where
T = (X—X 0 ) cos 9?o + (Z—Z 0 ) sin <p 0
N = — (X—Xq) sin (p 0 + (Z—Z 0 ) cos <po
7.1.9
7.1.10
When Z is constant for all points the following linear equation holds:
Z-Z 0
sin cpo a XQ + cos cpo dc H sin cp 0 axo
Z—Z 0 sin 9? 0
— cos 990 azo H a cp = 0 7.1.11
where «a is the coefficient for parameter k in 7.1.7 and 7.1.8. To be able to solve
for Xo and c, we must know one of Xq, Z 0 and <po to exclude the singularity. The
influence on x$ and c of an error in the known parameter can be determined
from 7.1.11.
The influence of the errors in the co-ordinates of the exterior projection cen
ter upon the interior orientation increases when the distance to the test plane
decreases. This can be seen from formulas 7.1.4—7.1.6.
7.2. ASYMMETRIC DISTORTION
Errors in the exterior orientation also affect the determination of radial dis
tortion, so that we obtain asymmetric radial distortion and so called tangential
distortion.
We assume an objective with lens elements that have coinciding optical axes,
i. e. there are no decentering errors or prismatic effects of the kind mentioned
in chapter 5.3. Moreover we assume the intersection of the optical axis and the
image plane to coincide with the principal point, i. e. the optical axis is per
pendicular to the image plane. The objective has radial distortion but no other
aberrations. In such an ideal camera the radial distortion changes the position
of the image points in directions toward or away from the principal point.
As demonstrated in formulas 7.1.4—7.1.6 we get errors in the determination
of the principal point. This incorrect point will hereinafter be called the center
point. A difference between principal and center points can also be obtained
in calibrations using a collimator technique, where the camera is oriented by
autocollimation and then a test picture is photographed. A central target is then
assumed to indicate the principal point. The same relations are obtained from
the goniometric method and in several field methods where the location of the
principal point is based on assumptions that may not be perfectly true.
