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det[G
r
]
det[G
l
]
=
g
11
g
22
−
g
12
G
11
G
22
U
u
V
v
− U
v
V
u
=
G
12
,
−
det[G
l
]
det[G
r
]
=
G
11
G
22
−
G
12
u
U
v
V
−
u
V
v
U
=
g
22
.
g
11
g
22
−
End of Theorem.
The proof is straightforward. For a better insight into the equivalence theorem of an equiareal
mapping, we recommend a detailed study of the next example.
1-13 One Example: Mapping from an Ellipsoid-of-Revolution
to the Sphere
One example for the equivalence theorem of equiareal mapping: the equiareal mapping from
an ellipsoid-of-revolution to the sphere.
A beautiful example for the equivalence theorem of equiareal mapping is the mapping of the
ellipsoid-of-revolution
7
, postulated by means of
Λ
1
Λ
2
=1tobearea
preserving. All notations are taken from Example
1.3
. First, by means of Box
1.40
,wesetup
the mapping equations
E
A
1
,A
1
,A
2
to the sphere
S
S
7
,namelyby
λ
=
Λ, φ
=
f
(
Φ
). Here, we compute the
left Cauchy-Green matrix C
l
as well as the left principal stretches
E
A
1
,A
1
,A
2
→
. Second, Box
1.41
illustrates the various steps to be taken in order to derive an equiareal map from the
canonical
postulate Λ
1
Λ
2
= 1. As soon as we transfer the general form of the principal stretches
{
Λ
1
,Λ
2
}
Λ
1
,Λ
2
}
into such a postulate, by means of separation of variables, we derive a first-order differential
equation, which is directly solved by integration. Third, with respect to standard integrals and
the boundary condition
φ
=
f
(
Φ
= 0) = 0, we find the classical formula for sin
φ
, where the
mapping function
φ
=
f
(
Φ
) is called
authalic latitude
(
Adams 1921
, p. 65;
Snyder 1982
, p. 19).
Fourth, we solve the problem how to choose the radius of the sphere
{
2
S
r
when only the semi-major
axis
A
1
or the relative eccentricity
E
2
=(
A
1
−
A
2
)
/A
1
,A
1
>A
2
of
2
A
1
,Λ
1
,A
2
are given. A first
choice is
A
1
=
r
, a second choice, also called
optimal
, is the identity of the
left global surface
element S
l
of
E
2
A
1
,A
1
,A
2
r
. As derived later, we give
E
and of the
right global surface element S
r
of
S
7
) in closed form. Accordingly, we have succeeded to solve
r
(
A
1
,E
). Step five, based upon Box
1.42
, summarizes the forward or direct equations of the
special equiareal mapping, called
authalic
,oftype
λ
=
Λ
and sin
φ
=sin
f
(
Φ
) for the optimal
equiareal choice of the radius
r
(
A
1
,E
). In addition, we have computed the left and right principal
stretches
{Λ
1
,Λ
2
}
and
{λ
1
,λ
2
}
for the
authalic mapping
.
2
A
1
,A
1
,A
2
both area (
E
)aswellasarea(
S
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