Geography Reference
In-Depth Information
Corollary G.1 (Pseudo-cylindrical mapping equations).
In the class of the pseudo-cylindrical mapping equations of type (
G.8
), only equiareal map projec-
tions are possible restricting the unknown functions to (
G.9
). Note that conformal map projections
are not in the class of pseudo-cylindrical mapping equations.
f
=
g
−
1
or
g
=
f
−
1
.
(G.9)
End of Corollary.
For the proof, we are going to construct the left Cauchy-Green deformation tensor and solve
its general eigenvalue problem in order to compute the principal distortions
{
Λ
1
,Λ
2
}
.Thetests
Λ
1
Λ
2
= 1 for equiareal and
Λ
1
=
Λ
2
for conformality are performed.
Proof.
The infinitesimal distance d
s
between two points
x
and
x
+
d
x
, both elements of the plane
{
R
2
,δ
μν
}
is by pullback transformed into a
{
Λ, Φ
}
coordinate representation, in particular
d
s
2
=d
x
2
+d
y
2
=
δ
μν
d
x
μ
d
x
ν
=
δ
μν
∂x
μ
∂U
A
∂x
ν
∂U
B
d
U
A
d
U
B
U
1
=
Λ, U
2
=
Φ,
∀
(G.10)
∂x
ν
∂U
B
.
Throughout, we apply the summation convention over repeated indices, for example,
a
μ
b
μ
=
a
1
b
1
+
a
2
b
2
and
a
i
b
i
=
a
1
b
1
+
a
2
b
2
+
a
3
b
3
. In addition, we adopt the notation (
G.11
), the symbols
for the meridional radius of curvature
M
(
Φ
) and the normal radius of curvature
N
(
Φ
), respectively.
∀ c
AB
:=
δ
μν
∂x
μ
d
s
2
=
c
AB
d
U
A
d
U
B
∂U
A
E
2
)
A
1
(1
−
M
(
Φ
):=
E
2
sin
2
Φ
)
3
/
2
,
(G.11)
(1
−
A
1
N
(
Φ
):=
E
2
sin
2
Φ
)
1
/
2
.
(1
−
A
1
,A
2
The normal or transverse radius of curvature of
E
is the curvature radius of a curve formed by
2
A
1
,A
2
the intersection of the normal or transverse plane
R
⊥
T
x
E
which is normal to the tangent
A
1
,A
2
A
1
,A
2
A
1
,A
2
space
T
x
E
of the ellipsoid
E
and is perpendicular to the meridian at a point
X
∈
E
.
x
=
x
(
Λ, Φ
)=
N
(
Φ
)
Λ
cos
Φg
(
Φ
)=
A
1
ΛP
(
Φ
)
g
(
Φ
)
,
(G.12)
y
=
y
(
Φ
)=
A
1
f
(
Φ
)
.
Thus, the left Cauchy-Green deformation tensor
c
AB
is generated by
c
11
=
∂x
∂Λ
2
+
∂y
∂Λ
2
=
A
1
P
2
g
2
,
c
12
=
c
21
=
∂x
∂Λ
∂Φ
+
∂y
∂x
∂y
∂Φ
=
A
1
ΛPg
(
P
g
+
Pg
)
,
(G.13)
∂Λ
c
22
=
∂x
2
+
∂y
∂Φ
2
=
A
1
Λ
2
(
P
g
+
Pg
)
2
+
A
1
f
2
,
∂Φ
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