Global Positioning System Reference
In-Depth Information
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{
P
1
(x
1
,y
1
) ,P
2
(x
2
,y
2
)
} ⇔ {
P
1
(ϕ
1
,
λ
1
) ,P
2
(ϕ
2
,
λ
2
)
}
↓
d
12
=
d (x
1
,y
1
,x
2
,y
2
)
t
12
=
t (x
1
,y
1
,x
2
,y
2
)
Figure 9.10
Inverse solution on the map.
is not affected by truncation errors because the conversion from the isometric latitude
to the geodetic latitude is done iteratively and is only limited by number of significant
digits carried by the computer.
[33
9.
2.4 Network Adjustment on the Map
The fact that plane trigonometry can be used makes network adjustments on the con-
formal plane especially attractive. The observed geodesic azimuth, angle, and dis-
tance
(
Lin
—
1
——
Lon
PgE
,
, s)
are further corrected by
(
s)
to obtain the respective observ-
ables on the map. During the adjustment, the current point of expansion, denoted by
the subscript 0 in the expressions below, should be used for computing the reduc-
tions. At any time during the adjustment, one may choose to deal with the geodetic
latitude and longitude or the mapping coordinates, because both sets are related by
the mapping equations. This scheme of reduction is shown in Figure 9.12. It requires
that the GML functions be used to compute the azimuth
α
δ
∆
t,
∆δ
,
∆
α
12
,
0
.
Just to be sure that there are no misunderstandings about the term
plane,
let
us review what created the situation that allows us to use plane trigonometry. The
conformal mapping model builds upon the 3D geodetic and 2D ellipsoidal models
as visualized by the transition of parameters
[33
{
ϕ,
λ
,h
}→{
ϕ,
λ}→{
x,y
}
and
δ
}→{
t,
δ
, d
α
, s
observables
. The height parameter and
the vertical observations are not present in the conformal mapping model.
Using again the (2-1-3) subscript notation for angles and standard adjustment
notation otherwise, the mapping observables are
{α
,
δ
,
β
,s,
∆
h,
∆
N
}→{
,
}
Figure 9.11
Accuracy of direct and inverse mapping for TM.
