I Mercator projection and polycylindric projection
Optimal Mercator projection and optimal polycylindric projection of conformal type. Case study Indonesia.
Universal Transverse Mercator Projection (UTM), Universal Polycylindric Projection (UPC).
As a conformal mapping of the sphere S
2
R or as a conformal mapping of the ellipsoid-of-revolution
E
2
A 1 ,A 2
, the Mercator projection maps the equator equidistantly while the transverse Mercator projection maps the transverse meta-equator, the meridian-of-reference, with equidistance. Accordingly,
the Mercator projection is very well suited to geographic regions which extend East–West along
the equator. In contrast, the transverse Mercator projection is appropriate for those regions which
have a South–North extension. Like the optimal transverse Mercator projection, which is also known
as the Universal Transverse Mercator Projection (UTM) and which maps the meridian-of-reference
Λ 0 with an optimal dilatation factor ρ = 0.999 578 with respect to the World Geodetic Reference
System (WGS 84) and a strip [Λ 0 − Λ W , Λ 0 − Λ E ] × [Φ S , Φ N ] = [−3.5
◦ , +3.5
◦ ] × [−80
◦ , +84
◦ ], we
construct an optimal dilatation factor ρ for the optimal Mercator projection, summarized as the
Universal Mercator Projection (UM), and an optimal dilatation factor ρ 0 for the optimal polycylindric projection for various strip widths which maps parallel circles Φ 0 equidistantly except for a
dilatation factor ρ 0 summarized as the Universal Polycylindric Projection (UPC). It turns out that
the optimal dilatation factors are independent of the longitudinal extension of the strip and depend
only on the latitude Φ 0 of the parallel circle-of-reference and the southern and northern extension,
namely the latitudes Φ S and Φ N , of the strip. For instance, for a strip [Φ S , Φ N ] = [−1.5
◦ , +1.5
◦ ] along
the equator Φ 0 = 0, the optimal Mercator projection with respect to WGS 84 is characterized by an
optimal dilatation factor ρ = 0.999 887 (strip width 3
◦ ). For other strip widths and different choices
of the parallel circle-of-reference Φ 0 , precise optimal dilatation factors are given. Finally, the UPC for
the geographic region of Indonesia is presented as an example.
Important!
The Mercator projection of the sphere S
2
R or of the ellipsoid-of-revolution E
2
A 1 ,A 2
is, amongst
conformality, characterized by the equidistant mapping of the equator. In contrast, the transverse Mercator projection is conformal and maps the transverse meta-equator, the meridianof-reference, equidistantly. Accordingly, the Mercator projection is very well suited to regions
which extend East–West along the equator, while the transverse Mercator projection fits well
to those regions which have a South–North extension. For geographic regions which are centered along lines neither equatorial, parallel circles, nor meridians, the oblique Mercator
projection according to J. Engels and E. Grafarend (1995) is the conformal mapping which
has to be preferred.
A typical feature of the Universal Transverse Mercator Projection (UTM) is the equidistant mapping
of the central meridian of a zone except for a dilatation factor ρ which is determined by an optimality criterion. As outlined in E. Grafarend (1995), the Airy criterion of a minimal average distortion
over the zone leads to an optimal value of the dilatation factor ρ depending on the strip width. An
Airy optimal dilatation factor ρ, in addition, sets the average areal distortion over the zone to zero,
which is quite a welcome result of an optimal map projection. Here we aim at a similar result for
the Universal Mercator Projection (UM) and for the Universal Polycylindric Projection (UPC): the
classical Mercator projection is designed Airy optimal for a finite zone along the equator. The equator
is equidistantly mapped except for an Airy optimal dilatation factor. In particular, we analyze the
Airy optimal dilatation factor as a function of the strip width. The UM strip is bounded by a southern
as well as a northern parallel circle. While UM with an Airy optimal dilatation factor is well suited
for geographic regions along the equator, the Airy optimal UPC has its merits for those territories
which extend along a parallel circle – as a case study, Indonesia has been chosen. For such a conformal
projection, a chosen parallel circle is equidistantly mapped except for a dilatation factor ρ 0 which is
designed Airy optimal for a zone bounded by a southern as well as a northern parallel circle. For both
types of optimal mapping, namely UM and UPC, the Airy criterion of a minimal average distortion
over the zone produces zero average areal distortion, too.
