152 3 Coordinates
It should be noted that the coordinates {A, B} of type oblique quasi-spherical longitude/latitude are
not orthogonal. Accordingly, the matrix of the metric of E
2
A 1 ,A 2
in terms of these coordinates contains
off-diagonal elements. Finally, we note that the terms up to order three of the corresponding Taylor
series expansions can be determined by resorting to the partial derivatives of Box 3.23.
Box 3.23 (Partial derivatives up to order three).
(tan Λ), A = +
cos
2 B cos I − sin A sin B cos B sin I
(cos A cos B cos Ω − sin A cos B sin Ω cos I + sin B sin Ω sin I)
2 ,
(tan Λ), B = −
cos A sin I
(cos A cos B cos Ω − sin A cos B sin Ω cos I + sin B sin Ω sin I)
2 ;
(3.148)
N = N (A, B) := cos
2 B cos I − sin A sin B cos B sin I ,
M = M (A, B) := cos A sin I ,
D = D(A, B) := cos A cos B cos Ω − sin A cos B sin Ω cos I + sin B sin Ω sin I ;
(3.149)
N, A = − cos A sin B cos B sin I ,
N, B = −2 sin B cos B − sin A cos
2 B sin I + sin A sin
2 B sin I ,
M, A = − sin A sin I , M, B = 0 ,
(3.150)
N, AA = + sin A sin B cos B sin I ,
N, AB = − cos A cos
2 B sin I + cos A sin
2 B sin I = N, BA ,
N, BB = −2 cos
2 B + 2 sin
2 B + 2 sin A sin B cos B sin I + 2 sin A sin B cos B sin I ,
M, AA = − cos A sin I , M, AB = 0 , M, BA = 0 , M, BB = 0 ,
(3.151)
D, A = − sin A cos B cos Ω − cos A cos B sin Ω cos I ,
D, B = − cos A sin B cos Ω + sin A sin B sin Ω cos I + cos B sin Ω sin I ,
D, AA = − cos A cos B cos Ω + sin A cos B sin Ω cos I ,
D, AB = + sin A sin B cos Ω + cos A sin B sin Ω cos I = D, BA ,
D, BB = − cos A cos B cos Ω + sin A cos B sin Ω cos I − sin B sin Ω sin I ;
(3.152)
(tan Λ), A =
N
D 2 , (tan Λ), B =
M
D 2 ,
(tan Λ), AA =
D
2 N, A −2DD, A N
D 4
, (tan Λ), BB =
D
2 M, B −2D
2 D, B M
D 4
,
(tan Λ), AAA =
D
4
`
2DD, A N, A +D
2 N, AA
´ − 8D
4 D,
2
A N
D 8
,
(tan Λ), BBB =
D
4
`
2DD, B M, B +D
2 M, BB
´ − 4D
6 D,
2
B M
D 8
,
(tan Λ), AB =
D
2 N, B −2DD, B N
D 4
,
(tan Λ), ABB =
D
4
`
2DD, B N, B +D
2 N, BB
´ − 8D
4 D,
2
B N
D 8
,
(tan Λ), AB = (tan Λ), BA , (tan Λ), ABB = (tan Λ), BAB = (tan Λ), BBA .
(3.153)
In the following chapter, let us close a gap and introduce a classification scheme that is needed in
the remaining chapters.
It should be noted that the coordinates {A, B} of type oblique quasi-spherical longitude/latitude are
not orthogonal. Accordingly, the matrix of the metric of E
2
A 1 ,A 2
in terms of these coordinates contains
off-diagonal elements. Finally, we note that the terms up to order three of the corresponding Taylor
series expansions can be determined by resorting to the partial derivatives of Box 3.23.
Box 3.23 (Partial derivatives up to order three).
(tan Λ), A = +
cos
2 B cos I − sin A sin B cos B sin I
(cos A cos B cos Ω − sin A cos B sin Ω cos I + sin B sin Ω sin I)
2 ,
(tan Λ), B = −
cos A sin I
(cos A cos B cos Ω − sin A cos B sin Ω cos I + sin B sin Ω sin I)
2 ;
(3.148)
N = N (A, B) := cos
2 B cos I − sin A sin B cos B sin I ,
M = M (A, B) := cos A sin I ,
D = D(A, B) := cos A cos B cos Ω − sin A cos B sin Ω cos I + sin B sin Ω sin I ;
(3.149)
N, A = − cos A sin B cos B sin I ,
N, B = −2 sin B cos B − sin A cos
2 B sin I + sin A sin
2 B sin I ,
M, A = − sin A sin I , M, B = 0 ,
(3.150)
N, AA = + sin A sin B cos B sin I ,
N, AB = − cos A cos
2 B sin I + cos A sin
2 B sin I = N, BA ,
N, BB = −2 cos
2 B + 2 sin
2 B + 2 sin A sin B cos B sin I + 2 sin A sin B cos B sin I ,
M, AA = − cos A sin I , M, AB = 0 , M, BA = 0 , M, BB = 0 ,
(3.151)
D, A = − sin A cos B cos Ω − cos A cos B sin Ω cos I ,
D, B = − cos A sin B cos Ω + sin A sin B sin Ω cos I + cos B sin Ω sin I ,
D, AA = − cos A cos B cos Ω + sin A cos B sin Ω cos I ,
D, AB = + sin A sin B cos Ω + cos A sin B sin Ω cos I = D, BA ,
D, BB = − cos A cos B cos Ω + sin A cos B sin Ω cos I − sin B sin Ω sin I ;
(3.152)
(tan Λ), A =
N
D 2 , (tan Λ), B =
M
D 2 ,
(tan Λ), AA =
D
2 N, A −2DD, A N
D 4
, (tan Λ), BB =
D
2 M, B −2D
2 D, B M
D 4
,
(tan Λ), AAA =
D
4
`
2DD, A N, A +D
2 N, AA
´ − 8D
4 D,
2
A N
D 8
,
(tan Λ), BBB =
D
4
`
2DD, B M, B +D
2 M, BB
´ − 4D
6 D,
2
B M
D 8
,
(tan Λ), AB =
D
2 N, B −2DD, B N
D 4
,
(tan Λ), ABB =
D
4
`
2DD, B N, B +D
2 N, BB
´ − 8D
4 D,
2
B N
D 8
,
(tan Λ), AB = (tan Λ), BA , (tan Λ), ABB = (tan Λ), BAB = (tan Λ), BBA .
(3.153)
In the following chapter, let us close a gap and introduce a classification scheme that is needed in
the remaining chapters.