Geography Reference
In-Depth Information
with the “
left Cauchy-Green deformation tensor
”
C
l
, namely by a simultaneous diagonalization of
the
pair
of positive definite matrices
, we have established the
third generic step
as the
general eigenvalue problem
(
23.64
) with respect to the matrices
C
l
given by (
23.62
)and(
23.63
),
respectively, and
G
l
given by (
23.65
). The
eigenvalues
{
C
l
,G
l
}
of type (
23.67
)and(
23.69
),
respectively, are computed by means of solving the
characteristic equations
(
23.66
)and(
23.68
),
respectively. The
fourth step
leads us to the postulate of
(
α
) a conformal mapping (conformeomorphism): Box
23.18
Λ
1
=
Λ
2
,
(
β
) an equiareal mapping (areomorphism): Box
23.19
Λ
1
Λ
2
=1,
(
γ
) an equidistant mapping: Box
23.20
Λ
2
=1.
{
Λ
1
,
Λ
2
}
Box 23.17 (The generic steps of a map projection for a given structure of the map (
23.55
)
and (
23.56
) of type circular cylinder and circular cone
2
C
0
).
The left Jacobi map
(23.57)
J
l
i
:=
x
U
x
V
=
10
(23.58)
0
f
y
U
y
V
J
l
ii
:=
α
U
α
V
=
n
0
(23.59)
0
g
r
U
r
V
Subject to the orientation conservation
|
f
s
J
l
|
=
x
U
y
V
−
x
V
y
U
>
0
↔
(23.60)
g
>
0
|
J
l
|
=
α
U
r
V
−
α
V
r
U
>
0
↔
TheleftCauchy
-
Green map
(23.61)
C
l
i
:=
J
l
i
G
r
i
J
l
i
=
10
0
f
2
∀
G
r
i
=
I
2
(23.62)
C
l
ii
:=
J
l
ii
G
r
ii
J
l
ii
=
n
2
g
2
0
0
g
2
∀G
r
ii
=
r
2
0
(23.63)
01
subject to the right metric G
r
i
=
I
2
of the developed cone
and
the right metric G
r
ii
=
Diag
(
r
2
,
1)
of the developed cone
The general eigenvalue problem
(23.64)
Λ
2
G
l
=0
subject to
(
23.61
)
G
l
=
V
2
01+4
V
2
C
l
−
0
(23.65)
2
G
l
=
1
=0
C
l
i
−
2
V
2
−
0
↔
(23.66)
f
2
2
(1 + 4
V
2
)
0
−
f
√
1+4
V
2
Λ
1
=
1
V
, Λ
2
=
(23.67)
Λ
2
G
l
=
n
2
g
2
=0
↔
C
l
ii
−
− Λ
2
V
2
0
(23.68)
0
g
2
− Λ
2
(1 + 4
V
2
)
g
Λ
1
=
ng
√
1+4
V
2
V
, Λ
2
=
(23.69)
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