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c
11
=
ρ
2
(
x
10
+(4
y
20
+6
x
10
x
30
)
l
2
+2
x
10
x
11
b
+
x
11
b
2
+O
l
(3))
,
(15.95)
c
22
=
ρ
2
(
y
01
+(
x
11
+2
y
01
y
21
)
l
2
+4
y
01
y
02
b
+(4
y
02
+6
y
01
y
03
)
b
2
+O
b
(3))
.
End of Corollary.
The proof of Corollary
15.5
4 is lengthy, namely for
c
12
=
c
21
= 0. Instead, we refer to the solution
of the general eigenvalue problem in Corollary
15.6
.
2
A
1
,A
1
,A
2
, principal distortions, Universal Transverse Mercator Projection
(UTM) modulo an unknown dilatation parameter).
Corollary 15.6 (
E
Under the mapping equations (
15.92
), which constitute the Universal Transverse Mercator Pro-
jection (UTM) modulo an unknown dilatation parameter
ρ
,the
principal distortion
or
factor of
conformality
, after a lengthy computation, amounts to
Λ
2
:=
Λ
1
=
Λ
2
=
c
11
G
11
=
c
22
(15.96)
G
22
or
Λ
2
=
ρ
2
1+cos
2
B
1+
E
2
cos
2
B
l
2
+O
Λ
2
(
l
4
)
.
E
2
(15.97)
1
−
End of Corollary.
In summarizing, we get the squared factor of conformality proportional to the order of squared
l
2
. In the following few passages, we determine the unknown dilation factor either by the postu-
late of
minimal total distance distortion
(Airy optimality) or by the postulate of
minimal total
areal distortion
. Results are collected in two corollaries, two examples (UTM and Gauss-Krueger
conformal coordinate systems) and five graphical illustrations.
Corollary 15.7 (Dilatation factor for an optimal transversal Mercator projection, minimal total
distance distortion, Airy optimum).
(i)
For a conformal map of the half-symmetric strip [
[
B
S
,B
N
] of type Universal Transverse
Mercator Projection (UTM), the unknown dilatation factor
ρ
is optimally designed under the
postulate of minimal total distance distortion if (
15.98
) accurate to the order O(
E
4
)holds.
−
l
E
,
+
l
E
]
×
sin
B
N
+
E
2
sin
B
N
E
5
sin
5
B
N
1
3
sin
3
B
N
−
−
1
6
l
E
ρ
=1
−
3
E
2
sin
3
B
S
+
(15.98)
sin
B
N
+
3
E
2
sin
3
B
N
−
2
sin
B
S
−
.
E
2
sin
B
S
+
3
sin
3
B
S
+
E
5
sin
5
B
S
sin
B
N
+
3
E
2
sin
3
B
N
−
+
−
sin
B
S
−
3
E
2
sin
3
B
S
+
···
2
sin
B
S
−
(ii)
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