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A Polynomial Equation for the Natural Earth Projection Bojan Šavrič

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A Polynomial Equation for the Natural Earth Projection Bojan Šavrič A Polynomial Equation for the Natural Earth Projection Bojan Šavrič, Bernhard Jenny, Tom Patterson, Dušan Petrovič, Lorenz Hurni ABSTRACT: The Natural Earth projection is a new projection for representing the entire Earth on small-scale maps. It was designed in Flex Projector, a specialized software application that offers a graphical approach for the creation of new projections. The original Natural Earth projection defines the length and spacing of parallels in tabular form for every five degrees of increasing latitude. It is a pseudocylindrical projection, and is neither conformal nor equal-area. In the original definition, piece-wise cubic spline interpolation is used to project intermediate values that do not align with the five-degree grid. This paper introduces alternative polynomial equations that closely approximate the original projection. The polynomial equations are considerably simpler to compute and program, and require fewer parameters, which should facilitate the implementation of the Natural Earth projection in geospatial software. The polynomial expression also improves the smoothness of the rounded corners where the meridians meet the horizontal pole lines, a distinguishing trait of the Natural Earth projection that suggests to readers that the Earth is spherical in shape. Details on the least squares adjustment for obtaining the polynomial formulas are provided, including constraints for preserving the geometry of the graticule. This technique is applicable to similar projections that are defined by tabular parameters. For inverting the polynomial projection the Newton-Raphson root finding algorithm is suggested. KEYWORDS: Projections, Natural Earth projection, Flex Projector The Natural Earth Projection he Natural Earth projection was The Natural Earth projection is an amalgam of developed by Tom Patterson in 2007 out the Kavraiskiy VII and Robinson projections, Tof dissatisfaction with existing projections with additional enhancements (Figure 1). for displaying physical data on small-scale world These two projections most closely fulfilled the maps (Jenny et al. 2008). Flex Projector, a requirement for representing small-scale physical freeware application for the interactive design data on world maps, but each had at least one and evaluation of map projections, was the Copyright (c) Cartography and Geographic Information Society. All rights reserved. undesirable characteristic (Jenny et al. 2008). means for creating the Natural Earth projection. The Kavraiskiy VII projection exaggerates the The graphical user interface in Flex Projector size of high latitude areas, resulting in oversized allows cartographers to adjust the length, shape, representation of polar regions. The Robinson and spacing of parallels and meridians of new projection, on the other hand, has a height-to- Delivered by Publishing Technology to: Oregon State University IP: 128.193.162.72 on: Thu, 01 Nov 2012 16:50:57 projections in a graphical design process (Jenny width ratio close to 0.5, resulting in a slightly too and Patterson 2007). wide graticule with outward bulging sides and Bojan Šavrič and Dušan Petrovič, Faculty of Civil and Geo- too much shape distortion near the map edges. detic Engineering, University of Ljubljana, Slovenia, Email: <[email protected]>, <[email protected]>; Bern- Creating the Natural Earth projection required hard Jenny, College of Earth, Ocean and Atmospheric Sciences, three major adjustments: Firstly, starting from Oregon State University, Corvallis, Oregon, USA, Email: <jennyb@ geo.oregonstate.edu>; Tom Patterson, US National Park Service, the Robinson projection, its vertical extension Harpers Ferry, West Virginia, USA, Email: <tom_patterson@nps. was slightly increased to give it more height. gov>; Lorenz Hurni, Institute of Cartography and Geoinformation, Secondly, using the Kavraiskiy VII as a template, ETH Zürich, Switzerland, Email: <[email protected]>. DOI: http://dx.doi.org/10.1559/15230406384363 the parallels were slightly increased in length. Cartography and Geographic Information Science, Vol. 38, No. 4, 2011, pp. 363-372 And thirdly, the length of the pole lines was Table 1. Parameters for the Natural Earth projection: Rela- decreased by a small amount to give the corners tive lengths of parallels and relative distance from the equator for every 5 degrees (after Jenny et al. 2008). at pole lines a rounded appearance. Designing Latitude Relative length of Relative distance of the Natural Earth projection in this way required [degrees] parallels parallels from equator trial-and-error experimentation and visual 0 1 0 assessment of the appearance of continents in 5 0.988 0.062 an iterative process (Jenny et al. 2008). The result of this procedure, the Natural Earth projection, 10 0.9953 0.124 is a true pseudocylindrical projection, i.e., a 15 0.9894 0.186 projection with regularly distributed meridians 20 0.9811 0.248 and straight parallels (Snyder 1993:189). As 25 0.9703 0.310 a compromise projection, the Natural Earth 30 0.9570 0.372 projection is neither conformal nor equal area, 35 0.9409 0.434 but its distortion characteristics are comparable 40 0.9222 0.4958 to other well known projections (Jenny et al. 45 0.9006 0.5571 2008). All three projections exaggerate the size 50 0.8763 0.6176 of high latitude areas (Figure 1). Appendix A 55 0.8492 0.6769 provides further details about the distortion 60 0.8196 0.7346 characteristics of the Natural Earth projection. 