Geography Reference
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Box 3.12 (From the equatorial frame of reference
{ E 1 , E 2 , E 3 |O}
to the meta-equatorial
(oblique) frame of reference
{
E 1 , E 2 , E 3 |O}
: the direct transformation).
(i) The first identity:
X = R cos B cos A
X
R cos B ,
cos A =
(3.72)
1
cos B (cos Φ cos Λ cos Ω +cos Φ sin Λ sin Ω ) ,
cos A = cos Φ
cos A =
cos B cos( Λ − Ω ) .
(ii) The second identity:
Y = R cos B sin A
Y
R cos B ,
sin A =
(3.73)
1
cos B ( cos Φ cos Λ sin Ω cos I +cos Φ sin Λ cos Ω cos I +sin Φ sin I ) ,
sin A =
sin A =
1
cos B (cos Φ cos I sin( Λ − Ω )+sin Φ sin I ) .
(iii) The third identity:
Z = R sin B
sin B = Z
R ,
(3.74)
sin B =cos Φ cos Λ sin Ω sin I
cos Φ sin Λ cos Ω sin I +sin Φ cos I,
sin B =
cos Φ sin I sin( Λ
Ω )+sin Φ cos I.
Box 3.13 (The forward problem of transforming spherical frames of reference. Input variables:
Λ, Φ, Ω, I . Output variables: A, B ).
(i) The first and second identities:
tan A = cos I sin( Λ
Ω )+tan Φ sin I
cos( Λ
.
(3.75)
Ω )
Alternatives :
sin A = cos Φ
cos B (cos I sin( Λ
Ω )+tan Φ sin I ) ,
 
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