optimum elevation Eo, which define the theodolite positions. Moreover, it 
gives the formulas that express the expected accuracy of the ground coordi- 
nates at any theodolite positions. 
II. FIELD MEASUREMENTS AND CALCULATIONS 
A one second theodolite is needed for measuring the directions and a 
tape or a subtance bar is needed for linear measurements. The field measure- 
ments and the calculations procedures can be summerized as follows. 
Field Measurements 
  
Two sets of measurements are required for calculating the targets coor- 
dinates. These are 
1. The angular measurements which are 
a) The horizontal direction of each target from the theodolite station 
# 1 (a;) and the theodolite station # 2 (az). The horizontal direc- 
tion ay and op are measured from the base direction. 
b) The vertical direction of each target from the theodolite station # 1 
(v1) and the theodolite station # 2 (y,). Station # : 
2. The linear measurements which are 
a) The horizontal distance between the two theodolite stations (Base 
distance) B. 
b) The difference of elevations between the two theodolites (V) at the 
two stations. 
The values of B and V can be measured directly or indirectly as given in 
Abdel-Aziz (1). 
The System of Coordinates 
  
The calculated target coordinates are referred to a three-dimensional 
rectangular coordinate system with an origin at the point of intersection of cu 
the theodolite axes at the left station. The X-axis is a horizontal line 
parallel to the base direction, the Y-axis is a horizontal line perpendicular 
to the base direction and the Z-axis is a vertical line determined by the 
theodolite vertical axes at the left station. These axes are shown in 
Figure 1. 
The Calculation Procedures 
  
  
  
The Xi, Yj and Zi coordinates of a target i are calculated from the H 
theodolite measurements (41, Y1, à? and Y2) and the distance measurements (B 
and V). The relations between Xi, Yi and Zi and the measured values Q1, Ce, 
Yi», Ya, B and V can be obtained from Figure 1. 
Referring to the geometric relationship in Figure 1, one gets 
X = Y: cot Oo; (1) 
;=B.=.Y, 2 
X. -B Y; cot a, (2) 
From equations (1) and (2), one gets 
Fig 
X - B cot oq (3) 
i COt O1 - cot à;