# A Polynomial Equation for the Natural Earth Projection

Savric, Bojan, Jenny, Bernhard, Patterson, Tom, Petrovic, Dusan, Hurni, Lorenz, Cartography and Geographic Information Science

The Natural Earth Projection

The Natural Earth projection was developed by Torn Patterson in 2007 out of dissatisfaction with existing projections for displaying physical data on small-scale world maps (Jenny et al. 2008). Flex Projector, a freeware application for the interactive design and evaluation of map projections, was the means for creating the Natural Earth projection. The graphical user interface in Flex Projector allows cartographers to adjust the length, shape, and spacing of parallels and meridians of new projections in a graphical design process (Jenny and Patterson 2007).

The Natural Earth projection is an amalgam of the Kavraiskiy VII and Robinson projections, with additional enhancements (Figure 1). These two projections most closely fulfilled the requirement for representing small-scale physical data on world maps, but each had at least one undesirable characteristic (Jenny et al. 2008). The Kavraiskiy VII projection exaggerates the size of high latitude areas, resulting in oversized representation of polar regions. The Robinson projection, on the other hand, has a height-to-width ratio close to 0.5, resulting in a slightly too wide graticule with outward bulging sides and too much shape distortion near the map edges. Creating the Natural Earth projection required three major adjustments: Firstly, starting from the Robinson projection, its vertical extension was slightly increased to give it more height. Secondly, using the Kavraiskiy VII as a template, the parallels were slightly increased in length.

And thirdly, the length of the pole lines was decreased by a small amount to give the corners at pole lines a rounded appearance. Designing the Natural Earth projection in this way required trial-and-error experimentation and visual assessment of the appearance of continents in an iterative process (Jenny et al. 2008). The result of this procedure, the Natural Earth projection, is a true pseudocylindrical projection, i.e., a projection with regularly distributed meridians and straight parallels (Snyder 1993:189). As a compromise projection, the Natural Earth projection is neither conformal nor equal area, but its distortion characteristics are comparable to other well known projections (Jenny et al. 2008). All three projections exaggerate the size of high latitude areas (Figure 1). Appendix A provides further details about the distortion characteristics of the Natural Earth projection. The shape of the graticule of any projection designed with Flex Projector is defined by tabular sets of parameters. For the Natural Earth projection, two parameter sets are used for specifying (1) the relative length of the parallels, and (2) the relative distance of parallels from the equator. Equation 1 defines the original Natural Earth projection, transforming spherical coordinates into Cartesian X/Y coordinates, and Table 1 provides the parameter values (Jenny et al. 2008; 2010):

X = R x s x [l.sub.[phi]] x [lambda] [l.sub.[phi]] [member of] [0, 1], (Eq. 1)

Y = R x s x [d.sub.[phi]] x k x [pi] [d.sub.[phi]] [member of] [-1, 1],

where:

X and Y are projected coordinates;

R is the radius of the generating globe;

s = 0.8707 is an internal scale factor;

[l.sub.[phi] is the relative length of the parallel at latitude [phi], with [phi] [member of] [-[pi]/2, [pi]/2], [l.sub.[phi]] = 1 for the equator and the slope of [l.sub.[phi]] is 63.883[degrees] at the poles;

[d.sub.[phi]] is the relative distance of the parallel at latitude [phi] from the equator, with [phi] [member of] [-[pi]/2, [pi]/2] and with [d.sub.[phi]] = [+ or -] 1 for the pole lines, and [d.sub.[phi]] = 0 for the equator;

[lambda] is the longitude with [lambda] [member of] [-[pi], [pi]]; and

k = 0.52 is the height-to-width ratio of the projection.

Arthur H. Robinson proposed the structure of Equation 1 and the associated graphical approach to the design of small-scale map projections when he developed his eponymous projection (Robinson 1974). …

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