2 
o?Y, s C [2X* - 4BXÄ ur (6B%+ AY?) X? - (4By? 4 4B')X, « (B*+ 2y?p2 
1 p? 1 1 1 a 1 x 1 
+ 2y;)] (18) 
^ 0.5v)? 
072, -» ——————— (0.25)o0? [2X* - 4BX? « (AY? « s.5B?) X? - (1.5B? 
1 2v2 i 1 j i 
B2Y 
+ 4Y7B) X. + B’Y" + 0.25B"+ 2Y*] + (0.25)0* [X? + 2Y? + 472 
1 + l 1 1 1 1 
zu 25 
PA ME gy IEEE OPE RA. (19) 
2 
(X « YD) X. - B) v7 
where © = go = Oy 
The values of Zi» (2; - 0.5V) and (2; - V) of equation (16) are taken as a 
common factor (Zi - 0.5V). This common factor produces a negligible error on 
the average values of 0Zj. 
Equations 17, 18 and 19 give the standard deviations oX., 0Yj and oZi of 
point i as functions of the standard deviations of the measured horizontal 
directions and the measured vertical directions. These equations (17, 18 
and 19) are very difficult to interprete and has no practical applications. 
Accordingly, the equations of the average standard deviations OXp, OYp and 
OZp of a plane are developed in the next section. 
IV. THE STANDARD DEVIATIONS eX eY AND Gé, OF A PLANE 
The standard deviations OXp, OYp and OZp represent the average accuracy 
of all the points on that plane. The average standard deviation OXp, OYp 
and OZp of all the points on that plane can be obtained by integrating 
equations 17, 18 and 19 for all the points on that plane. The integration of 
equations 17, 18 and 19 is possible if the position of the plane is known 
relative to the two theodolite stations. 
For simplicity, let us assume the two theodolites stations are on a line 
parallel to the plane and the two theodolites stations are on symmetrical 
positions relative to the plane as in Figure (2). Accordingly, the Yi coor- 
dinates of all the points on the plane are constant and equal to the plane 
distance D, the Xj-coordinates are range from (-L + 0.5B) (L + 0.5B) where 
2L is the length of the object and Zi-coordinates are range from (-E) to 
(H-E) where H is the height of the plane and E is the theodolite elevations 
above the base of the plane. The standard deviations oXp, 0Xp and OZp of a 
plane are obtained by integrating equations (17), (18) and (19) for the 
ranges of Xj and Zi. The integration of the above equations take these 
forms. 
g? 
aX = [ 1. lo (spy By pu (p. 
P B2p2 7 10 3 
B^ BS 
22 D. 2D. 
BD - zi L* 5 
  
2 
3 
  
  
B*p? D'5? 
* ] 
4 2 (20) 
G?Y 
9?7 
The 
the X, Y 
a plane : 
apart al 
rallel t: 
Table I. 
are calci 
the coor 
given ra 
T 
Theodo 
B (m) 
2. 
2. 
2. 
12. 
12. 
12. 
12. 
285. 
25. 
25. 
25. 
25. 
25. 
25. 
37. 
37. 
$7; 
37. 
50. 
50. 
50. 
50 
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