Furthermore, because K ny T 0 and T 0, equations (24), (25) and (26) reduce to: 
2 = 
XXV s YyYy Of 0 (50) 
LE (51) 
Soy, v f^ (52) 
Substituting the values for Xx and XV given in equations (47) and (48) into equation (50), 
we obtain: 
2 = 
(quy * D vt 0 (53) 
€ & . Solving equations (51), (52) and (53) for Y Vy and ÿ7> We get: 
va = $ £ V-G 4, + D | (54) 
-f? 
= = — (55) 
EXON Y, 
Once Ys and y, are found, Xx and x, can be determined from equations (47) and (48). Of 
2 
positive; if the camera points downward, y; is negative. The signs of the other coordinates 
course, x, = 0. The sign of y, must be decided. If the camera points upward, y, is 
must necessarily follow. Once all the vanishing point coordinates are found, the 
orientation matrix can be determined using equations (35) through (46). 
The foregoing analysis applies only to the special case in which the image is 
located at the principal point and vanishing point n, is on the y-axis. In general, j 
Z 
and n, lie elsewhere in the photograph, and further development of the analysis becomes 
necessary. This development is facilitated by introducing a new image coordinate system 
x'y'z' (Figüre 8). The origin of this system is on the ray Cj, or Cj extended, and at a 
  
  
  
  
Figure 8