Optimal Mercator projection and optimal polycylindric projection of conformal type. Case study Indonesia.
Universal Transverse Mercator Projection (UTM), Universal Polycylindric Projection (UPC).
As a conformal mapping of the sphere S
2
R or as a conformal mapping of the ellipsoid-of-revolution
E
2
A 1 ,A 2
, the Mercator projection maps the equator equidistantly while the transverse Mercator projection maps the transverse meta-equator, the meridian-of-reference, with equidistance. Accordingly,
the Mercator projection is very well suited to geographic regions which extend East–West along
the equator. In contrast, the transverse Mercator projection is appropriate for those regions which
have a South–North extension. Like the optimal transverse Mercator projection, which is also known
as the Universal Transverse Mercator Projection (UTM) and which maps the meridian-of-reference
Λ 0 with an optimal dilatation factor ρ = 0.999 578 with respect to the World Geodetic Reference
System (WGS 84) and a strip [Λ 0 − Λ W , Λ 0 − Λ E ] × [Φ S , Φ N ] = [−3.5
◦ , +3.5
◦ ] × [−80
◦ , +84
◦ ], we
construct an optimal dilatation factor ρ for the optimal Mercator projection, summarized as the
Universal Mercator Projection (UM), and an optimal dilatation factor ρ 0 for the optimal polycylindric projection for various strip widths which maps parallel circles Φ 0 equidistantly except for a
dilatation factor ρ 0 summarized as the Universal Polycylindric Projection (UPC). It turns out that
the optimal dilatation factors are independent of the longitudinal extension of the strip and depend
only on the latitude Φ 0 of the parallel circle-of-reference and the southern and northern extension,
namely the latitudes Φ S and Φ N , of the strip. For instance, for a strip [Φ S , Φ N ] = [−1.5
◦ , +1.5
◦ ] along
the equator Φ 0 = 0, the optimal Mercator projection with respect to WGS 84 is characterized by an
optimal dilatation factor ρ = 0.999 887 (strip width 3
◦ ). For other strip widths and different choices
of the parallel circle-of-reference Φ 0 , precise optimal dilatation factors are given. Finally, the UPC for
the geographic region of Indonesia is presented as an example.
Important!
The Mercator projection of the sphere S
2
R or of the ellipsoid-of-revolution E
2
A 1 ,A 2
is, amongst
conformality, characterized by the equidistant mapping of the equator. In contrast, the transverse Mercator projection is conformal and maps the transverse meta-equator, the meridianof-reference, equidistantly. Accordingly, the Mercator projection is very well suited to regions
which extend East–West along the equator, while the transverse Mercator projection fits well
to those regions which have a South–North extension. For geographic regions which are centered along lines neither equatorial, parallel circles, nor meridians, the oblique Mercator
projection according to J. Engels and E. Grafarend (1995) is the conformal mapping which
has to be preferred.
A typical feature of the Universal Transverse Mercator Projection (UTM) is the equidistant mapping
of the central meridian of a zone except for a dilatation factor ρ which is determined by an optimality criterion. As outlined in E. Grafarend (1995), the Airy criterion of a minimal average distortion
over the zone leads to an optimal value of the dilatation factor ρ depending on the strip width. An
Airy optimal dilatation factor ρ, in addition, sets the average areal distortion over the zone to zero,
which is quite a welcome result of an optimal map projection. Here we aim at a similar result for
the Universal Mercator Projection (UM) and for the Universal Polycylindric Projection (UPC): the
classical Mercator projection is designed Airy optimal for a finite zone along the equator. The equator
is equidistantly mapped except for an Airy optimal dilatation factor. In particular, we analyze the
Airy optimal dilatation factor as a function of the strip width. The UM strip is bounded by a southern
as well as a northern parallel circle. While UM with an Airy optimal dilatation factor is well suited
for geographic regions along the equator, the Airy optimal UPC has its merits for those territories
which extend along a parallel circle – as a case study, Indonesia has been chosen. For such a conformal
projection, a chosen parallel circle is equidistantly mapped except for a dilatation factor ρ 0 which is
designed Airy optimal for a zone bounded by a southern as well as a northern parallel circle. For both
types of optimal mapping, namely UM and UPC, the Airy criterion of a minimal average distortion
over the zone produces zero average areal distortion, too.