624 I Mercator projection and polycylindric projection
Section I-1.
In detail, Section I-1 focuses on the optimal Mercator projection of the ellipsoid-of-revolution E
2
A 1 ,A 2
with respect to the WGS 84. Figure I.1 displays the Airy optimal dilatation factor as a function of
the strip width, while Table I.1 lists various optimal dilatation factors for the given strip widths 3
◦ ,
6
◦ , 12
◦ , 20
◦ .
Section I-2.
In contrast, Section I-2 presents the optimal polycylindric projection of conformal type. Figure I.2
displays various Airy optimal dilatation factors for given parallel circles-of-reference parameterized by
the ellipsoidal latitude Φ 0 and the strip width Φ N −Φ S of northern and southern boundaries. Tables I.2
and I.3 are detailed lists of various optimal dilatation factors in different zones sorted 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 G. B. Airy (1861) for the Airy optimality criterion, to Snyder (1987)
with respect to the Mercator projection, to J. Engels and E. Grafarend (1995) with respect to the
oblique Mercator Projection, and to E. 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
2
1 − A
2
2 )/A
2
1 onto the developed circular cylinder C
2
ρA 1
of
radius ρA 1 is called a generalized Mercator projection if the equator of E
2
A 1 ,A 2
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 E
2
A 1 ,A 2
.
X ∈ E
2
A 1 ,A 2
:=
:=
X ∈ R
3
X
2 + Y
2
A 2
1
+
Z
2
A 2
2
= 1, A 1 ∈ R
+ , A 2 ∈ R
+ , A 1 > A 2
,
(I.1)
x = ρA 1 (Λ − Λ 0 ) ,
y = ρA 1 ln
tan
π
4
+
Φ
2
1 − E sin Φ
1 + E sin Φ
E/2
.
(I.2)
Λ 0 is called the surface normal longitude-of-reference. The plane covered by the chart {x, y}, Cartesian
coordinates, with an Euclidean metric, namely {R
2 , δ kl } (Kronecker delta, unit matrix) is generated
by developing the circular cylinder C
2
ρA 1
of radius ρA 1 .
End of Definition.
Section I-1.
In detail, Section I-1 focuses on the optimal Mercator projection of the ellipsoid-of-revolution E
2
A 1 ,A 2
with respect to the WGS 84. Figure I.1 displays the Airy optimal dilatation factor as a function of
the strip width, while Table I.1 lists various optimal dilatation factors for the given strip widths 3
◦ ,
6
◦ , 12
◦ , 20
◦ .
Section I-2.
In contrast, Section I-2 presents the optimal polycylindric projection of conformal type. Figure I.2
displays various Airy optimal dilatation factors for given parallel circles-of-reference parameterized by
the ellipsoidal latitude Φ 0 and the strip width Φ N −Φ S of northern and southern boundaries. Tables I.2
and I.3 are detailed lists of various optimal dilatation factors in different zones sorted 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 G. B. Airy (1861) for the Airy optimality criterion, to Snyder (1987)
with respect to the Mercator projection, to J. Engels and E. Grafarend (1995) with respect to the
oblique Mercator Projection, and to E. 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
2
1 − A
2
2 )/A
2
1 onto the developed circular cylinder C
2
ρA 1
of
radius ρA 1 is called a generalized Mercator projection if the equator of E
2
A 1 ,A 2
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 E
2
A 1 ,A 2
.
X ∈ E
2
A 1 ,A 2
:=
:=
X ∈ R
3
X
2 + Y
2
A 2
1
+
Z
2
A 2
2
= 1, A 1 ∈ R
+ , A 2 ∈ R
+ , A 1 > A 2
,
(I.1)
x = ρA 1 (Λ − Λ 0 ) ,
y = ρA 1 ln
tan
π
4
+
Φ
2
1 − E sin Φ
1 + E sin Φ
E/2
.
(I.2)
Λ 0 is called the surface normal longitude-of-reference. The plane covered by the chart {x, y}, Cartesian
coordinates, with an Euclidean metric, namely {R
2 , δ kl } (Kronecker delta, unit matrix) is generated
by developing the circular cylinder C
2
ρA 1
of radius ρA 1 .
End of Definition.