7 
‘S 
k _ U pk^ X ok" X o,7 ~ U pk^ Y ok' Y op , s 
PO ^ 4 V r O, “ U ^ V V ^ 4 
PJ pk pk pj 
u .(Y . -Y .) - v .(X , -X .) ... 2.C5 
k = ...EJ- , ok °J P.l ok o,l' / b N 
pk u . v . - u .v . ' ' 
pk po po pk 
Substitution of the above values for k . and k , into 2.01 (f) yields 
PJ pk v ' J 
rdinates 
(X -X .) (v .w . -v . w .) + (Y . -Y .) (u . w .-u .w . ) + 
ok oj' pj pk pk po 7 ok oj' pk pj pj pk' 
... 2.04 (b) 
(Z . -Z .) (u .v . -u , v .) = 0 
ok oj' pj pk pk pj' 
) 
which is the condition for intersection of rays R . and R , from 0,, and 0, . 
J PO Pk «3 k 
Condition equation 2.04aensures that rays R. and R. forming plane A 
0 
intersect at a point (Figure 2.02). Equation 2.(4b causes rays R^ and R^ 
... 2.02 
forming plane B to intersect at a point. To force and to coincide, a 
third equation is formed by equating k from plane A with k from plane B. 
k .. = k ... 2.06 
PjA poB 
k 
Substituting 2.03 (b) and 2.05 (a) into 2.06 gives 
. from 
[u . (Y .-Y . )-v . (X .-X . )1 (u v .-u .v ) 
L pi oj oi 1 pi v oj oi0 pk pj pj pk 
...2.04(c) 
- Ju . (Y . -Y .)-v . (X . -X A (u .v . -u .v ) = 0 
L pk ok oj pk v ok oj 3 pj pi pi pk 
which is the triplet equation. 
Equations 2.04, (a), (b), (c) are non-linear with respect to the eleven 
unknown parameters and the observed plate coordinates. Since a least 
squares adjustment involving redundant points is used, direct solution of 
this system of equations would be very difficult. Consequently, linearization 
of these equations is necessary. 
.04 (a) 
Linearization is performed using approximations for unknown parameters 
and plate coordinates in a Taylor series expansion neglecting second and 
higher degree terms. Thus, the linearized equations are in terms of corrections 
to unknown parameters and unknown residuals for measured plate coordinates. 
s