Geography Reference
In-Depth Information
For the symmetric strip [
−
l
E
,
+
l
E
]
×
[
−
B
N
,B
N
]
,
we specialize
6
l
E
(1 +
E
2
)sin
B
N
5
E
2
sin
5
B
N
sin
B
N
+
3
E
2
sin
2
B
N
1
3
sin
3
B
N
1
−
−
1
ρ
=1
−
·
(15.99)
(iii)
If
B
N
− B
S
=
π/
2uptoO(
E
4
)
,ρ
amounts to
1+
8
15
E
2
l
E
.
1
9
ρ
(
π/
2) = 1
−
(15.100)
End of Corollary.
3
.
5
◦
,
+3
.
5
◦
]
[80
◦
S
,
84
◦
N]).
Example 15.3 ([
−
l
E
,
+
l
E
]
×
[
B
S
,B
N
]=[
−
×
The classical UTM conformal coordinate system is chosen for a strip of 6
◦
width with 1
◦
overlays
and between
B
S
=
−
80
◦
of southern latitude and
B
N
=+84
◦
of northern latitude. Once we refer
to the Geodetic Reference System 1980 (
Moritz 1984
),
E
2
=0
.
00669438002290, in particular,
with
l
E
given by
l
E
=3
.
5
◦
=0
.
0610865rad, the dilatation parameter amounts to
ρ
=0
.
999578 (scale reduction factor 1 : 2370)
.
(15.101)
End of Example.
Example 15.4 ([
−l
E
,
+
l
E
]
×
[
B
S
,B
N
]=[
−
2
◦
,
+2
◦
]
×
[80
◦
S
,
80
◦
N]).
The classical Gauss-Krueger conformal coordinate system is chosen for a strip of 3
◦
width with
0
.
5
◦
overlays and between
B
S
=
80
◦
of southern latitude and
B
N
=+80
◦
of northern latitude
(Fig.
15.5
). Once we refer to the Geodetic Reference System 1980,
E
2
=0
.
00669438002290, in
particular, with
l
E
given by
l
E
=2
◦
=0
.
0349065rad, the dilatation parameter amounts to
−
ρ
=0
.
999864 (scale reduction factor 1 : 7 353)
.
(15.102)
End of Example.
For the proof, we start from the formula
Λ
2
(
l,b
) as a representation of formula (
15.97
), namely
the
principal distortion
as a function of the longitudinal difference
L
L
0
=:
l
and the latitude
B
.The
criterion of optimality
for the first design of the transverse Mercator projection modulo
an unknown dilatation factor
ρ
is the
minimal total distance distortion
over a meridian strip
[
l
W
,l
E
]
−
[
B
S
,B
N
] between a longitudinal extension
L
W
and
L
E
and a latitudinal extension
B
S
and
B
N
(namely the symbols S, N, E, and W as indices denote South, North, East, and West), in
particular, the
Airy
(
1861
)
distortion measure
(
15.103
) with respect to the principal distortions
Λ
1
and
Λ
2
and the spheroidal surface element, locally (
15.104
) and globally (
15.105
). The Airy
distortion minimization subject to
Λ
1
=
Λ
2
=
Λ
, the criterion for conformality, leads directly to
the representations (
15.98
)-(
15.100
).
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