PosterPDF Available

Derivation of a polynomial equation for the Natural Earth projection

Poster

Derivation of a polynomial equation for the Natural Earth projection

Abstract

The Natural Earth projection is a new projection for small-scale world maps. The projection defines the length and spacing of parallels in a tabular form for every five degrees of increased latitude and piece-wise cubic spline interpolation is used to project intermediate values. But its implementation into geospatial software requires considerable effort. This barrier prevents projections widespread use and because of that, the only software application where the Natural Earth projection can be computed is Flex Projector, where the projection was designed. This poster presents alternative polynomial equation for the Natural Earth projection, as a result of my graduation thesis. The proposed equation is easier to program and contains inverse projection. At the same time, the polynomial equation also improves the roundness of the corners, a distinguishing mark of the projection. Development of the equation was made in collaboration with designer of the projection, Tom Patterson. It is hope that the publication of this new formula will help the projection find its place in other cartographic projection libraries and software applications.
Polynomial Equations
A (black)
original Natural Eart projection,
B (red)
improved polynomial projection
Goals and Methods
1. Analytical expression for projecting spherical φ / λ
to Cartesian X / Y coordinates
A simple polynomial approximation was developed, which
involved least square adjustments. Preserving the dimensi-
ons of the graticule required the addition of two cons-
traints to these adjustments.
2. Smoothed corners at the end of the pole lines
Shortening the length of the pole lines and reducing the
slope of the meridians improved the roundness of the cor-
ners.
For inverting the projection, Newton's method is used for
computing the latitude φ from the Y coordinate. The lon-
gitude λ is then calculated from the X coordinate.
The Natural Earth Projection
The Natural Earth projection was developed in Flex Projec-
tor by Tom Patterson (U.S. National Park Service). Using a
graphical design approach, he defined the lengths and the
vertical distribution of parallels for every five degrees of
increasing latitude. In the Flex Projector implementation of
the projection, cubic spline interpolation determines the
position of intermediate points.
120°
80°
40°
60°
20°
1.0
2.0
5.0
1.5
Tissot’s indicatricesNatural Earth Isocols of angular distortion Isocols of areal distortion
This true pseudo-cylindrical projection has a distinguish-
ing characteristic – rounded corners where border me-
ridians meet the pole lines. The Natural Earth projection
is neither conformal nor equal area, but has distortion
characteristics comparable to other well-known projec-
tions.
A
B
where:
and are the projected coordinates,
and are the latitude and longitude in radians,
is the radius of the generating globe,
to and to are coefficients given below:
Coefficients for X Coefficients for Y
0.870700 1.007226
-0.131979 0.015085
-0.013791 -0.044475
0.003971 0.028874
-0.001529 -0.005916
Graduation Thesis by Bojan Šavrič
University of Ljubljana, Slovenia
Swiss Federal Institute of Technology Zurich
Institute of Cartography and Geoinformation
Derivation of a Polynomial Equation
for the Natural Earth Projection
Supervisor: Assist. Prof. Dr. Dušan Petrovič (University of Ljubljana, Slovenia)
Co-advisors: Dr. Bernhard Jenny (ETH Zurich, Switzerland, and Oregon State University)
Prof. Dr. Lorenz Hurni (ETH Zurich, Switzerland)
Zurich, June 2011
University of Ljubljana,
Faculty of Civil and Geodetic Engineering
Thesis
Full-text available
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 a tabular form for every five degrees of increased latitude. It is a true 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 graduation thesis introduces alternative polynomial equations that are considerably simpler to compute. The polynomial expression also improves the smoothness of the rounded corners where the meridians meet the horizontal pole lines, a distinguished mark of the Natural Earth projection which suggests to readers that the Earth is spherical in shape. An inverse projection is presented. The formulas are simple to implement in cartographic software and libraries. Distortion values of this new graticule are not significantly different from the original piece-wise projection. The development of the polynomial equations was inspired by a similar study of the Robinson projection. The polynomial coefficients were determined with least square adjustment of indirect observations with additional constraints to preserve the height and width of the graticule. The inverse procedure uses the Newton-Raphson method and converges in a few iterations.
Article
Full-text available
The recently introduced adaptive composite map projection technique changes the projection to the geographic area shown on a map. It is meant as a replacement for the commonly used web Mercator projection, which grossly distorts areas when representing the entire world. The original equal-area version of the adaptive composite map projection technique uses the Lambert azimuthal projection for regional maps and three alternative projections for world maps. Adaptive composite map projections can include a variety of other equal-area projections when the transformation between the Lambert azimuthal and the world projections uses Wagner’s method. To select the most suitable pseudocylindrical projection, the distortion characteristics of a pseudocylindrical projection family are analyzed, and a user study among experts in the area of map projections is carried out. Based on the results of the distortion analysis and the user study, a new pseudocylindrical projection is recommended for extending adaptive composite map projections. The new projection is equal-area throughout the transformation to the Lambert azimuthal projection and has better distortion characteristics then small-scale projections currently included in the adaptive composite map projection technique.
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