Geography Reference
In-Depth Information
The tests
Λ
1
Λ
2
=1foran
equiareal mapping
and
Λ
1
=
Λ
2
for a
conformal mapping
according to
Grafarend
(
1995
, p. 449) 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 pullback transformed into the
{Λ, Φ}
coordinate representation, in particular
d
s
2
=d
x
2
+d
y
2
=
δ
μν
d
x
μ
dx
ν
=
=
δ
μν
∂x
μ
∂U
A
∂x
ν
∂U
B
d
U
A
d
U
B
U
1
=
Λ, U
2
=
Φ,
∀
(F.9)
∂x
ν
∂U
B
.
Throughout, we apply the summation convention over repeated indices. For example, we here
apply
a
μ
b
μ
=
a
1
b
1
+
a
2
b
2
. In addition, we adopt the symbols provided by (
F.10
) as the symbols for
the
meridional radius of curvature M
(
Φ
)andthe
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
−
A
M
(
Φ
):=
E
2
sin
2
Φ
)
3
/
2
,N
(
Φ
):=
E
2
sin
2
Φ
)
1
/
2
.
(F.10)
(1
−
(1
−
Indeed,
M
(
Φ
) is the radius of curvature of the meridian, the coordinate line
Λ
=const.,
but
N
(
Φ
) as the transverse radius of curvature of a curve formed by the intersection of the normal
or transverse plane
2
2
A,B
which is normal to the tangent space
T
x
2
A,B
of the biaxial
P
⊥
T
x
E
E
2
A,B
. In contrast,
N
(
Φ
)cos
Φ
=
L
(
Φ
) is the radius of curvature of the parallel circle, the coordinate line
Φ
=const.
The principal curvature radii of the biaxial ellipsoid
E
2
A,B
and is perpendicular to the meridian at a point
ellipsoid
E
{
Λ, Φ
}∈
E
2
A,B
are
{M
(
Φ
)
,N
(
Φ
)
}
.
x
=
x
(
Λ, Φ
)=[
N
(
Φ
)cos
Φ
]
Λg
(
Φ
)=
L
(
Φ
)
Λg
(
Φ
)
,
(F.11)
y
=
y
(
Φ
)=
f
(
Φ
)
.
The left Cauchy-Green deformation tensor
c
AB
is generated by (
F.12
). (
x
Λ
,y
Λ
,x
Φ
,y
Φ
)denote
the partials of (
x, y
) with respect to (
Λ, Φ
)sothatweareledto(
F.13
).
c
11
=
x
Λ
+
y
Λ
,c
12
=
x
Λ
x
Φ
+
y
Λ
y
Φ
,c
21
=
c
12
,c
22
=
x
Φ
+
y
Φ
,
(F.12)
=
ΛLg
(
L
g
+
Lg
)
Λ
2
(
L
g
+
Lg
)
2
+
f
2
.
L
2
g
2
ΛLg
(
L
g
+
Lg
)
{
c
AB
}
(F.13)
The infinitesimal distance d
S
between two points
X
and
X
+d
X
, both elements of the biaxial
ellipsoid
E
2
A,B
, represented in terms of the first chart is
d
S
2
=
G
AB
d
U
A
d
U
B
∀
=
N
2
(
Φ
)cos
2
Φ
=
L
2
0
M
2
.
(F.14)
0
0
{
G
AB
}
M
2
(
Φ
)
0
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