65 0.7874 0.7903 The shape of the graticule of any projection 70 0.7525 0.8435 designed with Flex Projector is defined by 75 0.7160 0.8936 tabular sets of parameters. For the Natural 80 0.6754 0.9394 Earth projection, two parameter sets are used for specifying (1) the relative length of the parallels, 85 0.6270 0.9761 and (2) the relative distance of parallels from 90 0.5630 1 the equator. Equation 1 defines the original (Robinson 1974). In making the Natural Earth Natural Earth projection, transforming spherical projection, Jenny et al. (2010) provide numerical coordinates into Cartesian X/Y coordinates, and values for the tabular parameters that define lᵩ Table 1 provides the parameter values (Jenny et and dᵩ in Equation 1 for every five degrees. For al. 2008; 2010): intermediate spherical coordinates that do not X = R ∙ s ∙ lᵩ ∙ λ lᵩ ∈ [0, 1], (Eq. 1) align with the five-degree grid, values for lᵩ and Y = R ∙ s ∙ dᵩ ∙ k ∙ π dᵩ ∈ [-1, 1], dᵩ need to be interpolated. The Flex Projector where: application uses a piece-wise cubic spline X and Y are projected coordinates; interpolation, with each piece of the spline R is the radius of the generating globe; curve covering five degrees. While this type of s = 0.8707 is an internal scale factor; interpolation is rapid to evaluate, it is relatively lᵩ is the relative length of the parallel at latitude intricate to program and requires a large number Copyright (c) Cartography and Geographic Information Society. All rights reserved. φ, with φ ∈ [-π/2, π/2], lᵩ = 1 for the equator of parameters—factors that are likely to impede and the slope of lᵩ is 63.883° at the poles; the widespread implementation of the Natural dᵩ is the relative distance of the parallel at latitude Earth projection in geospatial software. Seeking φ from the equator, with φ ∈ [-π/2, π/2] and greater efficiency, the remainder of this paper with dᵩ = ±1 for the pole lines, and dᵩ = 0 for Delivered by Publishing Technology to: Oregon State University IP: 128.193.162.72 on: Thu, 01 Nov 2012 16:50:57 discusses a compact analytical expression that the equator; approximates Equation 1 with two simple poly- λ is the longitude with λ ∈ [-π, π]; and k = 0.52 is the height-to-width ratio of the nomial expressions. projection. Analytical Expressions for the Arthur H. Robinson proposed the structure of Robinson Projection Equation 1 and the associated graphical approach Robinson and Patterson used an identical to the design of small-scale map projections approach for the design of their pseudocylindrical when he developed his eponymous projection projections. Both defined their projection by Vol. 38, No. 4 364 Kavraiskiy VII Natural Earth Robinson Figure 1: The polynomial Natural Earth projection compared to the Kavraiskiy VII and Robinson projections (after Jenny et al. 2008). if deviations to the approximated values are small. Canters and Decleir (1989) present two adjusting the appearance of the projected five- polynomial equations for approximating the degree graticule in an iterative process—Robinson Robinson projection (Equation 2). For the X sketching the graticule with pen and paper, and coordinates they use even powers up to the order Patterson fine-tuning it in Flex Projector. In the four, and for the Y coordinates odd powers up past, various authors have tackled the problem to the order five. Each expression contains three of finding an analytical expression for the coefficients, and the constants k, s and π of Robinson projection. Since the two projections Equation 1 are integrated with lᵩ and dᵩ. Their are closely related, this section reviews existing solution contains only six parameters, and is fast mathematical models of Robinson’s projection. and simple to compute. Polynomial approximation is recommended, 2 4 X = R ∙ λ ∙ (A0 + A2 ∙ φ + A4 ∙ φ ) (Eq. 2) which is applied to the Natural Earth projection 3 5 Y = R ∙ (A1 ∙ φ + A3 ∙ φ + A5 ∙ φ ) in the next section. where: Two general approaches exist for mathemati- X and Y are projected coordinates; cally modeling graphically defined projections: φ and λ are the latitude and longitude; (1) interpolation and (2) approximation. The R is the radius of the generating globe; Robinson projection has had both approaches A0 = 0.8507; applied. A1 = 0.9642; Interpolating methods use a function that A2 = -0.1450; passes exactly through the reference points. A3 = -0.0013; Ipbüker (2004; 2005) presents a method based A4 = -0.0104; and on multiquadric interpolation for the forward A5 = -0.0129.
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