orientation of the image coordinate system, xyz, with respect to the object space coordinate 
system, XYZ. The scalar, k,, relates the length of the vector between the camera station, 
i 
C, and the image point, i, to the length of the vector between C and the corresponding 
object point, I. 
The elements of [M] are the cosines of the nine space angles between the three 
axes of the image coordinate system, xyz, and the three axes of the object space coordinate 
system, XYZ; that is, 
m1 p 744 cos Xx cos Yx cos Zx 
[M] = mi m, m4 - cos Xy cos Yy cos Zy (2) 
ma] May 043 cos Xz cos Yz cos Zz 
By dividing each of the first and second projective equations by the third, and 
multiplying through by -f, we obtain the well-known collinearity equations: 
mj, Oq7XQ * mj (YO t my n-20 
  
  
x,-x = -f = - = (3) 
i ^o um (X, XQ) + may (X, Ye) + Mag (Z, zo) 
m et oH GU tm) pO om IC (a) 
à 9 ma] (X,-X9) + m, Ye) + Mag (2-20) 
The subsequent photogrammetric analysis will make use of the projective equations 
and of the collinearity equations derived from them. This analysis will first examine the 
significance of vanishing points and show how they can be determined from the object geometry. 
Then it will proceed into the determination of the interior orientation of the photograph, 
followed by the determination of the exterior orientation matrix. Finally, it will develop 
the relationships which can be used to extract dimensional data. 
Vanishing Points 
Since this analysis is addressed to problems which involve relative positions 
and dimensions, rather than absolute positions, any convenient object space coordinate 
systemgsuch as the one shown in Figure 2, can be chosen. This particular coordinate system 
has the positive Z-axis directed vertically upward, the positive X-axis directed toward 
the right and away from the camera station, and the positive Y-axis directed toward the 
left and away from the camera station. 
The origin, J, of the object space coordinate system is imaged at j in the photo- 
graph, and points infinitely far along the X, Y and Z axes are imaged at By» Dy and nj» 
respectively. Point nx is located by intersecting the photograph with a ray through the 
camera station, C, directed parallel to the X-axis. Similarly, ny and n, are located by 
intersecting the photograph with rays through C directed parallel to the Y-axis and Z-axis, 
respectively.