3-4 The oblique frame of reference of the ellipsoid-of-revolution 145
Proof.
For the proof, we depart from the quadratic form which is characteristic for E
2
A 1 ,A 2
, namely we
replace the Cartesian coordinates {X
1 , X
2 , X
3
} = {X, Y, Z} via (3.105) by the Cartesian coordinates
{X
1
, X
2
, X
3
} = {X
, Y
, Z
} :
X
2 + Y
2
A 2
1
+
Z
2
A 2
2
= 1
⇔
A
2
1 A
2
2 =
=
X
2 + Y
2
A
2
2 + Z
2 A
2
1 =
= A
2
2 X
2 +
A
2
2 cos
2 I + A
2
1 sin
2 I
Y
2 +
A
2
2 sin
2 I + A
2
1 cos
2 I
Z
2 +
+2Y
Z
A
2
1 − A
2
2
sin I cos I ,
(3.110)
E
2 :=
A
2
1 − A
2
2
A 2
1
⇔
E
2 = 1 −
A
2
2
A 2
1
⇔
A
2
2 = A
2
1
1 − E
2
,
(3.111)
A
2
1
1 − E
2
=
=
1 − E
2
X
2 +
1 − E
2
cos
2 I + sin
2 I
Y
2 +
1 − E
2
sin
2 I + cos
2 I
Z
2 +
+2Y
Z
E
2 sin I cos I
and
A
2
1
1 − E
2
=
= X
2 + Y
2 + Z
2 − E
2
X
2 + Y
2 cos
2 I + Z
2 sin
2 I
+
+2Y
Z
E
2 sin I cos I ,
(3.112)
A
2
1
1 − E
2
R 2
=
= 1 − E
2
cos
2 A cos
2 B + sin
2 A cos
2 B cos
2 I + sin
2 B sin
2 I − 2 sin A sin B cos B sin I cos I
= 1 − E
2 [cos
2 A cos
2 B + (sin A cos B cos I − sin B sin I)
2 ]
⇒
(3.109) .
(3.113)
End of Proof.
Next, we have to compute the arc length of E
1
A 1 ,A 2 , i. e. that part of the oblique ecliptic equator
which ranges from the oblique quasi-spherical longitude zero to a fixed, but arbitrary value A.