396 18 “Sphere to cone”: pseudo-conic projections
Box 18.1 (General structure of equal area pseudo-conic mappings of the sphere).
Mapping equations:
α = g(∆)Λ cos ∆ = h(∆)Λ , r = f (∆) .
(18.6)
Left Cauchy–Green matrix:
C l = J
T
l G r J l =
"
f
2 h
2
f
2 hh
Λ
f
2 hh
Λ f
2 h
2 Λ
2 + f
2
#
,
(18.7)
det [C l ] = f
2 h
2 (f
2 h
2 Λ
2 + f
2 ) − f
4 h
2 h
2 Λ
2 = f
4 h
2 h
2 Λ
2 + f
2 h
2 f
2 − f
4 h
2 h
2 Λ
2 = f
2 f
2 h
2 . (18.8)
Left Jacobi matrix:
J l =
"
D Λ α D ∆ α
D Λ r D ∆ r
#
=
"
h(∆) Λh
(∆)
0
f
(∆)
#
.
(18.9)
Right metric tensor:
G r =
"
r
2
0
0 1
#
=
"
f
2
0
0 1
#
.
(18.10)
Left metric tensor:
G l =
"
R
2 sin
2 ∆ 0
0
R
2
#
, det [G l ] = R
4 sin
2 ∆ .
(18.11)
Postulate of an equal area mapping:
det [C l ] = det [G l ] ⇒ ff
g cos ∆ = +R
2 sin ∆
(only the + sign is here correct due to the orientation constance)
⇔
(18.12)
g = 2R
2 tan ∆
(f 2 ) = R
2 tan ∆
ff .
(18.13)
General structure
(equal area mapping: pseudo-conic):
α = α(Λ, ∆) = g(∆)Λ cos ∆ = 2R
2 sin ∆
[f 2 (∆)] Λ = R
2 sin ∆
ff Λ ,
r = r(∆) = f (∆) .
(18.14)