136 3 Coordinates
Box 3.14 (From the meta-equatorial (oblique) frame of reference {E 1 , E 2 , E 3 O} to the equatorial frame
of reference {E 1 , E 2 , E 3 O}: the inverse transformation).
(i) The first identity:
X = R cos Φ cos Λ ⇒ cos Λ =
X
R cos Φ
,
cos Λ =
1
cos Φ
`
cos B cos A cos Ω − cos B sin A sin Ω cos I + sin B sin Ω sin I
´
.
(3.78)
(ii) The second identity:
Y = R cos Φ sin Λ ⇒ sin Λ =
Y
R cos Φ
,
sin Λ =
1
cos Φ
`
cos B cos A sin Ω + cos B sin A cos Ω cos I − sin B cos Ω sin I
´
.
(3.79)
(iii) The third identity:
Z = R sin Φ ⇒ sin Φ =
Z
R
,
sin Φ = cos B sin A sin I + sin B cos I .
(3.80)
Box 3.15 (The backward problem of transforming spherical frames of reference. Input variables: A, B, Ω, I.
Output variables: Λ, Φ).
(i) The first and second identities:
tan Λ =
cos B cos A sin Ω + cos B sin A cos Ω cos I − sin B cos Ω sin I
cos B cos A cos Ω − cos B sin A sin Ω cos I + sin B sin Ω sin I
.
(3.81)
(ii) The third identity:
sin Φ = cos B sin A sin I + sin B cos I .
(3.82)
3-33 The transverse frame of reference of the sphere: part one
In the framework of the transverse aspect, we are aiming at establishing a special oblique frame of
reference by the inclination I = 90
◦ . We have to deal with two problems depending on the input data.
Forward problem.
Backward problem.
Input: Λ, Φ and Ω, I = π/2.
Input: A, B and Ω, I = π/2.
Output: A, B.
Output: Λ, Φ.
Within the forward problem, which is solved in Box 3.16, we depart from (i) given spherical longitude
Λ and spherical latitude Φ of a point in the sphere S
2
R and from (ii) given longitude of the ascending
node Ω and inclination I = π/2 of the meta-equatorial plane in order to derive (iii) meta-longitude
A and meta-latitude B of the homologous point in the sphere. Conversely, for solving the backward
problem, which is outlined in Box 3.17, we inject (i) meta-longitude A and meta-latitude B of a point
in the sphere S
2
R and (ii) longitude of the ascending node Ω and inclination I = π/2 of the metaequatorial plane in order to derive spherical longitude Λ and spherical latitude Φ of the homologous
point in the sphere. Consult Fig. 3.9, which is an illustration of the transverse aspect of the sphere.
Box 3.14 (From the meta-equatorial (oblique) frame of reference {E 1 , E 2 , E 3 O} to the equatorial frame
of reference {E 1 , E 2 , E 3 O}: the inverse transformation).
(i) The first identity:
X = R cos Φ cos Λ ⇒ cos Λ =
X
R cos Φ
,
cos Λ =
1
cos Φ
`
cos B cos A cos Ω − cos B sin A sin Ω cos I + sin B sin Ω sin I
´
.
(3.78)
(ii) The second identity:
Y = R cos Φ sin Λ ⇒ sin Λ =
Y
R cos Φ
,
sin Λ =
1
cos Φ
`
cos B cos A sin Ω + cos B sin A cos Ω cos I − sin B cos Ω sin I
´
.
(3.79)
(iii) The third identity:
Z = R sin Φ ⇒ sin Φ =
Z
R
,
sin Φ = cos B sin A sin I + sin B cos I .
(3.80)
Box 3.15 (The backward problem of transforming spherical frames of reference. Input variables: A, B, Ω, I.
Output variables: Λ, Φ).
(i) The first and second identities:
tan Λ =
cos B cos A sin Ω + cos B sin A cos Ω cos I − sin B cos Ω sin I
cos B cos A cos Ω − cos B sin A sin Ω cos I + sin B sin Ω sin I
.
(3.81)
(ii) The third identity:
sin Φ = cos B sin A sin I + sin B cos I .
(3.82)
3-33 The transverse frame of reference of the sphere: part one
In the framework of the transverse aspect, we are aiming at establishing a special oblique frame of
reference by the inclination I = 90
◦ . We have to deal with two problems depending on the input data.
Forward problem.
Backward problem.
Input: Λ, Φ and Ω, I = π/2.
Input: A, B and Ω, I = π/2.
Output: A, B.
Output: Λ, Φ.
Within the forward problem, which is solved in Box 3.16, we depart from (i) given spherical longitude
Λ and spherical latitude Φ of a point in the sphere S
2
R and from (ii) given longitude of the ascending
node Ω and inclination I = π/2 of the meta-equatorial plane in order to derive (iii) meta-longitude
A and meta-latitude B of the homologous point in the sphere. Conversely, for solving the backward
problem, which is outlined in Box 3.17, we inject (i) meta-longitude A and meta-latitude B of a point
in the sphere S
2
R and (ii) longitude of the ascending node Ω and inclination I = π/2 of the metaequatorial plane in order to derive spherical longitude Λ and spherical latitude Φ of the homologous
point in the sphere. Consult Fig. 3.9, which is an illustration of the transverse aspect of the sphere.