Geography Reference
In-Depth Information
Table 22.20
Equidistant mapping of the Parallel Circle
L
L
0
A
1
cos B
0
L
L
0
√
1
x
=
dL
=
N
(
B
0
)
cos B
0
dL
(22.127)
−
E
2
sin
2
B
0
x
=
ξ
(0
,l
)=
N
(
B
0
)cos
B
0
l
=
β
1
l
=
x
01
l
(22.128)
A
1
cos
B
0
β
1
:=
x
01
:=
N
(
B
0
)cos
B
0
√
1
(22.129)
−
E
2
sin
2
B
0
Table 22.21
Harmonic map, “Easting”, second boundary value problem,
l
α
6
l
6
+
O
l
7
=
β
1
l
2
α
2
l
2
−
β
3
l
3
+
α
4
l
4
+
β
5
l
5
−
x
=
ξ
(0
,l
)=
α
0
+
β
1
l
−
(22.130)
↔
α
0
=0
,α
2
=0
,α
2
=0
,β
3
=0
,α
4
=0
,β
5
=0
,α
6
=0
,etc.
“
symmetry
”
ξ
(
q,l
)=
ξ
(
−
q,l
)
↔
(22.131)
α
1
q
+
β
1
l
+2
β
2
ql
+
α
3
q
3
−
3
ql
2
+
β
4
4
q
3
l
4
ql
3
+
α
5
q
5
−
10
q
3
l
2
+5
ql
4
+
−
β
6
6
q
5
l
20
q
3
l
3
+6
ql
5
+
α
7
q
7
−
7
ql
6
21
q
5
l
2
+ 210
q
3
l
4
−
−
+
β
8
q
7
l
8
ql
7
+
O
(9) =
56
q
5
l
3
+56
q
3
l
5
−
−
(22.132)
2
β
2
ql − α
3
q
3
−
3
ql
2
− β
4
4
q
3
l −
4
ql
3
− α
5
q
5
−
10
q
3
l
2
+5
ql
4
−
−α
1
q
+
β
1
l −
β
6
(6
q
5
l
20
q
2
l
3
+6
ql
5
)
α
7
(
q
7
−
21
q
5
l
2
+ 210
q
3
l
4
−
7
ql
6
)
β
8
(8
q
7
l
56
q
5
l
3
+
−
−
−
−
6
q
3
l
5
−
8
ql
7
)
−
O
(9)
↔
(22.133)
α
1
=0
,β
2
=0
,α
3
=0
,β
4
=0
,α
5
=0
,β
6
=0
,α
7
=0
,β
8
=0
,etc.
x
(
b,l
)=
x
01
l
(22.134)
(
x
01
:
Eq.
(
22.129
))
distortion
energy density
tr
C
l
G
−
l
as well as the
total distortion energy
over a
Reference Meridian
Strip {L
W
≤ L ≤ L
E
,B
S
≤ B ≤ B
N
}
centered at (
L
0
,B
0
).
22-41 Tissot Distortion Analysis
By means of the
General Eigenvalue Problem
we are able to derive the basic elements of the
Tissot distortion analysis
. Accordingly by Table
22.22
we compute the coordinates of the
left
Cauchy-Green deformation tensor,
namely the matrix
C
l
∈
2
×
2
. The special harmonic map,
Eqs.(
22.114
)and(
22.134
), reproduces
{
c
11
,c
12
=
c
21
,c
22
}
as its elements, in particular
power
series
in “
b, l
” of type Eqs.(
22.136
)-(
22.138
).
Secondary we proceed to compute the two basic invariants
J
1
=tr
C
l
0
G
−
l
and
J
2
=det
C
l
G
−
l
.
Table
22.23
,Eqs.(
22.139
)and(
22.140
) collects the power series representation of the “
average
distortion
”tr
C
l
G
−
l
, the trace as the
first invariant
of the matrix
C
l
G
−
l
.Incontrast,Table
22.24
,
Eqs.(
22.141
)and(
22.142
), contain the power series representation of the “
determinant distortion
”
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