Abstract
Abstract. Seeking low distortion maps, it is usual to assume that areal and angular distortions are equally undesirable on the map. However, this might not be the case for certain map thematics. Should angular distortions be a bit less preferred to areal distortions, maps of unbalanced distortions may be developed. In this paper, the known analytic solution for the best cylindrical map projection is extended to such more general requirements by utilizing calculus of variations. The overall distortion of the resulted mappings are calculated and compared to each other to explore the distortion characteristics of these intentionally unbalanced map projections.
Highlights
A map projection is cylindrical if it maps parallel lines to parallel straight lines and meridians to proportionally spaced straight lines that are orthogonal to the images of parallels (Lapaine and Frančula, 2016)
The formula of the best cylindrical map projection was obtained for spherical zones even when the undesirability of areal and angular distortions were unequal
These mappings were found to be apt in traditional cartography
Summary
A map projection is cylindrical if it maps parallel lines to parallel straight lines and meridians to proportionally spaced straight lines that are orthogonal to the images of parallels (Lapaine and Frančula, 2016). All three variants may have two distortion-free parallels symmetrical to the Equator, the optimal choice (resulting in the least distortion possible) was observed to be the same for all three variants (Frančula, 1971; Grafarend and Niermann, 1984) While these solutions are well-known if areal and angular distortions are undesired, one may try to avoid areal distortion more than angular distortion. Former studies all optimized maps using distortion criteria that aimed to balance all kinds of distortions, but the cartographer may want to fine-tune the ratio of areal and angular distortions if he would like to avoid one of them more than the other In this paper, these results will be extended: the formula of the cylindrical map projections with the least distortion possible will be developed even if areal and angular distortions are not undesired. This will show a symmetrical behaviour of areal and angular distortions on cylindrical mappings
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