— E 7 
GV-18 PHOTOGRAMMETRIC ENGINEERING 
Zp 
P~CELESTIAL POLE 
X 
FıG. 6. Astronomic direction angles. 
D. INCLINATION OF THE CAMERA OPTICAL AXIS TO THE X AXIS 
If the optical axis of the camera is inclined to the X axis, the astronomic 
coordinates of each exposure define a secondary circle with respect to the X 
axis. A secondary circle is one which is perpendicular to a pole and parallel to 
an equator but does not pass through the center of a sphere. Great circles can- 
not define parallel planes. Secondary circles can. Thus, while the path traced [ x 
by the exposure is a line with any orientation whatsoever with respect to the 
celestial coordinate system, it is a secondary circle which defines equal angles 
to the great circle that is perpendicular to the inclined X axis. A E 
In other words, the arcs defined by the X axis produced to the celestial 
sphere and each principal point produced to the celestial sphere are equal. With 
this geometry in mind, we may write an equation for each exposure: 
COS a, COS ax + cos B., cos Bx + cos ys, cos yx = cos I, 
1 1 1 
COS a, COS ax + COS B., cos Bx + cos y4, cos yx — cos I» 
COS a, COS ax 4- cos B,, cos Qx d- cos ya, Cos yx = COS In 
where ax, Bx, yx are the direction angles of the X axis, and / is the angle 
defined by the camera z axis and the X axis. The following equality would exist 
if the X axis had no mechanical rotational error: 
ht ltr hd, 
0 
n à * 
Let it be assumed that the Least Square value is I,, which is the value corre- 
sponding to a perfectly precise horizontal, or X, axis. Division of the above equa- 
tions by cos I, gives 
cos av + cos BH + cosy.» = 1 
COS av + COS Boop + cos yao» — 1 
— 
cos œ,3v + cos B.3u + COS "y zv