22
(7) Steps (1) through (6) are repeated until the discrepancies among the
sub-blocks and between them and the control reach preassigned values
depending on the required level of accuracy.
The transformation equations used in this section allow for shift,
rotation and scale-change of each sub-block. They are linear in terms of
the unknown coefficients and, therefore, they do not impose any deformations
on the three-dimensional model representing the sub-block. Under certain
circumstances, errors in sub-block relative orientation may introduce small
deformations in the sub-block which cannot be removed by equation 4.36.
Consequently, non-linear three-dimensional transformations would be required
for such cases. In the following section an attempt is made to implement
equation 4.36 by including higher degree terms,
(3 9 )
4.4.2 Three-Dimensional Transformation of Higher Degree *
It has been the general practice in analytical photogrammetry to
separate the horizontal adjustment from the adjustment of heights. The only
simultaneous three-dimensional transformation in current use is the one
expressed by formula 4.32. This formula represents linear conformal trans
formation, which transforms small local elements without deformation.
Conformal transformations are generally preferred for this kind of applica
tion because of that property.
( 9 )
Attempts have been made independently by the author and
Schut^ 10 ^ to develop formulae for simultaneous three-dimensional conformal
transformation with terms of higher than first degree. However, it has been
found that all terms of second and higher degrees do not exist for the
three-dimensional case (although they exist for the four-dimensional case.
See reference 9). For this reason, a new set of equations for non-linear
three-dimensional transformation, that is as nearly conformal as possible,
is developed next.
The general form of the equations for the second-degree three-
dimensional transformation between two systems (u, v, w) and (U, V, W),
is given by:
