a mq (X--Xc) + 1, (0) + m3 (0) (5) 
Pilon 5 pog (X =X.) + m,» (0) + m4 (0) 5 
i o ma] Gt 7X9) + may (0) + m4 (0) 
Since the image of I must be formed at the vanishing point, n the image coordinates must 
x 
be those of ny. Designating the coordinates of ny by xy and yy» We obtain from equations 
(5) and (6): 
Ye =F =F -— (8) 
eo lr! 
In a similar manner, the image coordinates of vanishing points n, and n, can be 
Y Z 
obtained.  Designating the image coordinates of ny by Xy and yy» we have: 
m 
12 
= x mf--— (9) 
d o ma, 
m2 
y, =y -f (10) 
Y 
o m3» 
and designating the image coordinates of n, by x, and ÿ7> We have: 
m 
X, 7 X - f Hi (11) 
33 
m 
2 
= - f (12) 
33 
e & The six equations (7) through (12) relate the six image coordinates of the three 
vanishing points (n; Dy and n,) to the three elements of the interior orientation (xs 
Yo? f) and the nine elements of the orientation matrix [M]. Since the nine elements of 
[M] can be expressed as functions of three independent angles, such as the familiar o, 6 
and k, or azimuth, tilt and swing, equations (7) through (12) actually relate 12 independent 
quantities —- the six image coordinates of the vanishing points, the three parameters of 
the interior orientation, and the three parameters of the orientation matrix. Given any 
six of these quantities, the six equations (7) through (12) can be used to solve for the 
remaining six quantities. In practice, the solution of these six equations can become 
quite complicated, and may be circumvented, as will be shown very shortly. Nevertheless, 
these six equations do serve the purpose of pointing out the feasibility of solving for 
the elements of the interior orientation and exterior orientation matrix using only the 
vanishing points of the object space coordinate axes..