Geography Reference
In-Depth Information
Box 23.9 (The generic steps of a map projection for a given structure of the map (
23.15
)of
type circular cylinder
C
1
).
TheleftJacobimap
(23.16a)
J
l
:=
x
U
x
V
=
10
(23.16b)
0
f
y
U
y
V
subject to the orientation conservation
f
>
0
|
J
l
|
=
x
U
y
V
−
x
V
y
U
>
0
↔
(23.17a)
The left Cauchy-Green map
(23.17b)
C
l
:=
J
l
G
r
J
l
=
10
0
f
2
(23.17c)
subject to the right metric C
r
=
I
2
of the developed circular cylinder
The general eigenvalue problem
(23.18)
Λ
2
G
l
=0
subject to
(
23.17b
)
G
l
=
cosh
2
V
0
C
l
−
(23.19)
0 s
V
Λ
2
G
l
=
1
C
l
Λ
2
cosh
2
V
−
0
−
↔
(23.20)
f
2
Λ
2
cosh 2
V
0
−
f
1
cosh
V
, Λ
2
=
√
cosh 2
V
Λ
1
=
(23.21)
Finally based upon
case
(
γ
), the identity of the second eigenvalue (left principal stretches)
Λ
2
=
1, the “
canonical postulate
” generates the
first order differential equation
(
23.42
). Solved by
direct integration (
23.43
), the integration constant
c
is fixed by the
boundary condition
(
23.44
)
such that the final
mapping equation
(
23.45
)ofBox
23.14
appears. Its basis is laid by the integral
1
tanh
2
x
−
1
/
2
(1 +
tanh
2
x
)
−
1
/
2
dx
outlined in Box
23.15
, in particular by a series expansion
of
1
− tanh
2
x
−
1
/
2
(1+
tanh
2
x
)
−
1
/
2
subject to
|
tanh
x| <
1by(
23.46
), (
23.47
) and termwise inte-
gration (
23.48
)-(
23.50
) which is permitted for
uniform convergent
series (
23.47
). Equation (
23.51
)
collects the result of termwise integration
up to the order seven
of tanh
x
.
−
2
:=
2
onto the
Box 23.10 (Conformal mapping of the rotational symmetric hyperboloid
M
H
1
).
circular cylinder
M
f
=
√
cosh 2
V
cosh
V
=
1+
tanh
2
V
Λ
1
(
V
)=
Λ
2
(
V
)
∀
V
↔
(23.22)
f
(
V
)=
1+
tanh
2
V
dV
+
c
(23.23)
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