16-2 The oblique reference frame 365
e3
e 3
Ω
i
e 1
e1
e2
e 2
Fig. 16.3. Oblique reference frame {e 1 , e 2 , e 3 , O} with respect to the normal reference frame {e 1 , e 2 , e 3 , O}
along the principal axes 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
+ }.
Proof.
[ ( x
1 )
2 + (x
2 )
2 ]A
−2
1 + (16.23)
⇒
[ ( x
1 )
2 + (x
2 )
2 ]A
−2
1 = A
−2
1 [ x
2 + y
2 cos
2 i + z
2 sin
2 i − 2y
z
sin i cos i] ,
(16.26)
[ ( x
3 )
2 ]A
−2
2 + (16.23)
⇒
[ ( x
3 )
2 ]A
−2
2 = A
−2
2 [ y
2 sin
2 i + z
2 cos
2 i] ,
(16.27)
if x
= 0, then
[ ( x
1 )
2 + (x
2 )
2 ]A
−2
1 + [ ( x
3 )
2 ]A
−2
2 =
x
2
A 2
1
+
cos
2 i
A 2
1
+
sin
2 i
A 2
2
y
= 1 ,
A
2
2 = A
2
1 (1 − E
2 )
⇒
x
2
A 2
1
+
y
2
A 2
1 (1 − E 2 )
[
(1 − E
2 ) cos
2 i + sin
2 i]=
x
2
A 2
1
+
y
2
A 2
1 (1 − E 2 )
(1 − E
2 cos
2 i) = 1 .
(16.28)
End of Proof.
In the plane {x
, y
} ∈ {x
∈ R
2 Ax
+ By
+ C = 0}, we introduce circle-reduced meta-longitude α
in order to parameterize E
1
A
1 ,A
2
, namely b
y (16.29), illustrated b
y F ig. 16.4.
x
= A
1 cos α = A
1 sin α
∗ , α
∗ =
π
2 − α ,
y
= A
2 sin α = A
2 cos α
∗ , α =
π
2 − α
∗ .
(16.29)
In terms of circle-reduced metalongitude α or of circular reduced meta-pole distance α
∗ = π/2 − α,
we are ab le to represent the arc length of E
1
A
1 ,A
2
as an elliptic integral of the second k ind.