Geography Reference
In-Depth Information
[
a
21
t
x
+
a
22
t
y
+
a
23
t
z
+
a
24
α
+
a
25
β
+
a
27
s
]+
+
ln tan
π
1
E/
2
4
+
Φ
E
sin
Φ
1+
E
sin
Φ
−
2
+
A
1
1
a
28
δa
+
E
2
cos
Φ
−
1
(21.100)
E
2
sin
2
Φ
1
−
+
A
1
1
2
ln
1
E
sin
Φ
1+
E
sin
Φ
−
E
sin
Φ
−
E
2
sin
2
Φ
1
−
+2
EA
1
1
a
29
δe
+
E
2
cos
Φ
−
1
E
2
sin
2
Φ
1
−
+O
2
y
.
21-52 Numerical Results
In order to test the algorithm for computing coordinates of the universal Mercator projection as
a function of global coordinates (GPS, GLONASS) and extended datum parameters, we present
some numerical examples. Special emphasis is on the estimation of the order of magnitude of the
nonlinear terms in (
21.96
)and(
21.97
). Let us begin with a set of extended datum parameters as
given in Table
21.12
. These represent the transformation of global curvilinear coordinates given in
Table
21.14
into local curvilinear coordinates as given in Table
21.13
.Bymeansof(
21.94
), we have
computed Easting and Northing for the five points given in Table
21.15
. In contrast, by means of
Table
21.1
7, we have computed the terms
of
(
21.96
)and(
21.97
), which sum up to Easting and Northing in Table
21.17
. Obviously, the bilinear
term
x
3
:=
δaδΛ
accounts for approximately 2cm, while the terms
y
5
(quadratic in
δe
2
)0
.
2cm,
y
6
(quadratic in
δΦ
2
) 0.4cm,
y
7
(bilinear in
δaδe
) 0.8cm, and finally
y
8
(bilinear in
δΦδe
)5cm.
As a computational test, we have compared the difference between Easting and Northing in local
coordinates and global coordinates (columns 4 and 5 of Table
21.17
), namely in the submillimeter
range. If we neglect the quadratic-bilinear terms of (
21.96
)and(
21.97
), respectively, we document
errors of the order of 40cm according to Tables
21.18
and
21.19
.
{
x
0
,x
1
,x
2
,x
3
}
as well as
{
y
0
,y
1
,y
2
,y
3
,y
4
,y
5
,y
6
,y
7
,y
8
}
Table 21.12
Datum parameters
t
x
=
−
584
.
911 m,
t
y
=
−
66
.
121 m,
t
z
=
−
402
.
257 m
α
= 0”.0038,
β
=0
.
0013,
γ
=
−
2”.3960
10
−
6
ppm
s
=
−
10
.
11
×
a
1
=6
,
377
,
397
.
155 m,
A
1
=6
,
378
,
137
.
000 m
e
2
=0
.
0066743700,
E
2
=0
.
00669438
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