3-4 The oblique frame of reference of the ellipsoid-of-revolution 151
3-46 Inverse transformation of oblique quasi-spherical longitude/latitude
Let us depart from the representation (3.108) of oblique Cartesian coordinates {X
, Y
, Z
} in terms
of oblique quasi-spherical longitude/latitude {A, B}. The inverse map relates these oblique Cartesian
coordinates to oblique quasi-spherical longitude/latitude:
tan A =
Y
X ,
tan B =
Z
X 2 + Y 2
.
(3.143)
As soon as we implement the transformation of normal Cartesian coordinates {X, Y, Z} into oblique
Cartesian coordinates {X
, Y
, Z
} of type (3.104) and (3.105) as well as the surface normal ellipsoidal
longitude/latitude {Λ, Φ} into normal Cartesian coordinates {X, Y, Z} of type (3.129), we are led to
X
= X cos Ω + Y sin Ω =
=
A 1
1 − E 2 sin
2 Φ
cos Φ (cos Λ cos Ω + sin Λ sin Ω) =
=
A 1
1 − E 2 sin
2 Φ
cos Φ cos(Λ − Ω) ,
(3.144)
Y
= −X sin Ω cos I + Y cos Ω cos I + Z sin I =
=
A 1
1 − E 2 sin
2 Φ
− cos Φ cos Λ sin Ω cos I + cos Φ sin Λ cos Ω cos I +
1 − E
2
sin Φ sin I
=
=
A 1
1 − E 2 sin
2 Φ
+ cos Φ cos I sin(Λ − Ω) +
1 − E
2
sin Φ sin I
,
(3.145)
Z
= X sin Ω sin I − Y cos Ω cos I + Z cos I =
=
A 1
1 − E 2 sin
2 Φ
cos Φ cos Λ sin Ω sin I − cos Φ sin Λ cos Ω cos I +
1 − E
2
sin Φ cos I
,
(3.146)
such that
tan A =
− cos Φ cos I sin(Λ − Ω) +
1 − E
2
sin Φ sin I
cos Φ cos(Λ − Ω)
,
tan B =
cos Φ cos Λ sin Ω sin I − cos Φ sin Λ cos Ω cos I +
1 − E
2
sin Φ cos I
cos 2 Φ cos 2 (Λ − Ω) + [cos Φ cos I sin(Λ − Ω) + (1 − E 2 ) sin Φ sin I]
2
.
(3.147)
Let us here additionally collect the result of the transformation {Λ, Φ} → {A, B} by the following
Corollary 3.10.
Corollary 3.10 (The change from one chart to another chart: cha-cha-cha, the surface normal ellipsoidal
longitude/latitude versus the oblique quasi-spherical longitude/latitude).
Given the longitude of the ascending node Ω as well as the inclination I of the oblique equatorial plane,
then the transformation of surface normal ellipsoidal longitude/latitude into oblique quasi-spherical
longitude/latitude is represented by (3.147).
End of Corollary.
3-46 Inverse transformation of oblique quasi-spherical longitude/latitude
Let us depart from the representation (3.108) of oblique Cartesian coordinates {X
, Y
, Z
} in terms
of oblique quasi-spherical longitude/latitude {A, B}. The inverse map relates these oblique Cartesian
coordinates to oblique quasi-spherical longitude/latitude:
tan A =
Y
X ,
tan B =
Z
X 2 + Y 2
.
(3.143)
As soon as we implement the transformation of normal Cartesian coordinates {X, Y, Z} into oblique
Cartesian coordinates {X
, Y
, Z
} of type (3.104) and (3.105) as well as the surface normal ellipsoidal
longitude/latitude {Λ, Φ} into normal Cartesian coordinates {X, Y, Z} of type (3.129), we are led to
X
= X cos Ω + Y sin Ω =
=
A 1
1 − E 2 sin
2 Φ
cos Φ (cos Λ cos Ω + sin Λ sin Ω) =
=
A 1
1 − E 2 sin
2 Φ
cos Φ cos(Λ − Ω) ,
(3.144)
Y
= −X sin Ω cos I + Y cos Ω cos I + Z sin I =
=
A 1
1 − E 2 sin
2 Φ
− cos Φ cos Λ sin Ω cos I + cos Φ sin Λ cos Ω cos I +
1 − E
2
sin Φ sin I
=
=
A 1
1 − E 2 sin
2 Φ
+ cos Φ cos I sin(Λ − Ω) +
1 − E
2
sin Φ sin I
,
(3.145)
Z
= X sin Ω sin I − Y cos Ω cos I + Z cos I =
=
A 1
1 − E 2 sin
2 Φ
cos Φ cos Λ sin Ω sin I − cos Φ sin Λ cos Ω cos I +
1 − E
2
sin Φ cos I
,
(3.146)
such that
tan A =
− cos Φ cos I sin(Λ − Ω) +
1 − E
2
sin Φ sin I
cos Φ cos(Λ − Ω)
,
tan B =
cos Φ cos Λ sin Ω sin I − cos Φ sin Λ cos Ω cos I +
1 − E
2
sin Φ cos I
cos 2 Φ cos 2 (Λ − Ω) + [cos Φ cos I sin(Λ − Ω) + (1 − E 2 ) sin Φ sin I]
2
.
(3.147)
Let us here additionally collect the result of the transformation {Λ, Φ} → {A, B} by the following
Corollary 3.10.
Corollary 3.10 (The change from one chart to another chart: cha-cha-cha, the surface normal ellipsoidal
longitude/latitude versus the oblique quasi-spherical longitude/latitude).
Given the longitude of the ascending node Ω as well as the inclination I of the oblique equatorial plane,
then the transformation of surface normal ellipsoidal longitude/latitude into oblique quasi-spherical
longitude/latitude is represented by (3.147).
End of Corollary.