364 16 “Ellipsoid-of-revolution to cylinder”: oblique aspect
16-2 The oblique reference frame
Oblique reference frame and normal reference frame, central oblique plane, circle-reduced meta-longitude
and circle-reduced meta-pole.
In the following discussion, let us orientate a set of orthonormal b
ase vectors {e 1 , e 2 , e 3 } along the
principal ax es of E
2
A 1 ,A 2
:= {x ∈ R
3 [ ( x
1 )
2 + (x
2 )
2 ]A
−2
1 + (x
3 )
2 A
−2
2 = 1, A 1 ∈ R
+ , A 2 ∈ R
+
}. Against
this frame of reference {e 1 , e 2 , e 3 , O} (consisting of the b
ase vectors e i , and the origin O), we introduce
the ob liq ue one {e 1 , e 2 , e 3 , O} b
y means of (16.20) illustrated b
y F igure 16.3.
⎡
⎣
e 1
e 2
e 3
⎤
⎦ = R 1 (i)R 3 (Ω)
⎡
⎣
e 1
e 2
e 3
⎤
⎦ .
(16.20)
The rotation around the 3 ax is, we have denoted b
y Ω, the “right ascension of the ascending node”,
while the rotation around the intermediate 1 ax is b
y i, the “inclination”. R 1 (i) and R 3 (Ω) are orthonormal matrices such that (16.21) holds.
R 1 (i)R 3 (Ω) =
⎡
⎣
cos Ω
sin Ω
0
− sin Ω cos i + cos Ω cos i sin i
+ sin Ω sin i − cos Ω sin i cos i
⎤
⎦ ∈ R
3×3 .
(16.21)
Accordingly, (16.22) is a representation of the placement vector x in the orthonormal b
ases {e 1 , e 2 , e 3 , O}
and {e 1 , e 2 , e 3 , O}, respectively. N ote that (16.23) and (16.24) hold.
x =
3
i=1
e i x
i =
3
i =1
e i x
i
,
(16.22)
x
1 = x
1
cos Ω − x
2
sin Ω cos i + x
3
sin Ω sin i ,
x
2 = x
1
sin Ω + x
2
cos Ω cos i − x
3
cos Ω sin i ,
x
3 = x
2
sin i + x
3
cos i ,
(16.23)
x
1
= +x
1 cos Ω
+ x
2 sin Ω
=: x
,
x
2
= −x
1 sin Ω cos i + x
2 cos Ω cos i + x
3 sin i =: y
,
x
3
= +x
1 sin Ω sin i − x
2 cos Ω sin i + x
3 cos i =: z
.
(16.24)
Corollary 16.1 (Intersection of E
2
A 1 ,A 2
and L
2
O ).
The intersection of the ellipsoid-of-revolution E
2
A 1 ,A 2
and the central ob liq ue plane L
2
O (two-dimensional
linear manifold through the origin O) is the ellipse (16.25) of semi-maj or ax is A
1 = A 1 and semi-minor
ax is A
2 = A 1
√
1 − E 2 /
√
1 − E 2 cos 2 i.
E
1
A
1 ,A
2
:=
:=
x ∈ R
2 x
2
A
1
2 +
y
2
A
2
2 = 1, A
1 = A 1 , A
2 = A 1
√
1−E 2
√
1−E 2 cos 2 i
, A
1 > A
2
.
(16.25)
End of Corollary.
16-2 The oblique reference frame
Oblique reference frame and normal reference frame, central oblique plane, circle-reduced meta-longitude
and circle-reduced meta-pole.
In the following discussion, let us orientate a set of orthonormal b
ase vectors {e 1 , e 2 , e 3 } along the
principal ax es of E
2
A 1 ,A 2
:= {x ∈ R
3 [ ( x
1 )
2 + (x
2 )
2 ]A
−2
1 + (x
3 )
2 A
−2
2 = 1, A 1 ∈ R
+ , A 2 ∈ R
+
}. Against
this frame of reference {e 1 , e 2 , e 3 , O} (consisting of the b
ase vectors e i , and the origin O), we introduce
the ob liq ue one {e 1 , e 2 , e 3 , O} b
y means of (16.20) illustrated b
y F igure 16.3.
⎡
⎣
e 1
e 2
e 3
⎤
⎦ = R 1 (i)R 3 (Ω)
⎡
⎣
e 1
e 2
e 3
⎤
⎦ .
(16.20)
The rotation around the 3 ax is, we have denoted b
y Ω, the “right ascension of the ascending node”,
while the rotation around the intermediate 1 ax is b
y i, the “inclination”. R 1 (i) and R 3 (Ω) are orthonormal matrices such that (16.21) holds.
R 1 (i)R 3 (Ω) =
⎡
⎣
cos Ω
sin Ω
0
− sin Ω cos i + cos Ω cos i sin i
+ sin Ω sin i − cos Ω sin i cos i
⎤
⎦ ∈ R
3×3 .
(16.21)
Accordingly, (16.22) is a representation of the placement vector x in the orthonormal b
ases {e 1 , e 2 , e 3 , O}
and {e 1 , e 2 , e 3 , O}, respectively. N ote that (16.23) and (16.24) hold.
x =
3
i=1
e i x
i =
3
i =1
e i x
i
,
(16.22)
x
1 = x
1
cos Ω − x
2
sin Ω cos i + x
3
sin Ω sin i ,
x
2 = x
1
sin Ω + x
2
cos Ω cos i − x
3
cos Ω sin i ,
x
3 = x
2
sin i + x
3
cos i ,
(16.23)
x
1
= +x
1 cos Ω
+ x
2 sin Ω
=: x
,
x
2
= −x
1 sin Ω cos i + x
2 cos Ω cos i + x
3 sin i =: y
,
x
3
= +x
1 sin Ω sin i − x
2 cos Ω sin i + x
3 cos i =: z
.
(16.24)
Corollary 16.1 (Intersection of E
2
A 1 ,A 2
and L
2
O ).
The intersection of the ellipsoid-of-revolution E
2
A 1 ,A 2
and the central ob liq ue plane L
2
O (two-dimensional
linear manifold through the origin O) is the ellipse (16.25) of semi-maj or ax is A
1 = A 1 and semi-minor
ax is A
2 = A 1
√
1 − E 2 /
√
1 − E 2 cos 2 i.
E
1
A
1 ,A
2
:=
:=
x ∈ R
2 x
2
A
1
2 +
y
2
A
2
2 = 1, A
1 = A 1 , A
2 = A 1
√
1−E 2
√
1−E 2 cos 2 i
, A
1 > A
2
.
(16.25)
End of Corollary.