Geography Reference
In-Depth Information
by the strip widths of 3
◦
and 6
◦
. As a detailed example, the optimal UPC for the geographic
region of Indonesia is presented as a case-study.
Particular reference is made to
Airy
(
1861
) for the Airy optimality criterion, to
Snyder
(
1987a
)
with respect to the Mercator projection, to
Engels and Grafarend
(
1995
) with respect to the
oblique Mercator Projection, and to
Grafarend
(
1995
) for a review of the Tissot distortion analysis
of a map projection and for the optimal transverse Mercator projection.
I-1 The Optimal Mercator Projection (UM)
Here we present three definitions which relate to the generalized Mercator projection, the Airy
optimal generalized Mercator projection (UM) and finally the generalized Mercator projection
of least total areal distortion. Three lemmas and one corollary describe in detail the optimal
Mercator projection which is finally illustrated by one table, one figure and two examples with
respect to WGS 84.
Definition I.1 (Generalized Mercator projection, mapping equations).
The conformal mapping of the ellipsoid-of-revolution (
I.1
) with semi-major axis
A
1
, semi-minor
axis
A
2
, and relative eccentricity squared
E
2
:= (
A
1
−
A
2
)
/A
1
onto the developed circular cylinder
ρA
1
A
1
,A
2
C
is mapped
equidistantly except for a dilatation factor
ρ
such that the mapping equations (
I.2
) hold with
respect to surface normal coordinates (longitude
Λ
, latitude
Φ
) which parameterize
of radius
ρA
1
is called a
generalized Mercator projection
if the equator of
E
A
1
A
2
E
.
2
A
1
,A
2
X
∈
E
:=
3
:=
X
+
,A
1
>A
2
,
X
2
+
Y
2
A
1
+
Z
2
+
,A
2
∈
R
A
2
=1
,A
1
∈
R
∈
R
(I.1)
x
=
ρA
1
(
Λ
−
Λ
0
)
,
y
=
ρA
1
ln
tan
π
E/
2
.
1
4
+
Φ
E
sin
Φ
1+
E
sin
Φ
−
(I.2)
2
Λ
0
is called the surface normal longitude-of-reference. The plane covered by the chart
{
x, y
}
,
2
,δ
kl
}
Cartesian coordinates, with an Euclidean metric, namely
{
R
(Kronecker delta, unit matrix)
2
ρA
1
is generated by developing the circular cylinder
C
of radius
ρA
1
.
End of Definition.
Lemma I.2 (Generalized Mercator projection, principal stretches).
With respect to the left Tissot distortion measure represented by the matrix C
l
G
−
l
of the left
Cauchy-Green deformation tensor C
l
=J
l
G
r
J
l
multiplied by the inverse of the left metric tensor
G
l
, the matrix of the metric tensor of
2
A
1
,A
2
E
, the left principal stretches of the generalized Mercator
projectionaregivenby
Λ
1
=
Λ
2
=
ρ
1
E
2
sin
2
Φ
cos
Φ
−
.
(I.3)
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