  The major sources causing deterioration of optical quality in extremely large
optical telescopes are misadjustments of the mirrors, deformations of
monolithic mirrors, and misalignments of segments in segmented mirrors. For
active optics corrections, all three errors, which can partially compensate
each other, are measured simultaneously. It is therefore of interest to
understand the similarities and differences between the three corresponding
types of modes which describe these errors. The first two types are best
represented by Zernike polynomials and elastic modes respectively, both of them
being continuous and smooth functions. The segment misaligment modes, which are
derived by singular value decomposition, are by their nature not smooth and in
general discontinuous. However, for mirrors with a large number of segments,
the lowest modes become effectively both smooth and continuous. This paper
derives analytical expressions for these modes, using differential operators
and their adjoints, for the limit case of infinitesimally small segments. For
segmented mirrors with approximately 1000 segments, it is shown that these
modes agree well with the corresponding lowest singular value decomposition
modes. Furthermore, the analytical expressions reveal the nature of the segment
misalignment modes and allow for a detailed comparison with the elastic modes
of monolithic mirrors. Some mathematical features emerge as identical in the
two cases.
