144 3 Coordinates
3-42 The intersection of the ellipsoid-of-revolution and the central oblique plane
In order to obtain an oblique equatorial plane, namely an oblique equator, we intersect the ellipsoidof-revolution E
2
A 1 ,A 2
of semi-major axis A 1 and semi-minor axis A 2 and the central oblique plane L
2
O
(two-dimensional linear manifold through the origin O). Subsequently, the oblique equatorial plane as
well as its normal enables us to establish an oblique quasi-spherical coordinate system. Our first result
is summarized in Corollary 3.7.
Corollary 3.7 (The 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 oblique plane L
2
O is the ellipse
of semi-major axis A 1 = A 1 and semi-minor axis A 2 = A 1
√
1 − E 2 /
√
1 − E 2 cos 2 I with respect to
the relative eccentricity E
2 :=
A
2
1 − A
2
2
/A
2
1 :
E
1
A 1 ,A 2 :=
:=
x
∈ R
2
x
A 2
1
+
y
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
.
(3.106)
End of Corollary.
For short, the proof of Corollary 3.7 has been given in J. Engels, E. Grafarend (1995, pp. 42–43).
Compare with Fig. 3.8, which illustrates the oblique elliptic equator as well as the orthogonal projection
of a point X ∈ E
2
A 1 ,A 2
onto the oblique equatorial plane, respectively. Note that in a following section,
the oblique equator is used to establish the following elliptic cylinder:
C
2
A 1 ,A 2 :=
:=
X
∈ R
3 X
2
A 2
1
+
Y
2
A 2
2
= 1, Z
∈ R
.
(3.107)
We here note that the points of E
2
A 1 ,A 2
are conformally mapped just to lay down the foundation of a
cylindric map projection.
3-43 The oblique quasi-spherical coordinates
With respect to the oblique equatorial plane and its normal vector, namely the oblique orthonormal
frame
E 1 , E 2 , E 3
O
, let us introduce oblique quasi-spherical coordinates by means of
X
= R(A, B) cos A cos B ,
Y
= R(A, B) sin A cos B ,
Z
= R(A, B) sin B .
(3.108)
A (A ∈ [0, 2π[) usually is called oblique quasi-spherical longitude, B (B ∈ [−π/2, +π/2]) usually is
called oblique quasi-spherical latitude, and R(A, B) is the oblique radius, which in turn is a function
of A and B. Corollary 3.8 gives the answer, how this radial function can be expressed.
Corollary 3.8 (The oblique radial function R(A, B)).
If a point X ∈ E
2
A 1 ,A 2
is given in terms of oblique quasi-spherical coordinates of type (3.108), its
radial function is represented by
R(A, B) =
A 1
√
1 − E 2
1 − E 2 [cos 2 A cos 2 B + (sin A cos B cos I − sin B sin I) 2 ]
,
(3.109)
where the angle I characterizes the inclination of the oblique equatorial plane.
End of Corollary.
3-42 The intersection of the ellipsoid-of-revolution and the central oblique plane
In order to obtain an oblique equatorial plane, namely an oblique equator, we intersect the ellipsoidof-revolution E
2
A 1 ,A 2
of semi-major axis A 1 and semi-minor axis A 2 and the central oblique plane L
2
O
(two-dimensional linear manifold through the origin O). Subsequently, the oblique equatorial plane as
well as its normal enables us to establish an oblique quasi-spherical coordinate system. Our first result
is summarized in Corollary 3.7.
Corollary 3.7 (The 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 oblique plane L
2
O is the ellipse
of semi-major axis A 1 = A 1 and semi-minor axis A 2 = A 1
√
1 − E 2 /
√
1 − E 2 cos 2 I with respect to
the relative eccentricity E
2 :=
A
2
1 − A
2
2
/A
2
1 :
E
1
A 1 ,A 2 :=
:=
x
∈ R
2
x
A 2
1
+
y
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
.
(3.106)
End of Corollary.
For short, the proof of Corollary 3.7 has been given in J. Engels, E. Grafarend (1995, pp. 42–43).
Compare with Fig. 3.8, which illustrates the oblique elliptic equator as well as the orthogonal projection
of a point X ∈ E
2
A 1 ,A 2
onto the oblique equatorial plane, respectively. Note that in a following section,
the oblique equator is used to establish the following elliptic cylinder:
C
2
A 1 ,A 2 :=
:=
X
∈ R
3 X
2
A 2
1
+
Y
2
A 2
2
= 1, Z
∈ R
.
(3.107)
We here note that the points of E
2
A 1 ,A 2
are conformally mapped just to lay down the foundation of a
cylindric map projection.
3-43 The oblique quasi-spherical coordinates
With respect to the oblique equatorial plane and its normal vector, namely the oblique orthonormal
frame
E 1 , E 2 , E 3
O
, let us introduce oblique quasi-spherical coordinates by means of
X
= R(A, B) cos A cos B ,
Y
= R(A, B) sin A cos B ,
Z
= R(A, B) sin B .
(3.108)
A (A ∈ [0, 2π[) usually is called oblique quasi-spherical longitude, B (B ∈ [−π/2, +π/2]) usually is
called oblique quasi-spherical latitude, and R(A, B) is the oblique radius, which in turn is a function
of A and B. Corollary 3.8 gives the answer, how this radial function can be expressed.
Corollary 3.8 (The oblique radial function R(A, B)).
If a point X ∈ E
2
A 1 ,A 2
is given in terms of oblique quasi-spherical coordinates of type (3.108), its
radial function is represented by
R(A, B) =
A 1
√
1 − E 2
1 − E 2 [cos 2 A cos 2 B + (sin A cos B cos I − sin B sin I) 2 ]
,
(3.109)
where the angle I characterizes the inclination of the oblique equatorial plane.
End of Corollary.