As a consequence of the cross-disperser dispersion relation, a given line repeated in two overlapped regions of the spectrum must be found at the same y position if spectrum and detector are perfectly aligned.

In the following, we will call a pair two occurences of the same line in the overlapped parts of two adjacent orders.

The above condition provides a geometrical way to estimate the angle, as:
,
with
the difference in *y*
position of the two occurences of the pairs and
the difference
of their *x* positions. This method requires overlaps between orders
and is used in the calibration method `PAIR`.

Another approach consists of estimating the rotation angle as the parameter for which regression analysis provides the smallest residual for a given set of observations.

- such as the standard-error is minimized

The determination of the optimum is performed analytically and is
involved in the calibration method `ANGLE`. This one-dimensional
representation of the dispersion relation is used as a starting point
in the loop and is replaced by a bivariate solution
as soon as a sufficient number of identifications has been performed.