v 
b 
vw 
\ 
X 
AntX Y, 2p) s 1451) 
À y x 
ax I. 
> S 
— ei 
Fig. (1) 
assume D, and without loss of generality, 
the equation of M may take the form 
AX 1 BY : CZ 4 * = 5 (5) 
Substituting (3) into (5), the value of 
ti - t associated with the image point a 
is found to be 
AX .+=BY.+ CZu+ 1 
3 3 i 
ti = (8) 
ACX -Xi) + B(Y 14 + CCZ_—Z,) 
  
which emn now be substituted inde (339 to 
get the.space coordinates of the 1mage 
2.4%, .% Za 
à 3° à? 1 
y= T+ (Y-% 
i 1 s 3 x or 
3 3 ST A A s ü 
Yit Y; LT, Yt. 
where i - 1,2,3, ....,Hn 
Now, the distance 1, connecting two 
image points ai and a may have the form 
2: 2 vgs Ser up yt (8) 
whils ly can, as mentioned before, be 
calculated or messured. 
Equation (8) can take the form 
D AT = = 2 
$9 3 £u 2C x = m4 3 .—YX 3 
Fy (X.Y QA A.B.C) (X Xi) + (Y Yi) 
23 
x Ly xy. Fun = 7 X 
(1 Zi) 1, 0 CS 
with k s 1,2,3....:m. 
Taking into considerstion all point pairs, 
we got m - nn 13 ‘equations 
similar to (9), which form a system of 
nonlinear equations in the six unknowns 
(X, Y oZ V ABICO). 
S 
Determining the Unknowns 
  
For the case when n-5 (min. value), the 
system (9) contains 10 nonlinear equations 
in the six unknowns and can be solved 
using the least squares technique to 
compensate for any probable measuring 
errors. 
each equation 
using Taylor's 
To simplify the procedure, 
in (9) can be linearized 
theorem as follows :- 
  
  
  
  
  
  
IF aF, 
pF. dE x (59 Y + 
of kie ax Ban ar ‘=n 
3F F OF 
(—3-) dz, + (—E-) dA + (—2-) dB + 
aZ. = SA eB 
ôF, 
( i ) dC * terms of higher order (10) 
ar 
IF, 
where Ge n E Qe neis are value of 
Fo snd its partial derivatives at suitable 
chosen initial values of the unknowns 
X Y- LE .A.,B ;C .The.derivatives can 
so o o © 
so’ so 
be found by differentiating (9) partially 
with respect to the unknowns. For example: 
  
  
  
  
  
  
  
  
A 
OF on «x. ax. S 
dieu CEE 31 SR is > + C7 +1? 
ax 3 ox ax, 3 
2 5 5 
rure de s 
( ut à + 2(2,-8,3C a > (115 
ex ox » ox ox 
s 5 = 5 
The new derivatives in (11) can be found 
from (7), such as 
>» 
ax. 0t 
z (X -Xi.) — 35 t, (32 
ax. T SX E 
> pe 
ti is previously found in (65, then 
8t. -ACAX.+BY.+CZ.+ 1) 
$ s i i i . 
= £313 
Xe [ACX, -X.,o «BOY -Y,o 300-7 xz 
wr y Ss ris u = d Les "i 
Similar formulas can be found for other 
derivatives. 
The system of equation (10) can now be 
approximated by considering only the 
linear term. The new system will have the 
398