Geography Reference
In-Depth Information
Table 21.11
Algorithm for computing coordinates of the universal Mercator projection as a function of global
coordinates (GPS, GLONASS) and extended datum parameters
Step one.
Collect global coordinates of type
by means of GPS, GLONASS,
or other satellite positioning system.
Step two.
{
Λ,Φ,H
}
Collect the elements of a curvilinear datum transformation,
namely three translation parameters
{
t
x
,t
y
,t
z
}
, three rotation parameters
{
α, β, γ
}
,
δa,δe
2
,
=2
eδe
one scale parameter
s
, and two ellipsoidal form parameters
{
}
.
Step three.
Compute
x
(
Λ
)bymeansof(
21.99
) as Easting (“Rechswert”).
Step four.
Compute
y
(
Φ
)bymeansof(
21.100
) as Northing (“Hochwert”).
2
a
1
,a
2
to a local ellipsoid-of-revolution
E
is called a
universal Mercator proje
ction
if (
2
1.94
)holds,
where
a
1
denotes the semi-major axis,
a
2
the semi-minor axis, and
e
=
1
a
2
/a
1
the relative
−
2
a
1
,a
2
eccentricity of
E
.
x
=
a
1
λ,
y
=
a
1
ln
tan
π
e/
2
.
1
4
+
ϕ
e
sin
ϕ
1+
e
sin
ϕ
−
(21.94)
2
End of Definition.
In order to transform Mercator coordinates which are given in a global datum with respect to
the ellipsoid-of-revolution
into Mercator coordinates which are given in a local datum with
respect to the ellipsoid-of-revolution
E
E
A
1
,A
2
a
1
,a
2
, we take advantage of the Taylor expansion of second
order, namely
a
1
=
A
1
+
δa, e
=
E
+
δe,
(21.95)
λ
=
Λ
+
δΛ, ϕ
=
Φ
+
δΦ
so that
x
(
λ, a
1
)=
x
(
Λ
+
δΛ,Φ
+
δΦ,A
1
+
δA
)=
A
1
Λ
+
Λδa
+
A
1
δΛ
+
δAδΛ
+O
3
x
,
(21.96)
x
:=
x
0
+
x
1
+
x
2
+
x
3
+O
3
x
and
y
(
ϕ, a
1
,e
)=
=
y
(
Φ
+
δΦ,A
1
+
δa,E
+
δe
)=
=
A
1
ln
tan
π
E/
2
+
1
4
+
Φ
E
sin
Φ
1+
E
sin
Φ
−
2
+ln
tan
π
E/
2
δa
+
A
1
1
1
−
E
sin
Φ
1+
E
sin
Φ
2
ln
1
−
E
sin
Φ
4
+
Φ
2
1+
E
sin
Φ
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