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X
+
2
sgn(
Y
)sgn(
X
)+1
180
◦
,
Λ
(
X
) = arctan
Y
1
2
sgn(
Y
)
−
1
−
1
Z
√
1
− E
2
√
X
2
+
Y
2
.
Φ
(
X
) = arctan
(G.3)
In order to establish the second set of local coordinates, in particular
{
meta-longitude, meta-
3
=
3
,δ
ij
}
latitude
where
δ
ij
is the Kronecker symbol for a unit matrix (
i, j
=1
,
2
,
3), into the transverse orthonormal
triad
}
, we transform the orthonormal triad
{
e
1
,
e
2
,
e
3
}
,whichspans
E
{
R
{
e
1
,
e
2
,
e
3
}
by means of the rotation matrices R
3
(
Λ
0
)andR
1
(
π/
2).
[
e
1
,
e
2
,
e
3
]=[
e
1
,
e
2
,
e
3
]R
3
(
Λ
0
)R
1
(
π/
2)
,
(G.4)
X
∈
E
2
A
1
,A
2
:=
:=
X
∈
R
3
+
,A
1
>A
2
,
X
2
+
Z
2
A
1
+
Y
2
+
,A
2
∈
R
A
2
=1
,A
1
∈
R
(G.5)
X
(
α, β
)=
E
2
)sin
α
cos
β
A
1
cos
α
cos
β
A
1
(1
−
A
1
sin
β
1
− E
2
sin
2
α
cos
2
β
1
− E
2
sin
2
α
cos
2
β
1
− E
2
sin
2
α
cos
2
β
=
e
1
+
e
2
+
e
3
.
(G.6)
For surface normal ellipsoidal
{
meta-longitude
α
, meta-latitude
β
}
,wechoosetheopenset0
<
α<
2
π
and
π/
2
<β<
+
π/
2 in order to avoid any
coordinate singularity
once we differentiate
X
(
α, β
) with respect to meta-longitude and meta-latitude. In order to ensure
bijectivity
of the
mapping
−
X
,Y
,Z
} →{
{
α,β
}
,weuse
X
+
2
sgn(
Y
)sgn(
X
)
180
◦
,
1
Y
1
1
α
(
X
) = arctan
2
sgn(
Y
)
−
−
(G.7)
1
−
E
2
Z
(1
− E
2
)
X
2+
Y
2
.
β
(
X
) = arctan(1
E
2
)
−
X
(
α,β
) covers the entire biaxial ellipsoid
A
1
,A
2
The union of the two charts
X
(
L, B
)
,thus
generating a
minimal atlas
. (Surfaces which are topological similar to the sphere, for example, the
biaxial ellipsoid, are uniquely described by a minimal atlas of two charts. An alternative example
is the torus whose minimal atlas is generated by
three charts
.)
Second, we set up
pseudo-cylindrical mapping equations
of the biaxial ellipsoid
∪
E
E
A
1
,A
2
onto the
plane
R
2
in terms of surface normal ellipsoidal
{
longitude
Λ
, latitude
Φ}
, in particular
cos
Φ
x
=
x
(
Λ, Φ
)=
A
1
Λ
1
g
(
Φ
)
,
(G.8)
E
2
sin
2
Φ
−
y
=
y
(
Φ
)=
A
1
f
(
Φ
)
,
2
x
:=
{
x
∈
R
|
ax
+
by
+
c
=0
}
.
The structure of the pseudo-cylindrical mapping equations with unknown functions
f
(
Φ
)and
g
(
Φ
) is motivated by the postulate that for
g
(
Φ
) = 1 parallel circles of
2
E
A
1
,A
2
should be mapped
2
. Note that for zero relative eccentricity
E
= 0, we arrive at the pseudo-
cylindrical mapping equations of the sphere
equidistantly onto
R
2
R
.
Third, by means of Corollary
G.1
, we want to show that in the class of pseudo-cylindrical
mapping equations (
G.8
), only equiareal map projections are possible.
S
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