3-3 The oblique frame of reference of the sphere 137
Ω
I = π/2
O
Fig. 3.9. The transverse aspect of the sphere.
Box 3.16 (The forward problem of transforming spherical frames of reference: the transverse aspect. Input
variables: Λ, Φ, Ω, I = π/2. Output variables: A, B).
(i) Meta-longitude:
tan A =
tan Φ
cos(Λ − Ω)
,
cos A =
cos Φ
cos B
cos(Λ − Ω) =
=
cos Φ cos(Λ − Ω)
p
1 + cos Φ sin(Λ − Ω)
p
1 − cos Φ sin(Λ − Ω)
,
sin A =
sin Φ
cos B
=
=
sin Φ
p
1 + cos Φ sin(Λ − Ω)
p
1 − cos Φ sin(Λ − Ω)
.
(3.83)
(ii) Meta-latitude:
sin B = − cos Φ sin(Λ − Ω) .
(3.84)
(iii) Substitutions:
cos A =
1
√
1 + tan
2 A
, sin A =
tan A
√
1 + tan
2 A
,
cos A =
cos(Λ − Ω)
p
cos 2 (Λ − Ω) + tan
2 Φ
, sin A =
tan Φ
p
cos 2 (Λ − Ω) + tan
2 Φ
,
1
cos B
=
1
p
1 − cos 2 Φ sin
2 (Λ − Ω)
=
1
p
1 − cos Φ sin(Λ − Ω)
p
1 − cos Φ sin(Λ − Ω)
.
(3.85)