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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