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In-Depth Information
subject to
Λ
∗
(
Λ
=0)=0
, Φ
∗
(
Φ
=0)=0
,
(H.63)
Λ
∗
=
c
3
Λ,
(H.64)
artanh(
E
sin
Φ
∗
)
2
E
sin
Φ
∗
+
E
2
sin
2
Φ
∗
)
=
(H.65)
2(1
−
=
c
4
artanh(
E
sin
Φ
)
.
sin
Φ
+
2
E
E
2
sin
2
Φ
)
2(1
−
If
E
=0
,
then
sin
Φ
∗
=
c
4
sin
Φ.
(H.66)
If
E
=0
,
then
sin
Φ
∗
=
=
c
4
sin
Φ
1+
2
15
E
4
sin
4
Φ
(9
−
20
c
4
+11
c
4
)+O(
E
6
)
.
3
E
2
sin
2
Φ
(1
− c
4
)+
1
(H.67)
As prepared by Boxes
H.2
and
H.3
, we can finally present by Lemma
H.5
the equiareal mapping of
the biaxial ellipsoid with respect to a transverse frame of reference and a change of scale, in short
the
Hammer projection of the biaxial ellipsoid
. In particular, the transfer of the four characteristic
terms
,namely(
H.34
), being functions of
Λ
∗
=
c
3
Λ
and sin
Φ
∗
(sin
Φ
;
c
4
), has to be
made. While case (i) of Corollary
H.6
highlights the general ellipsoidal Hammer projection, case
(ii) is its specific form for zero relative eccentricity,
E
= 0, namely its spherical counterpart. For
the choice
c
1
=2
,c
2
=1
,c
3
=1
/
2
,c
4
= 1 of case (iii), we receive by means of (
H.89
)-(
H.92
)the
ellipsoidal mapping equations of special equiareal projection in the Hammer gauge. In contrast,
case (iv) specializes, for
E
=0.(
H.96
) to the spherical mapping equations in the Hammer gauge,
indeed the original Hammer mapping equations (
Hammer 1892
). Various alternative variants of
the ellipsoidal mapping equations of equiareal type can be chosen, for different gauge constants
{
t
1
,t
2
,t
3
,t
4
}
{
as long as they fulfill
c
1
c
2
c
3
c
4
= 1. In particular, they refer to a pointwise map of
the North Pole or not or to other criteria.
c
1
,c
2
,c
3
,c
4
}
Lemma H.5 (The equiareal mapping of the biaxial ellipsoid with respect to a transverse frame
of reference and a change of scale (the Hammer projection of
E
A
1
,A
2
)).
2
A
1
∗
,A
2
∗
subject to
A
1
∗
=
A
1
,A
2
∗
=
A
2
onto the transverse tangent plane normal to
E
3
and with respect
to a change of scale is equiareal if
2
A
1
,A
2
The mapping of the right biaxial ellipsoid
E
with respect to left biaxial ellipsoid
E
x
=
c
1
r
(
Λ, Φ
;
c
3
,c
4
)cos
α
(
Λ, Φ
;
c
3
,c
4
)
,
(H.68)
y
=
c
2
r
(
Λ, Φ
;
c
3
,c
4
)sin
α
(
Λ, Φ
;
c
3
,c
4
)
,
(H.69)
subject to
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