3-3 The oblique frame of reference of the sphere 141
Box 3.19 (Individual equations of transforming oblique frames of reference).
Equation (i):
R 3 (ω)R 1 (I)R 3 (Ω) =
=
2
4
cos ω cos Ω − sin ω sin Ω cos I
cos ω sin Ω + sin ω cos Ω cos I
sin ω sin I
− sin ω cos Ω − cos ω sin Ω cos I − sin ω sin Ω + cos ω cos Ω cos I cos ω sin I
sin Ω sin I
− cos Ω sin I
cos I
3
5 .
(3.95)
Equation (ii):
R 2
“ π
2
− φ 0
”
R 3 (λ 0 ) =
=
2
4
sin φ 0 cos λ 0 sin φ 0 sin λ 0 − cos φ 0
− sin λ 0
cos λ 0
0
cos φ 0 cos λ 0 cos φ 0 sin λ 0
sin φ 0
3
5 .
(3.96)
Equation (iii):
cos ω = 0 , sin ω = 1 ⇒ ω = 90
◦ .
(3.97)
Equation (iv):
cos Ω = sin λ 0 , sin Ω = − cos λ 0 ⇒ Ω = 270
◦ + λ 0 , λ 0 = 90
◦ + Ω .
(3.98)
Equation (v):
cos I = sin φ 0 , sin I = − cos φ 0 ⇒ I = 270
◦ + φ 0 , φ 0 = 90
◦ + I .
(3.99)
Lemma 3.4 (The transformation of the first oblique frame of reference to the second oblique frame of
reference: λ 0 , φ 0 → ω, I, Ω).
If the first oblique frame of reference is given by defining a meta-North P
ole {λ 0 , φ 0 }, then the second
oblique frame of reference is determined by the orbital K epler elements ω = 90
◦ , I = 270
◦ + φ 0 , and
Ω = 270
◦ + λ 0 .
End of Lemma.
Lemma 3.5 (The transformation of the second oblique frame of reference to the first oblique frame of
reference: ω, I, Ω → λ 0 , φ 0 ).
If the second oblique frame of reference is given by defining the orbital K epler elements {ω, I, Ω},
then the first oblique frame of reference is determined by the meta-North P
ole λ 0 = 90
◦ + Ω and
φ 0 = 90
◦ + I, subject to ω = 90
◦ .
End of Lemma.
For the transverse frame of reference, the inclination of the ascending node I is chosen ninety degrees,
i. e. I = 90
◦ . A ccordingly, the transformation of reference frames leads us to Corollary 3.6.
Corollary 3.6 (Transformation of reference frames, transverse aspect, I = 90
◦ ).
If the second transverse frame of reference is given by defining the orbital K epler elements as
{ω, I, Ω} = {90
◦ , 90
◦ , Ω}, then the first transverse frame of reference is determined by the meta-North
P
ole λ 0 = 90
◦ + Ω and φ 0 = 0
◦ .
End of Corollary.
Box 3.19 (Individual equations of transforming oblique frames of reference).
Equation (i):
R 3 (ω)R 1 (I)R 3 (Ω) =
=
2
4
cos ω cos Ω − sin ω sin Ω cos I
cos ω sin Ω + sin ω cos Ω cos I
sin ω sin I
− sin ω cos Ω − cos ω sin Ω cos I − sin ω sin Ω + cos ω cos Ω cos I cos ω sin I
sin Ω sin I
− cos Ω sin I
cos I
3
5 .
(3.95)
Equation (ii):
R 2
“ π
2
− φ 0
”
R 3 (λ 0 ) =
=
2
4
sin φ 0 cos λ 0 sin φ 0 sin λ 0 − cos φ 0
− sin λ 0
cos λ 0
0
cos φ 0 cos λ 0 cos φ 0 sin λ 0
sin φ 0
3
5 .
(3.96)
Equation (iii):
cos ω = 0 , sin ω = 1 ⇒ ω = 90
◦ .
(3.97)
Equation (iv):
cos Ω = sin λ 0 , sin Ω = − cos λ 0 ⇒ Ω = 270
◦ + λ 0 , λ 0 = 90
◦ + Ω .
(3.98)
Equation (v):
cos I = sin φ 0 , sin I = − cos φ 0 ⇒ I = 270
◦ + φ 0 , φ 0 = 90
◦ + I .
(3.99)
Lemma 3.4 (The transformation of the first oblique frame of reference to the second oblique frame of
reference: λ 0 , φ 0 → ω, I, Ω).
If the first oblique frame of reference is given by defining a meta-North P
ole {λ 0 , φ 0 }, then the second
oblique frame of reference is determined by the orbital K epler elements ω = 90
◦ , I = 270
◦ + φ 0 , and
Ω = 270
◦ + λ 0 .
End of Lemma.
Lemma 3.5 (The transformation of the second oblique frame of reference to the first oblique frame of
reference: ω, I, Ω → λ 0 , φ 0 ).
If the second oblique frame of reference is given by defining the orbital K epler elements {ω, I, Ω},
then the first oblique frame of reference is determined by the meta-North P
ole λ 0 = 90
◦ + Ω and
φ 0 = 90
◦ + I, subject to ω = 90
◦ .
End of Lemma.
For the transverse frame of reference, the inclination of the ascending node I is chosen ninety degrees,
i. e. I = 90
◦ . A ccordingly, the transformation of reference frames leads us to Corollary 3.6.
Corollary 3.6 (Transformation of reference frames, transverse aspect, I = 90
◦ ).
If the second transverse frame of reference is given by defining the orbital K epler elements as
{ω, I, Ω} = {90
◦ , 90
◦ , Ω}, then the first transverse frame of reference is determined by the meta-North
P
ole λ 0 = 90
◦ + Ω and φ 0 = 0
◦ .
End of Corollary.