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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