Geography Reference
In-Depth Information
ellipsoidal longitude
Λ
∗
(
Λ, Φ
;
c
3
,c
4
) and latitude
Φ
∗
(
Λ, Φ
∗
;
c
3
,c
4
) of the right biaxial ellipsoid are
in consequence related to surface normal ellipsoidal longitude/latitude
{
Λ, Φ
}
of the left biax-
ial ellipsoid, particularly postulating d
Λ
∗
/
d
Λ
=
c
3
,
d
Φ
∗
/
d
Φ
=
c
4
(1
E
2
sin
2
Φ
∗
)
2
cos
Φ/
[(1
−
−
E
2
sin
2
Φ
)
2
cos
Φ
∗
], where the scale constants
are chosen in such a way to guaran-
tee an areomorphism by
c
1
c
2
c
3
c
4
= 1. The final form of the mapping equations generating the
ellipsoidal Hammer projection
is achieved by the inversion of an odd homogeneous polynomial
equation for sin
Φ
∗
outlined in Sect.
H-5
.
{
c
1
,c
2
,c
3
,c
4
}
H-21 The Equiareal Mapping from a Left Biaxial Ellipsoid to a Right
Biaxial Ellipsoid
A
1
∗
,A
2
∗
which is
outlined by Box
H.1
is of preparatory nature for the following section. We assume that pointwise
the surface normal ellipsoidal longitude
{Λ, Λ
∗
}
of types left and right coincide, but the func-
tion which relates surface normal ellipsoidal latitude from the left to the right, namely
Φ
∗
(
Φ
), is
unknown. Based upon the structure of the mapping equations (
H.41
), the postulate of an equiareal
mapping (
H.23
), in particular det [C
l
G
−
l
] = 1, leads to the left Cauchy-Green deformation ten-
sor (
H.42
) with respect to the left metric tensor (
H.43
)of
A
1
,A
2
The equiareal mapping of a left biaxial ellipsoid
E
to a right biaxial ellipsoid
E
. The equivalence of det [C
l
G
−
l
]
with det[C
l
]=det[G
l
] leads to the differential equation (
H.44
) for the unknown function
Φ
∗
(
Φ
).
For the identities of (
H.45
)and(
H.46
), we have used only the positive preserving diffeomorphism
[d
Λ
∗
,
d
Φ
∗
]
T
=J[d
Λ,
d
Φ
]
T
,
E
A
1
,A
2
>
0, namely a positive determinant of the Jacobi matrix J. Left
and right integration of (
H.46
) with respect to the condition
Φ
∗
(
Φ
= 0) = 0 leads finally to the
mapping equations in (
H.47
) of equiareal type from a left biaxial ellipsoid
|
J
|
2
A
1
,A
2
E
to a right biaxial
A
1
∗,A
2
∗
ellipsoid
E
.
Box H.1 (Equiareal mapping from a left biaxial ellipsoid to a right biaxial ellipsoid).
Λ
∗
=
Λ
∗
(
Λ
)
, Φ
∗
=
Φ
∗
(
Φ
)
,
(H.41)
=
A
1
∗
cos
2
Φ
∗
,
C
l
=
G
11
Λ
∗
Λ
1
−E
∗
sin
2
Φ
∗
Λ
∗
Λ
0
0
(H.42)
G
22
Φ
∗
Φ
A
1
∗
(1
−E
∗
)
2
0
(1
−E
∗
sin
2
Φ
∗
)
3
Φ
∗
Φ
0
G
l
=
A
1
cos
2
Φ
,
0
1
−E
2
sin
2
Φ
(H.43)
A
1
(1
−E
2
)
2
(1
−E
2
sin
2
Φ
)
3
0
A
1
∗
(1
E
2
)
2
cos
2
Φ
∗
Φ
∗
Φ
=
A
1
(1
E
2
)
2
cos
2
Φ
−
−
∗
det [C
l
]=det[G
l
]
⇔
,
(H.44)
sin
2
Φ
∗
)
4
E
2
sin
2
Φ
)
4
(1
−
E
2
(1
−
∗
sin
2
Φ
∗
)
2
Φ
Φ
=
d
Φ
∗
d
Φ
=
(1
− E
2
cos
Φ
cos
Φ
∗
A
1
(1
−
E
2
)
A
1
∗
(1
∗
)
,
(H.45)
E
2
sin
2
Φ
)
2
(1
−
−
E
2
∗
)
ar tanh(
E
∗
sin
Φ
∗
)
2
E
∗
=
sin
Φ
∗
A
1
∗
(1
− E
2
+
∗
sin
2
Φ
∗
)
2(1
−
E
2
=
A
1
(1
− E
2
)
ar tanh(
E
sin
Φ
)
∗
.
sin
Φ
+
(H.46)
E
2
sin
2
Φ
)
2
E
2(1
−
Search WWH ::
Custom Search