Natural Earth projection

The natural Earth is defined by the following formulas:

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

where

• ${\displaystyle x\in [-2.73539,2.73539]}$ and ${\displaystyle y\in [-1.42239,1.42239]}$ are the Cartesian coordinates;
• ${\displaystyle \lambda \in [-\pi ,\pi ]}$ is the longitude from the central meridian in radians;
• ${\displaystyle \varphi \in [-\pi /2,\pi /2]}$is the latitude in radians;
• ${\displaystyle l(\varphi )}$ is the length of the parallel at latitude ${\displaystyle \varphi }$;
• ${\displaystyle d(\varphi )}$ is the distance of the parallel from the equator at latitude ${\displaystyle \varphi }$.

${\displaystyle l(\varphi )}$ and ${\displaystyle d(\varphi )}$ 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]

• Robinson projection
• Winkel tripel projection
• Kavrayskiy VII
• Equal Earth projection