1 
i 
4 
  
  
  
  
The process to determine definite values for the tilt elements is 
similar to the scale-tilt-height procedure. 
o 
(b) HEIGHT-DETERMINATION 
After evaluation of the planimetric triangulation, the height trian- 
gulation is finally carried out. This triangulation is based on the 
last equation of (1i) 
2; 72 = À 
A. = V 
X 
is determined by the foregoing planimetric triangulation. 
i Wij (8) 
in which 
  
p» 
2 2 
$a eL y 
Rn 
i 
11 
pte 
Solution of the planimetric normal equations. 
The straightforward approach from equation (2) gives very large 
systems of equations containing two groups of unknowns: coordinates 
of the points to be determined and 4 n planimetric elements of 
orientation. Because of the special structure of the coefficient 
matrix, one group of unknowns can easily be eliminated; for example, 
the coordinates. 
Let 
Mx = 1 (9) 
be the partially reduced system of m normal equations in m unknowns 
(m = 4n), in which M is a coefficient matrix, symmetrical and non- 
singular, x and 1 being vectors composed of the values of the plani- 
metric parameters and of the constant terms, respectively. 
If matrix M is symmetric, the matrix can be resolved into the pro- 
duct of two triangular matrices of which one is the transpose of the 
other. Thus if S is an upper triangular matrix and ST its trans- 
pose 
M = STS (10) 
The solution of the system is reduced then to the solution of two 
triangular system 
cl 2 1 1 
SKK = 1 and Sx = k (11) 
because the two systems (11) are equivalent to system (9). 
In view of the rule of matrix multiplication, the elements of matrix 
t 
i 
S are simple functions of the elements of matrix M.