Natural Earth projection

https://en.wikipedia.org/wiki/Natural_Earth_projection
Natural Earth projection of the world.
The natural Earth projection with Tissot's indicatrix of deformation

The Natural Earth projection is a pseudocylindrical map projection designed by Tom Patterson and introduced in 2012. It is neither conformal nor equal-area, but a compromise between the two.

It was designed in Flex Projector, a specialized software application that offers a graphical approach for the creation of new projections. [1] [2]

Definition

The natural Earth is defined by the following formulas:

{\displaystyle {\begin{aligned}x&=l(\varphi )\times \lambda ,\\y&=d(\varphi ),\end{aligned}}},

where

• 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: [3]

{\displaystyle {\begin{aligned}l(\varphi )&=0.870700-0.131979\times \varphi ^{2}-0.013791\times \varphi ^{4}+0.003971\times \varphi ^{10}-0.001529\times \varphi ^{12},\\d(\varphi )&=\varphi \times (1.007226+0.015085\times \varphi ^{2}-0.044475\times \varphi ^{6}+0.028874\times \varphi ^{8}-0.005916\times \varphi ^{10}).\end{aligned}}}

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". [1]