Geography Reference
In-Depth Information
Fig. 5.29.
Images of coordinate lines under four categories of mapping, special case: polar coordinates
Category A :
Category B :
Category C :
Category D :
α
=
Λ,
α
=
g
(
Λ, Δ
)
,
α
=
g
(
Λ
)
,
α
=
g
(
Λ, Δ
)
,
(5.137)
r
=
f
(
Δ
)
.
r
=
f
(
Δ
)
.
r
=
f
(
Λ, Δ
)
.
r
=
f
(
Λ, Δ
)
.
Box 5.21 (Polar pseudo-azimuthal projections “sphere to plane”, left principal stretches).
Parameterized mapping:
α
=
g
(
Λ, Δ
)
,r
=
f
(
Δ
)
,x
=
r
cos
α
=
f
(
Δ
)cos
g
(
Λ, Δ
)
,
y
=
r
sin
α
=
f
(
Δ
)sin
g
(
Λ, Δ
)
.
(5.138)
Left Jacobi matrix:
J
l
=
D
Λx
D
Δx
−
.
fg
Δ
sin
g
+
fg
Λ
cos
gf
sin
g
+
fg
Δ
cos
g
fg
Λ
sin
gf
cos
g
−
(5.139)
D
Λy
D
Δy
Left Cauchy-Green matrix:
G
r
=I
2
,
C
l
=J
l
G
r
J
l
=J
l
J
l
=
f
2
g
Λ
f
2
g
Λ
g
Δ
.
(5.140)
f
2
g
Λ
g
Δ
f
2
+
f
2
g
Δ
Left principal stretches:
Λ
2
G
l
|
Λ
1
,
2
=
Λ
2
|
C
l
−
=0
⇔
±
tr [C
l
G
−
l
]
4det[C
l
G
−
l
]
,
(tr[C
l
G
−
l
])
2
=
1
2
±
−
(5.141)
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