The optimal Mercator projection and the optimal polycylindric projection of conformal type

The optimal Mercator projection and the optimal polycylindric projection of conformal type – case-study Indonesia

Grafarend, E. W. / Syffus, R.

Abstract. As a conformal mapping of the sphere S 2R or of the ellipsoid of revolution E 2A,B the Mercator projection maps the equator equidistantly while the transverse Mercator projection maps the transverse metaequator, 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 known as the Universal Transverse Mercator Projection (UTM), which maps the meridian of reference $ Λ_{0} $ with an optimal dilatation factor &ρcirc;=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 &ρcirc;=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.