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e
2
sin
2
b
0
)
5
/
2
(1
5
e
2
sin
2
b
0
+4
e
2
sin
4
b
0
)
b
21
=
−
(1
−
−
,
2
a
1
(1
−
e
2
)
2
cos
2
b
0
e
2
sin
2
b
0
)
5
/
2
(1
2sin
2
b
0
−
5
e
2
sin
2
b
0
+6
e
2
sin
4
b
0
)
b
03
=
−
e
2
(1
−
−
.
2
a
1
(1
− e
2
)
3
As soon as we implement (
21.79
)into(
21.70
), we gain the final transformation of Cartesian con-
formal coordinates
}
in a global frame of reference. Indeed, via the coecients of the bivariate polynomials (
21.82
)of
Box
21.25
listed in Boxes
21.26
and
21.27
, the final transformation depends on the parameters
of a curvilinear datum transformation, namely three parameters
{
x, y
}
in a local frame of reference into Cartesian conformal coordinates
{
X,Y
of translation, three
parameters
{α,β,γ}
of rotation, one scale parameter
s
and the change of the form parameter
δE
2
=
E
2
{
t
x
,t
y
,t
z
}
e
2
. The bivariate polynomial representation (
21.82
) is given up to order three due to
the limited space in printing the lengthy coe
cients in Boxes
21.26
and
21.27
. Indeed, the final
transformation
X
(
x, y, ρ, t
x
,t
y
,t
z
,α,β,γ,s,δE
2
),
Y
(
x, y, p, t
x
,t
y
,t
z
,α,β,γ,s,δE
2
) highlights the
key result of a datum transformation from local conformal coordinates of type Gauss-Krueger or
UTM to global conformal coordinates of type Gauss-Krueger or UTM. We therefore summarize
the results as follows.
−
Box 21.25 (Polynomial representation of global conformal coordinates
{X,Y }
in terms of
local conformal coordinates
{x, y}
due to a curvilinear datum transformation, Gauss-Krueger
conformal mapping or UTM, polynomial degree three, Easting
X,x
, Northing
Y,y
).
X
(
x, y, ρ, t
x
,t
y
,t
z
,α,β,γ,s,δE
2
)=
=
ρ
x
00
+
x
10
x
ρ
+
x
01
y
ρ
− y
00
+
x
20
x
2
y
ρ
− y
00
+
x
11
x
ρ
ρ
+
x
02
y
y
00
2
ρ
−
+
y
00
3
+
x
30
x
ρ
3
+
x
21
x
ρ
2
y
ρ
−
y
00
+
x
12
x
ρ
y
ρ
−
y
00
2
+
x
03
y
ρ
−
+O(4)
,
(21.82)
Y
(
x, y, ρ, t
x
,t
y
,t
z
,α,β,γ,s,δE
2
)=
=
ρ
y
00
+
y
10
x
ρ
+
y
01
y
y
00
+
y
20
x
ρ
2
y
ρ
−
y
00
+
y
11
x
ρ
ρ
−
+
y
02
y
y
00
2
ρ
−
+
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