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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