Geography Reference
In-Depth Information
Proof.
ρ
=
c
1
/c
2
is proven by the following procedure.
J
l
A
=
c
0
−
2
c
1
ρ
+
c
2
ρ
2
=min
ρ
.
(I.20)
Necessary :
d
J
l
A
d
ρ
(
ρ
=
ρ
)=
−
2
c
1
+
c
2
ρ
= 0
(I.21)
⇔
ρ
=
c
1
/c
2
.
Sucient :
d
2
J
l
A
d
ρ
2
(
ρ
=
ρ
)=2
c
2
>
0
.
(I.22)
(
I.19
) directly follows from (
I.10
)and
ρ
=
c
1
/c
2
.
End of Proof.
Before we go into numerical computations of the optimal dilatation factor
ρ
for the generalized
Mercator projection, let us here briefly present a result for zero total areal distortion as it is
outlined by
Grafarend
(
1995
).
Definition I.6 (Generalized Mercator projection, optimal with respect to areal distortion).
The generalized Mercator projection of the ellipsoid-of-revolution
E
A
1
,A
2
onto the developed cir-
cular cylinder
of radius
ρA
1
is called
optimal with respect to areal distortion
if the deviation
from an equiareal mapping
Λ
1
Λ
2
−
C
ρA
1
1 in terms of the left principal stretches (
Λ
1
,Λ
2
) averaged over
a mapping area of interest, namely the total areal distortion (
I.23
), is minimal with respect to
the unknown dilatation factor
ρ
.
J
l
:=
1
S
(
Λ
1
Λ
2
−
1)d
S
=min
ρ
.
(I.23)
S
End of Definition.
Corollary I.7 (Generalized Mercator projection, dilatation factor).
For a generalized Mercator projection of the half-symmetric strip [
Λ
W
=
Λ
0
−
ΔΛ, Λ
0
+
ΔΛ
=
Λ
E
]
[
Φ
S
,Φ
N
], the postulates of minimal Airy distortion energy (minimal total distance distortion)
and of minimal total areal distortion lead to the same unknown dilatation factor
ρ
of (
I.19
)by
first-order approximation. The total areal distortion amounts to zero.
×
End of Corollary.
Proof.
We start from the representation of the left principal stretches for a mapping of conformal type
implemented into, firstly,
J
l
A
, secondly, J
l
. The squared left principal stretches are assumed to be
given by 1 plus a small quantity
μ
except for the dilatation factor
ρ
:
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