Determination of the Orientation Matrix of the Photograph 
The orientation matrix can be easily determined if the elements of the interior 
orientation are known in addition to the coordinates of the vanishing points, ny» ny and 
E 
The directions of lines cn,» cn, and Cn, (Figure 2) in the image coordinate system, 
Xyz, are given by the respective vectors: 
Normalization of these vectors provides the direction cosines of Cnys Cn, and Cn, in the 
image coordinate system. Let Le be the length of Cn, Ly be the length of Cn, and 2, be 
the length of Cn,. Then: 
  
  
  
Z 
o Vox? e (ry )2 + £2 (35) 
ty = Vioxx)? + G,-y,)? * £? (36) 
m - 2 - 2 2 
2, V, x) + ly, y + f (37) 
The normalized vectors are thus: 
x = x x T x 
ty 2, 
2 Yy = Yo , and y; Yo 
by 27 
=f =f 
by tz 
  
The elements of these normalized vectors are the direction cosines of Cn Cn, and Cn 
X? 2” 
respectively. 
In accordance with the object space coordinate system shown in Figure 2, cn, 
is always in the direction of the positive X axis and cn, is always in the direction of 
the positive Y axis. Hence, the direction cosines of the positive X and Y axes are always 
identical to the direction cosines of cn, and Cn, respectively. The direction of Cn,, 
however, is the same as that of the positive Z axis only when the camera points 
upward; when the camera points downward, the direction of Cn, is opposite to that 
of the positive Z axis. Hence, the direction cosines of the positive Z axis are identical 
to the direction cosines of Cn, only when the camera points upwards. If the camera points 
downward, the direction cosines of the positive Z axis are equal in magnitude but oppsite 
in sign with respect to the direction cosines of Cn,. 
= 12 -