additional field-dependent
central obscuration by the holes and baffles of the WFC leads to complex pupil geometries. The combination of these is
a complicated dynamically varying non-circular telescope pupil. This unique problem to the WFS on the HET needs to
be dealt with by choosing an appropriate set of orthonormal aberration polynomials during wavefront reconstruction. In
this paper, three ways of computing orthonormal aberration polynomials and their coefficients are discussed. These are
based on the Gram-Schmidt (GS) process, but differ in the way of computing key integrals during the GS process. The
first method analytically computes the integrals, where a computer algebra program is used. The second uses the
Gaussian quadrature over triangulated pupil geometries that approximate the true pupil shape. The last uses indirect
numerical estimates of the integrals, which turned out to be natural by-products of the usual least-square Zernike
polynomials fit. It is shown that the first method is limited to cases of simple pupil shapes, while the second can be
applied to more general pupil shapes. However, when dealing with complicated dynamically varying non-circular
pupils, the last method can be vastly more efficient than the second and enables the possibility of estimating
orthonormal aberration coefficient on the fly. Also noticed is that the last method naturally takes into account the
pixelation effect of pupil geometries due to pixel-based imaging sensors (e.g. CCDs). With these benefits, the last
method can be used as a viable tool in real-time wavefront analysis over dynamically changing pupils as in the Hobby-
Eberly Telescope, which is otherwise vastly inefficient with analytic methods used in past studies.
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