Geography Reference
In-Depth Information
1
12
+
c
f
→
1
12
f
(0) = 0 =
c
f
=
−
(23.86)
g
(0) = 0 =
1
1
12
n
6
n
+2
c
g
→
c
g
=
−
(23.87)
√
1+4
V
2
f
(
V
)=
1+4
V
2
12
1
12
−
(23.88)
(1 + 4
V
2
)
3
/
2
−
1
g
(
V
)=
(23.89)
6
n
2
:=
2
onto the circular cylinder
1
Box 23.20 (Equidistant mapping of the paraboloid
M
P
C
0
).
and the circular cone
C
f
=
√
1+4
V
2
Λ
2
(
V
)=1
∀
V
↔
(23.90)
g
=
√
1+4
V
2
Λ
2
(
V
)=1
∀
V
↔
(23.91)
2
V
√
1+4
V
2
+
1
f
(
V
)=
g
(
V
)=
1
4
arcsinh
2
V
+
c
(23.92)
f
(0) = 0
→
c
= 0
(23.93)
2
V
√
1+4
V
2
+
1
f
(
V
)=
g
(
V
)=
1
4
arcsinh
2
V
(23.94)
For the
case
(
α
)ofa
conformal mapping
the “
canonical postulate
”
Λ
1
=
Λ
2
,theidentityof
the eigenvalues (left principal stretches) leads us to (
23.70
)and(
23.71
)as
first order differential
equations,
which are solved by (
23.72
), (
23.73
)aswellas(
23.74
), respectively. The integration
constants
c
f
and
c
g
, respectively, such that the final
mapping equations
(
23.77
)and(
23.80
)appear.
For the
case
(
β
)ofan
equiareal mapping
the “
canonical postulate
”
Λ
1
Λ
2
=1, the product identity
of the eigenvalues (left principal stretches), the
first order differential equations
(
23.81
)and(
23.82
)
are generated which are solved by (
23.83
), (
23.84
)and(
23.85
), respectively. The integration con-
stants
c
f
and
c
g
, respectively, are fixed by the
boundary conditions
(
23.86
)and(
23.87
), respec-
tively. Thus we are led to the mapping equations (
23.88
)and(
23.89
).
Finally based upon
case
(
γ
), the identity of the second eigenvalue (left principal stretch)
Λ
2
=1,
the “
canonical postulate
” generates the
first order differential equations
(
23.90
)aswellas(
23.91
).
Solved by direct integration (
23.92
) the integration constant
c
is fixed by the
boundary condi-
tion
(
23.93
) leading to the final
mapping equation
(
23.94
).
Boxes
23.21
and
23.22
are a collection of the final mapping equation of a paraboloid
2
onto
P
1
and a
circular cone
0
which are developed, namely of type
conformal,
a
circular cylinder
C
C
equiareal and equidistant.
Figures
23.16
,
23.17
,
23.18
,
23.19
,
23.20
,and
23.21
are a visualization of various parabolic map
projections where as a
cardoid
onto a paraboloid
2
has been mapped as an object.
This contribution is based on
Grafarend and Syffus
(
1998a
).
P
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