Bas, Hüseyin Gazi 
  
as targets, stabilitiy of the theodolite stations, lighting, environmental conditions, the redundancy 
of the measurements, the skill of the operator, and the computing method also affect the accuracy 
of the object space coordinates. 
In this study, a mathematical formulation is developed for estimating the acuracy of object space 
coordinates which can be achieved using a theodolite as a measuring tool and the parallax 
equations of the normal case of photogrammetry as the mathematical model. 
2 GEOMETRY 
The basic geometry of measuring with a theodolite is the same as that of an intersection. On the 
other hand such a theodolite can be assumed as a ficititious camera positioned on a station to 
take a photograph. The relationships between such a fictitious camera and a theodolite are as 
follows: 
e The perspective center of the fictitious camera coincides with the points of intersection 
of the theodolite horizontal and vertical axes. 
e The optical axis of that camera is parallel to the plane of the theodolite horizontal circle 
and perpendicular to the base line between the two theodolite stations. 
e The principal distance, f, of fictitious camera can be assumed equal to any value 
(Abdel-Aziz, 1979). 
The respective x axes of theodolite images, which are parallel to the base line, are parallel, and the 
respective y axes of the theodolite images, which are parallel to the theodolite vertical axes at the 
theodolite stations, are parallel in this system. The angle between the optical axis f and the ray to 
object point i is the horizontal angle ai. The angle between the horizontal plane and the ray to 
object point i is the vertical angle of inclination Bj . The zero direction of the theodolite 
horizontal circle is any direction when the theodolite is set up at the station. The relationship 
between theodolite image coordinates xij yj and the horizontal direction ai; and vertical direction 
pj is shown in Figure 1. These relationship can be expressed mathematically according to Figure 
1, in general terms, in the following way: 
x; = f.tanaj 
(1a) 
y; = f.tanB; / cosa 
(1b) 
where où = (100 — do; + aj), ao and a are the horizontal direction of other theodolite station 
and horizontal direction to image i, and the subscripts i and j represent point number and 
theodolite station number, respectively. As shown in equations (1), the acuracy of theodolite image 
coordinates x; and y; is a function of horizontal and vertical angles measured with a theodolite. 
Applying the principles of error propagation to the above equations and assuming an errorless 
focal length, one gets estimated standart deviations, mj ,myij as a function of observed angles. In 
this error propagation xiand yi are assumed uncorrelated. 
mxj 7 f(1/cos'aj) Mai 
(2a) 
my = f f1/ cos’ Pay J(1/ cosa; ) mm. * tanBg (sin aj / cos'og ) maj ] 
(2b) 
  
46 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 
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