redundancy 
he accuracy 
jbject space 
he parallax 
tion. On the 
a station to 
lolite are as 
intersection 
izontal circle 
o any value 
lel, and the 
] axes at the 
1d the ray to 
id the ray to 
1e theodolite 
relationship 
ical direction 
ing to Figure 
dolite station 
number and 
odolite image 
| a theodolite. 
an errorless 
red angles. In 
s Où ) maj } 
Bas, Hüseyin Gazi 
in which m?; = (m? ij + m20j)!/2 where m zij and mo; are estimates for standart deviations in 
directions ij and oj, and mg; is an estimate for the standart deviation in the vertical angle f. 
Further, if one assumes that observations are generally carried out under the same conditions, 
with similiar instruments and by the same operator, mg; can be assumed equal to my; , that is my; 
- mgij - my .This assumpsion will be used all subsequent developments. 
theodolite vertical 
  
  
  
  
  
  
  
  
  
axis 
A | 
In 
vertical diredtion 
um = 
E | 
| L^ ped Ps 
v 
D 
* | 
Qj 
I s. : Qo 
VE E 
  
Figure 1. The relationship between the theodolite image coordinates and the horizontal and the 
vertical angles 
3 MATHEMATICAL PRINCIPLES 
In this measuring method, the left hand theodolite is assumed as fixed and the right hand 
theodolite is oriented with respect to the left one. The origin of the coordinate system is the station 
of the left theodolite. The X axis is pararllel to the direction of B, and the Y axis used in this study 
is a vertical line to the plumb bob direction at the origin. The Z axis is parallel to the optical axis 
of the fictitious camera. In this coordinate system, the X, Y, Z values of the origin are held fixed at 
zero and the rotations around axes are fixed at zero during the set up procedure. Object space 
coordinates are computed using the following equations obtained from the normal case of 
photogrammetry. 
Xi 7 xi B/( xi - xi») Yi-7yu -B/(xui- xo) Zi - f -B/(xi — Xi) 
(3) 
where B is the horizontal distance between the two theodolite stations and is assumed as an 
errorless value. 
4 THE ACCURACY OF OBJECT SPACE COORDINATES 
As shown in the above equations (3), object space coordinates are a function of the theodolite 
image coordinates xi, yi. To obtain the standart deviations of the object space coordinates mx, my, 
mz as a function of the image coordinates mys, myij, the error is propageted through equations (3): 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 47