7 
A i 
a* 
d 
b c 
b’ c * 
e 
the orthogonal matrix expressing the 
orientation of the photographic system 
of coordinates into the selected space 
coordinate system 
Equation 3.01 contains nine parameters, six positional and orientational 
elements of the exposure station and three positional elements of the 
object point. As it stands, equation 3.01 is non-linear with respect to 
these parameters, and its direct application in the least squares solution 
is a rather difficult operation. Therefore, equation 3.01 is first 
linearized by applying Taylor's expansion, neglecting terms of second and 
higher order. 
For the application of Taylor's expansion, it is assumed that 
approximate values for the unknown elements of the nine parameters are 
available. Using these approximations, equation 3.01 can be converted 
to linear form in terms of corrections to the parameters rather than the 
parameters themselves. The linearized form is written in matrix notation 
as : 
V ij + B 
ID 
- C 
1 D 
= -F 
ID 
.... 3.02 
where : 
C v x v y] i;5 
b 11 ... b 16 
b 21 ... b 26 
ij 
residuals of the measured plate 
coordinates 
a coefficient matrix whose elements 
are the partial derivatives of 
equation 3.01 with respect to the 
exposure station parameters 
D 
t 
i 
[dw d$ ... dZ q j. 
the vector of corrections to the 
exposure station parameters 
coefficient matrix whose elements are 
the partial derivatives with respect 
to the object coordinates 
Ü- 
[dX dY dZ]j 
corrections to the object coordinates