I 
I 
37 
The asymmetric radial and so called tangential distortions and the ensuing 
difficulties with point of best symmetry and its determination are immediately 
overcome, when the location of the exterior projection center is determined in 
the same procedure as the interior projection center and the radial distortion. 
For infinitely distant objects the angles of the bundle of rays in the photography 
are independent of the camera position and this solves the problem (stellar cali 
bration). But for cameras intended for close-up photography, and when the cali 
bration shall be done under operational conditions a three-dimensional test- 
object is required to make this possible. 
7.3. THREE-DIMENSIONAL TEST-OBJECTS 
To establish this type of test object we can begin with a base plane and add 
points which are not in the plane. If we add just one point with coordinates X 1 
Yj Z 1 and transform them by means of formulas 5.1.3—5.1.5 they become 
(X\) 9 (YJ and (Z\). All other points have transformed coordinates (X), (Y) 
and (Z), (Z) being constant for all points in the plane. In this case the following 
linear relation between the coefficients in equations 5.5.5 and 5.5.6 holds 
(Xi) c 
(Zi) (Z) 
+ 
(XJ 
(Zi) ax ° 
+ 
(Yj) c 
(Zi) (Z) 
+ 
+ 
(Yj) 
(Zi) 
tfYO + tfZo — 
7.3.1 
To overcome this singularity we must know one of X 0 , Y 0 and Z 0 to be able to 
solve for the interior orientation. 
The singularity also disappears when we add two or more points which are 
not in the base plane. If these points are close to each other the determinant of 
the normal equation matrix is almost zero, which means a very weak solution 
with great weight coefficients for the determined parameters. Larger distances 
between the points provide a stronger solution, the strongest possible being 
obtained when the points are in the corners of the picture. We can, then, also 
conclude that wide angle cameras can be more accurately calibrated than those 
with narrower angles. This statement is based on the variation of the weight 
coefficients, which are one part of the accuracy. The other part is the standard 
error of unit weight. A function of the two parts yield the standard error of 
the parameter as in formula 6.1.24. 
The singularities in formulas 7.1.1—7.1.3 depend on the constant (Z) — va 
lues. For small differences in (Z) the solution is rather weak, but it becomes 
stronger and stronger as the differences in (Z) become greater. This means 
that the extension of the test object in the direction of photography should be