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Λ
1
,Λ
2
}
of type (
23.8d
)and(
23.8f
), respectively, are computed by means of solving the
characteristic
equations
(
23.8c
)and(
23.8e
), respectively. The
fourth step
leads us to the postulates of
(
by (
23.7b
)and(
23.7c
), respectively, and
C
l
given by (
23.8b
). As a result the
eigenvalues
{
α
) a conformal mapping (conformeomorphism): Box
23.3
Λ
1
=
Λ
2
,
(
β
) an equiareal mapping (areomorphism): Box
23.4
Λ
1
Λ
2
=1,
) an equidistant mapping: Box
23.5
Λ
2
=1
.
For the
case
(
α
)ofa
conformal mapping
the “
canonical postulate
”
Λ
1
=
Λ
2
,theidentityof
the eigenvalues (left principal stretches) leads us to (
23.9a
)and(
23.9b
)as
first order differential
equations,
which are solved by (
23.9c
), (
23.9d
)aswellas(
23.9e
), respectively. The integration
constants
c
f
and
c
g
, respectively, are fixed by
boundary conditions
(
23.9f
)and(
23.9h
), respectively,
such that the final
mapping equations
(
23.9g
)and(
23.9i
)appear.
(
γ
Box 23.2 (The generic steps of a map projection for a given structure of the map (
23.4
)of
type plane parallel to
T
π/
2
M
2
and (
23.5
) of type circular cylinder
C
A
+
B
).
The left Jocabi map
(23.6a)
J
l
:=
x
U
x
V
=
−
f
sin
Uf
cos
U
f
cos
Uf
sin
U
(23.6b)
y
U
y
U
J
l
:=
x
U
x
V
=
A
+
B
0
(23.6c)
g
y
U
y
U
0
Subject to
the orientation conservation
f
<
0
g
>
0
J
l
=
x
U
y
V
−
x
V
y
U
>
0
↔
(23.6d)
TheleftCauchy
-
Green map
(23.7a)
C
l
:=
J
l
G
r
J
l
=
f
2
0
f
2
0
(23.7b)
C
l
:=
J
l
G
r
J
l
=
(
A
+
B
)
2
0
g
2
(23.7c)
0
Subject to
the right metric G
r
=
I
2
of the tangent plane
and
the developed circular cylinder
The general eigenvalue problem
(23.8a)
Λ
2
G
l
=0
subject to
(
23.7a
)
G
l
=
(
A
+
B
cos
V
)
2
B
2
C
l
−
0
(23.8b)
0
Λ
2
G
l
=
Λ
2
B
2
2
C
l
− Λ
2
(
A
+
B
cos
V
)
2
f
2
0
−
=0
↔
(23.8c)
f
2
0
−
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