There are two solutions for O : Let the direction ratios of the x-sxis 
a) 9-289 0% 6. « 190" relative to the spsce s 
= 6, > 
T 
sud b> 8 = = 6, 
The coordinates of P(x ,y,)can now be 
x) 
zm fb: gg) (25) 
which sre unknown till now. This vector 
is perpendiculsr to £z and. subtends an 
angle = S with the vector 7 slong 8,85 
found 
+ 8) 
x, + 15cos Co, 
: given by 
v = vit l5sin fa, + e 
X= (5% 00 (7-7, v (2-0. (28) 
To reveal the ambiguity about the values » = s 
of €, the above procedure is repeated 
using a third point äg with s4. One of The two equations 
the new solution of (x oy should > = = Ix| . [vl cos a, (27) 
coincide with one of the first ones. This 
is chosen to be the correct answer. It is + 
to be noted that if all image points are end 2°> x = 0 (28) 
taken into consideration, the least 
squares technique can be applied to find 
the most probable position of the can be solved simultaneously for a : b : o. 
principal point p. They yield & quadratic equation, which can 
be easily solved. 
Determination of The  Angular Orientation 
  
Parameters o, à and x 
  
There are three rotations necessary to 
transform the space coordinate system O, 
X, Y, 2 into a position parallel to the 
image plane system o, X, y, z, fig.(3)}. 
At first the space system is rotated 
&bout the Y-sxis through an angle # to the 
position X, > Y= Xx. Zi - The X, axis is 
parallel to d, the line of intersection of 
H with the XZ-plane. Next, the new system 
is rotated about the X,-axis through an 
angle c to the position X,= Xi, as Za 
a x: o£ 
such that 2, is parallel to Z. Finally, fis. (3) 
  
the second system is rotated through an 
angie x about the 2, axis to the position 
  
  
  
  
  
X x. Ya // v; 13 : La // z. Hence Finally, the rotation angles can be 
calculated as follows 
, + — 
= À ca,x) sus dz o. uo C (28) 
i Kar {21} idi. 1X1 Y A? 4 p? 
w={ (Z,.2) 
pe > 
2 rudy Asc 
The values of these angles depend on the SOS wm zi 0 rar” (30) 
direction of the vectors along the Iz] . Zi | y A*4C? / A?^*B^4c* 
different lines. epa v 
The direction ratios of Z is the same as and 
the normal to H, hence m oe 
s d x = aC - cA 
+ gos à = — = 2 y s (31) 
ZC ATE C9 (223 | d| ix 1 en 1 Be 
Those of d are perpendicular to z and Y : 
> = 23 EVALUATION OF THE ACCURACY 
d=ix = (C0 v8 5 CEO) : 
F, is also perpendicular to both Ÿ and d, The developed method has been verified 
th ® using both = mathematical model and = 
en PS : x : 
s a 1 ; practical approach. A mathematical 
Z,*dxY3€«4-:0:0»5 (24) simulation of a photograph of an object 
400