Geography Reference
In-Depth Information
F
(
Φ
)=
B
sin
Φ, G
(
Φ
)=
B
cos
Φ.
−
(14.59)
Coordinates of the metric tensor:
G
l
=
(
A
+
B
cos
Φ
)
2
B
2
.
0
(14.60)
0
The mapping equations are provided by the following formulae. As “equator”, let us define the
coordinate line
Φ
= 0 in the
X,Y
plane.
⎡
⎤
⎡
⎤
X
Y
Z
(
A
+
B
)cos
Λ
(
A
+
B
)sin
Λ
0
⎣
⎦
Φ
=0
⎣
⎦
,
=
(14.61)
(
X
2
+
Y
2
)
Φ
=0
=(
A
+
B
)
2
,
(
X
2
+
Y
2
)
Φ
=0
=
A
+
B, F
(0) =
A
+
B,
(14.62)
x
=(
A
+
B
)
Λ, y
=
f
(
Φ
)
.
(14.63)
Most notable, we could have alternatively chosen the “equator“ as
Φ
=
π
. From this, we conclude
the special case
F
(
π
)=
A
B
.Inaddition,werefertoFig.
14.2
illustrating the geometry of the
torus, namely its
vertical section
. As a case study, we present the special forms of the deformation
tensor for the torus as well as its left principal stretches.
−
C
l
=
F
2
(0)
=
(
A
+
B
)
2
.
0
0
(14.64)
f
2
(
Φ
)
f
2
(
Φ
)
0
0
F
2
(
Φ
)+
G
2
(
Φ
)
=
f
(
Φ
)
f
(
Φ
)
Λ
1
=
F
(0)
A
+
B
A
+
B
cos
Φ
, Λ
2
=
F
(
Φ
)
=
.
(14.65)
B
The special case
normal cylindric mapping
, equidistant on the equator and the set of parallel
circles, the special case
normal conformal cylindric mapping
, equidistant on the equator, and
the special
normal equiareal cylindric mapping
, equidistant on the equator are summarized in
Box
14.4
.
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