lated coor- 
mated from 
Columns 7, 8 
estimated 
lues obtain- 
FOR A 
ine object 
id the theo- 
> values of 
je values of 
d the 
he optimum 
ch minimize 
ue at 
not have Bo 
ordingly, 
y be taken 
ity between 
ghly 
er than 
he accuracy 
ctive of 
d the opti- 
d OYp for 
ze the 
and Y-axis 
form 
(23) 
.) one gets 
B^ 
8 
(24) 
Equating F = 0, one gets one equation with two unknowns B and D. Accordingly, 
there are infinite solutions for the values of B, D, oXp and oYp which satis- 
fy the condition oXp = oYp. The optimum values for B and D are the values of 
Bo and Do which satisfy the condition oXp = oYp at the minimum values of oXp 
or OYp. Unfortunately, there is no algebraic solution for estimating the 
values of Bo and Do from equation (24). The values of Bo and Do are estimat- 
ed numerically where different values of B are assumed and the corresponding 
values of D (from equation 24) and OYp (from equation 21) are calculated. 
The values of B and D which give the minimum value of cYp are called the 
optimum base distance Bo and the optimum object distance Do. The estimated 
value of Bo is 1.4L and the estimated value of Do is 0.26L. If the observer 
takes the base distance to be 1.40L and the object distance to be 0.26L he 
will have the maximum accuracy of OXp and OoYp which satisfy the condition 
oXp = OYp. If the field condition do not permit B to be 1.40L or/and D to 
be 0.26L, the observer has to set-up the distances B and D on the field to 
satisfy equation 24. 
V-2 The Optimum Theodolite Elevation (Eo) That Mazimize the Accuracy Along 
Z-axis 
The standard deviation oZp (equation 22) is a function of B,D and E 
while the standard deviations oXp and oYp are functions of B and D only. 
Accordingly, the two parameters Bo and Do were chosen to maximumze the accu- 
racy of OXp and oYp for oXp = oYp. The theodolite elevation E is chosen to 
maximize the accuracy of 0Zp. The optimum theodolite elevation Eo, that 
maximize the accuracy of OZp, can be calculated from this equation. 
- 0 
QQ Qa 
t] t3 
The above equation takes this form 
23i 4D? Bi, VETE -H) (H-E)"- E* 
[s M rete s D ] [od + (2E-H) + [R7] 
[tzn ^! Eo Seen EUR = 9 (25) 
Solving equation 25, one gets the optimum theodolite elevation Eo. The esti- 
mated value of Eg is 0.5H. Accordingly, it is recommended to keep theodolite 
elevation as close as possible to the middle of the plane object. 
VI. THE OPTIMUM THEODOLITE POSITION FOR 3-DIMENSIONAL OBJECT 
The standard deviation oXp, oYp and oZp which are given in equations 20, 
21, and 22 give the average standard deviation of the object points if they 
were well distributed on a plane at a distance D from the theodolite stations. 
The standard deviation of three dimensional object (o0Xt, oYt and oZt) may be 
obtained mathematically by integrating the standard deviation equations 20, 
21 and 22 for the parameter D over the limits (Do) to (Do + W), where W is 
the depth of the object and Do is the distance between the front edge of the 
object and the two theodolite stations. The values of oXt, oYt and oZt give 
the average accuracy of point-coordinates which are well distributed inside 
a parallelpiped shape of length 2L, height H and depth W. The values of oXt, 
OYt and oZt have no practical applications even if the object has a parallel- 
es piped shape because the photogrammetric and the theodolite measurements 
can be taken only for points on one or maximum two surfaces of the 
9