Geography Reference
In-Depth Information
H-22 The Explicit Form of the Mapping Equations Generating
an Equiareal Map
We here consider the explicit form of the mapping equations generating an equiareal map from a
left biaxial ellipsoid to a right biaxial ellipsoid with respect to a transverse frame of reference and
a change of scale. Let us begin with the setup (
H.47
)ofBox
H.2
of general mapping equations
x
=
c
1
x
∗
(
Λ
∗
,Φ
∗
)and
y
=
c
2
y
∗
(
Λ
∗
,Φ
∗
) in a transverse frame of reference and under a change of
scale with respect to gauge constants
Λ
∗
,Φ
∗
}
which characterize a
point on the right ellipsoid-of-revolution are related via (
H.48
)
{Λ
∗
(
Λ
)
,Φ
∗
(
Φ
)
}
to the coordinates
{Λ, Φ}
characteristic for a point on the left ellipsoid-of-revolution. Note that right ellipsoidal
longitude/latitude depend only on left ellipsoidal longitude/latitude. In addition, we assume a
coincidence between the semi-major axis
A
1
∗
and
A
1
, respectively, the semi-minor axis
A
2
∗
and
A
2
between right and left
E
{
c
1
,c
2
}
. Those coordinates
{
A
1
,A
2
, respectively, expressed by (
H.49
). Next, by (
H.50
)
and (
H.51
), we subscribe the differential relations d
Λ
∗
/
d
Λ
=
c
3
subject to
Λ
∗
(
Λ
=0)=0and
d
Φ
∗
/
d
Φ
=
c
4
(1
A
1
∗
,A
2
∗
and
E
E
2
sin
2
Φ
∗
)
2
cos
Φ/
[(1
E
2
sin
2
Φ
)
2
cos
Φ
∗
] subject to
Φ
∗
(
Φ
= 0) = 0 with respect
−
−
to gauge constants
.(
H.45
) has motivated (
H.51
). The detailed computation of the left
Cauchy-Green tensor via (
H.52
), (
H.53
), (
H.54
), and (
H.55
) leads us to the postulate (
H.56
)of
an equiareal mapping. Indeed, we take advantage via (
H.56
), (
H.57
), and (
H.58
)ofthefactthat
{
{
c
3
,c
4
}
x
∗
(
Λ
∗
,Φ
∗
)
,y
∗
(
Λ
∗
,Φ
∗
)
}
is already an equiareal mapping. Thus, we may consider the transfor-
x
∗
,y
∗
}→{
mation
as a change from one equiareal chart to another equiareal chart (a:a:
cha-cha-cha). (
H.59
) is a representation of the postulate of an areomorphism which leads by
subscribing d
Λ
∗
/
d
Λ
=
c
3
to the explicit form of d
Φ
∗
/
d
Φ
of type (
H.52
), too. Indeed, we do not
have to postulate (
H.51
)! In order to guarantee an equiareal mapping
{
x, y
}
{
Λ, Φ
}→{
x, y
}
, the gauge
constants have to fulfill (
H.60
), namely
c
1
c
2
c
3
c
4
=1.
Box H.2 (The equiareal mapping from a left biaxial ellipsoid to a right biaxial ellipsoid with
respect to a transverse frame of reference and a change of scale (the Hammer projection of
the ellipsoid-of-revolution)).
x
=
c
1
x
∗
(
Λ
∗
,Φ
∗
)
,y
=
c
2
y
∗
(
Λ
∗
,Φ
∗
)
,
(H.47)
Λ
∗
(
Λ
)
,Φ
∗
(
Φ
)
,
(H.48)
subject to
A
1
∗
=
A
1
, A
2
∗
=
A
2
, E
∗
=
E,
(H.49)
Λ
A
=
d
Λ
∗
d
Λ
=
c
3
,
(H.50)
Φ
Φ
=
d
Φ
∗
E
2
sin
2
Φ
∗
)
2
d
Φ
=
c
4
(1
−
cos
Φ
cos
Φ
∗
.
(H.51)
E
2
sin
2
Φ
)
2
(1
−
The left Cauchy-Green deformation tensor:
∂Λ
=
c
1
∂x
∗
d
Λ
∗
∂Φ
=
c
1
∂x
∗
d
Φ
∗
x
Λ
=
∂x
d
Λ
=
c
1
Λ
Λ
x
Λ
∗
,x
Φ
=
∂x
d
Φ
=
c
1
Φ
Φ
x
Φ
∗
,
(H.52)
∂Λ∗
∂Φ
∗
∂Λ
=
c
2
∂y
∗
d
Λ
∗
∂Φ
=
c
2
∂y
∗
d
Φ
∗
y
Λ
=
∂y
d
Λ
=
c
2
Λ
Λ
y
Λ
∗
,y
Φ
=
∂y
d
Φ
=
c
2
Φ
Φ
y
Φ
∗
,
∂Λ
∗
∂Φ
∗
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