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are mapped onto transverse tangent plane of the biaxial ellipsoid has to be performed. For an
up-to-date reference of the Hammer projection of the sphere under a general gauge
{
c
1
,c
2
,c
3
,c
4
}
,
we refer to
Hoschek
(
1984
)and
Wagner
(
1962
).
H-1 The Transverse Equiareal Projection of the Biaxial Ellipsoid
The first constituent of the Hammer projection is the transverse equiareal projection, now being
developed for the ellipsoid-of-revolution. First, we set up the transverse reference frame which
leads to an elliptic meta-equator and a circular zero meta-longitude meta-meridian. In particular,
we derive the mapping equations from surface normal ellipsoidal longitude/latitude to surface
normal ellipsoidal meta-longitude/meta-latitude. Second, we derive the differential equation for
a transverse equiareal projection of the biaxial ellipsoid and find its integral in terms of meta-
longitude/meta-latitude. Third, we express the mapping equations which generate a transverse
equiareal diffeomorphism in terms of ellipsoidal longitude/latitude. Mathematical details are pre-
sented in Sects.
H-3
and
H-4
.
H-11 The Transverse Reference Frame
First, let us orientate a set of orthonormal base vectors
{
E
1
,
E
2
,
E
3
}
along the principal axes
of (
H.1
). Against this frame of reference
{
E
1
,
E
2
,
E
3
;0
}
consisting of the orthonormal base vectors
{
E
1
,
E
2
,
E
3
}
and the origin 0, we introduce the oblique frame of reference
{
E
1
,
E
2
,
E
3
;0
}
by
means of (
H.2
).
2
A
1
,A
2
3
(
X
2
+
Y
2
)
/A
1
+
Z
2
/A
2
=1
,
+
+
E
:=
{
X
∈
R
|
R
A
1
>A
2
∈
R
}
,
(H.1)
⎡
⎤
⎡
⎤
E
1
E
2
E
3
E
1
E
2
E
3
⎣
⎦
=R
1
(
I
)R
3
(
Ω
)
⎣
⎦
.
(H.2)
The rotation around the 3 axis, we have denoted by
Ω
, the right ascension of the ascending node,
while the rotation around the intermediate 1 axis by
I
, the inclination. R
1
and R
3
, respectively,
are orthonormal matrices such that (
H.3
)holds.
⎡
⎤
cos
Ω
sin
Ω
0
⎣
⎦
∈
R
3
×
3
.
R
1
(
I
)R
3
(
Ω
)=
−
sin
Ω
cos
I
+cos
Ω
cos
I
sin
I
(H.3)
+sin
Ω
sin
I
−
cos
Ω
sin
I
cos
I
Accordingly, (
H.4
) is a representation of the placement vector
X
in the orthonormal bases
{
E
1
,
E
2
,
E
3
}
and
{
E
1
,
E
2
,
E
3
}
, respectively. We aim at a transverse orientation of the oblique
frame of reference
{
E
1
,
E
2
,
E
3
;0
}
, which is characterized by a base vector
E
3
in the equatorial
2
(
X,Y
)andthebasevectors
2
(
plane
P
{
E
1
,
E
2
}
in the rotated plane
P
−
Y,
−
Z
).Suchanorien-
tation of the transverse frame of reference
{
E
1
,
E
2
,
E
3
;0
}
is achieved choosing the inclination
I
= 270
◦
(cos
I
=0
,
sin
I
=
−
1). for instance, namely (
H.5
)or(
H.6
).
3
3
E
i
X
i
,
E
i
X
i
=
X
=
(H.4)
i
=1
i
=1
E
1
=
E
1
cos
Ω
+
E
2
sin
Ω,
E
2
=
−
E
3
,
E
3
=
−
E
1
sin
Ω
+
E
2
cos
Ω,
(H.5)
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