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(iii) The third identity (spherical side cosine lemma):
sin β =cos ψ = z 0
z 0 = r sin β = r cos ψ
r ,
sin β =sin ψ =cos φ cos φ 0 cos λ cos λ 0 +cos φ cos φ 0 sin λ sin λ 0 +sin φ sin φ 0 ,
(3.50)
sin β =sin ψ =cos φ cos φ 0 cos( λ
λ 0 )+sin φ sin φ 0 .
Box 3.6 (The forward problem of transforming spherical frames of reference. Input variables:
λ, φ, λ 0 0 . Output variables: α,β ).
(i) The first and second identities:
cos φ sin( λ
λ 0 )
tan α =
sin φ cos φ 0 ,
cos φ sin φ 0 cos( λ
λ 0 )
sin( λ
λ 0 )
tan α =
sin φ 0 cos( λ − λ 0 ) tan φ cos φ 0 .
(3.51)
Alternatives:
1
cos β cos φ sin( λ − λ 0 ) ,
sin α =
1
cos β (cos φ sin φ 0 cos( λ
cos α =
λ 0 )
sin φ cos φ 0 ) .
(3.52)
(ii) The third identity:
sin β =sin ψ =cos φ cos φ 0 cos( λ
λ 0 )+sin φ sin φ 0 .
(3.53)
2
Box 3.7 (Representation of a placement vector x
S
r in both the equatorial frame of refer-
ence
{
e 1 , e 2 , e 3 |O}
and the meta-equatorial (oblique) frame of reference
{
e 1 0 , e 2 0 , e 3 0 |O}
).
e 1 0 =+ e 1 sin φ 0 cos λ 0 + e 2 sin φ 0 sin λ 0
e 3 cos φ 0 ,
e 2 0 =
e 1 sin λ 0 + e 2 cos λ 0 ,
(3.54)
e 3 0 =+ e 1 cos φ 0 cos λ 0 + e 2 cos φ 0 sin λ 0 + e 3 sin φ 0 ,
x = e 1 x + e 2 y + e 3 z = e 1 0 x 0 + e 2 0 y 0 + e 3 0 z 0 =
= e 1 ( x 0 sin φ 0 cos λ 0 − y 0 sin λ 0 + z 0 cos φ 0 cos λ 0 )+
 
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