Natural Earth projection
This article is missing information about distortion, software support and the polynomial.(February 2017)
The natural Earth is defined by the following formulas:
- x∈[-2.73539, 2.73539] and y∈[-1.42239, 1.42239] are the Cartesian coordinates;
- 𝜆∈[-π, π] is the longitude from the central meridian in radians;
- φ∈[-π/2, π/2] is the latitude in radians;
- l(φ) is the length of the parallel at latitude φ;
- d(φ) is the distance of the parallel from the equator at latitude φ.
l(φ) and d(φ) are given as polynomials: 
In the original definition of the projection, planar coordinates were lineally interpolated from a table of 19 latitudes and then multiplied by other factors. The authors of the projection later provided a polynomial representation that closely matches the original but improves smoothness at the "corners". 
- Šavrič, Bojan; Jenny, Bernhard; Patterson, Tom; Petrovič, Dušan; Hurni, Lorenz (February 17, 2012). "A Polynomial Equation for the Natural Earth Projection" (PDF). Oregon State University. Archived from the original (PDF) on 2016-03-03. Retrieved January 24, 2020.
- Jenny, Bernhard; Patterson, Tom; Hurni, Lorenz (2008). "Flex Projector–Interactive Software for Designing World Map Projections". Cartographic perspectives. Retrieved January 24, 2020.
"Natural Earth Projection: Home". www.shadedrelief.com. Archived from
the original on 2012-04-07. Retrieved 2017-02-12.
It was originally designed in Flex Projector using graphical methods and now exists as a polynomial version.