Geography Reference
In-Depth Information
B
2
arctan
A
]sin
U
2
B
−
B
B
2
tan
V
√
A
2
√
A
2
y=(
A
+
B
)
exp
[
2
−
−
B
2
arctan
A
]
A
+
B
A
+
B
cos
V
exp
[
2
B
√
A
2
−
B
B
2
tan
V
Λ
1
=
Λ
2
=
√
A
2
2
−
−
(ii) equiareal:
x
=
2
ABV
+2
B
2
sin
V
+(
A
+
B
)
2
cos
U
y=
2
ABV
+2
B
2
sin
V
+(
A
+
B
)
2
sin
U
Λ
1
=(
A
+
B
cos
V
)
−
1
2
ABV
+2
B
2
sin
V
+(
A
+
B
)
2
A
+
B
cos
V
2
ABV
+2
B
2
sin
V
+(
A
+
B
)
2
Λ
2
=
(iii) equidistant:
x=(A+B+BV)cos
U
y=(A+B+BV)sin
U
Λ
1
=
A
+
B
+
BV
A
+
B
cos
V
,
2
=1
)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
(
23.10a
)
and (
23.10b
) are generated which are solved by (
23.10c
), (
23.10d
)and(
23.10e
), respectively.
The integration constants
c
f
and
c
g
, respectively, are fixed by the
boundary conditions
(
23.10f
)
and (
23.10h
), respectively. Thus we are led to the mapping equations (
23.10g
)and(
23.10i
).
Finally based upon
case
(
For the
case
(
β
), the identity of the second eigenvalue (left principal stretch)
Λ
2
=
1, the “canonical postulate” generates the
first order differential equations
(
23.11a
)aswell
as (
23.11b
). Solved by direct integration (
23.11c
)aswellas(
23.11d
) their integration constant
c
f
and
c
g
s, respectively, are fixed by the
boundary conditions
(
23.11e
)and(
23.11g
), respectively,
leading to the final
mapping equations
(
23.11f
)and(
23.11h
), respectively.
Boxes
23.6
and
23.7
are a collection of the final mapping equations of the torus
T
γ
2
A,B
onto a
central plane
which is parallel to
T
π/
2
M
2
and onto the circular cylinder
C
A
+
B
which is developed,
namely of type
conformal, equiareal
and
equidistant.
Figures
23.3
,
23.4
,
23.5
,
23.6
,
23.7
,and
23.8
are a visualization of various toroidal map projec-
tions where as a theme a
cardoid
onto a torus
T
2
A,B
has been mapped as an object.
2
2
Box 23.7 (Mapping the torus
T
A,B
onto a circular cylinder
C
A
+
B
: (i) conformal, (ii) equiareal,
(iii) equidistant, left principal stretches
Λ
1
,Λ
2
).
(i) conformal:
Search WWH ::
Custom Search