# Projections

## Plate Carrée (Equirectangular, φ1 = 0°)

$\mathit{x}=\frac{\mathit{\lambda }}{2\cdot \mathit{\pi }}$ $\mathit{y}=\frac{\mathit{\phi }}{2\cdot \mathit{\pi }}$

## Equirectangular, φ1 = 15°

$\mathit{x}=\frac{\mathit{\lambda }}{2\cdot \mathit{\pi }}$ $\mathit{y}=\frac{\mathit{\phi }\cdot \sqrt{2}\cdot \left(\sqrt{3}-1\right)}{2\cdot \mathit{\pi }}$

## Equirectangular, φ1 = 30°

$\mathit{x}=\frac{\mathit{\lambda }}{2\cdot \mathit{\pi }}$ $\mathit{y}=\frac{\mathit{\phi }\cdot \sqrt{3}}{3\cdot \mathit{\pi }}$

## Plate Carrée (Equirectangular, φ1 = 45°)

$\mathit{x}=\frac{\mathit{\lambda }}{2\cdot \mathit{\pi }}$ $\mathit{y}=\frac{\mathit{\phi }\cdot \sqrt{2}}{2\cdot \mathit{\pi }}$

## Equirectangular, φ1 = 60°

$\mathit{x}=\frac{\mathit{\lambda }}{2\cdot \mathit{\pi }}$ $\mathit{y}=\frac{\mathit{\phi }}{\mathit{\pi }}$

## Mercator

$\mathit{x}=\frac{\mathit{\lambda }}{2\cdot \mathit{\pi }}$ $\mathit{y}=\frac{\mathrm{ln}\left(\mathrm{tan}\left(\frac{\mathit{\pi }}{4}+\frac{\mathit{\phi }}{2}\right)\right)}{2\cdot \mathit{\pi }}$

## Gall Stereographic

$\mathit{x}=\frac{\mathit{\lambda }}{2\cdot \mathit{\pi }}$ $\mathit{y}=\frac{\mathrm{tan}\left(\frac{\mathit{\phi }}{2}\right)\cdot \left(\sqrt{2}+1\right)}{2\cdot \mathit{\pi }}$

## Miller

$\mathit{x}=\frac{\mathit{\lambda }}{2\cdot \mathit{\pi }}$ $\mathit{y}=\frac{5\cdot \mathrm{ln}\left(\mathrm{tan}\left(\frac{\mathit{\pi }}{4}+\frac{2\cdot \mathit{\phi }}{5}\right)\right)}{8\cdot \mathit{\pi }}$

## Lambert (Cylindrical Equal Area, φ1 = 0°)

$\mathit{x}=\frac{\mathit{\lambda }}{2\cdot \mathit{\pi }}$ $\mathit{y}=\frac{\mathrm{sin}\left(\mathit{\phi }\right)}{2\cdot \mathit{\pi }}$

## Behrmann (Cylindrical Equal Area, φ1 = 30°

$\mathit{x}=\frac{\mathit{\lambda }}{2\cdot \mathit{\pi }}$ $\mathit{y}=\frac{2\cdot \mathrm{sin}\left(\mathit{\phi }\right)}{3\cdot \mathit{\pi }}$

## Hobo-Dyer (Cylindrical Equal Area, φ1 = 37°30'

$\mathit{x}=\frac{\mathit{\lambda }}{2\cdot \mathit{\pi }}$ $\mathit{y}=\frac{\mathrm{sin}\left(\mathit{\phi }\right)}{{\left(\mathrm{cos}\left(\frac{5\cdot \mathit{\pi }}{24}\right)\right)}^{2}\cdot 2\cdot \mathit{\pi }}$

## Gall-Peters (Cylindrical Equal Area, φ1 = 45°)

$\mathit{x}=\frac{\mathit{\lambda }}{2\cdot \mathit{\pi }}$ $\mathit{y}=\frac{\mathrm{sin}\left(\mathit{\phi }\right)}{\mathit{\pi }}$

## Balthasart (Cylindrical Equal Area, φ1 = 50°)

$\mathit{x}=\frac{\mathit{\lambda }}{2\cdot \mathit{\pi }}$ $\mathit{y}=\frac{\mathrm{sin}\left(\mathit{\phi }\right)}{{\left(\mathrm{cos}\left(\frac{5\cdot \mathit{\pi }}{18}\right)\right)}^{2}\cdot 2\cdot \mathit{\pi }}$

## Sinusoidal

$\mathit{x}=\frac{\mathit{\lambda }\cdot \mathrm{cos}\left(\mathit{\phi }\right)}{2\cdot \mathit{\pi }}$ $\mathit{y}=\frac{\mathit{\phi }}{2\cdot \mathit{\pi }}$

## Mollweide

$2\cdot \theta +\mathrm{sin}\left(2\cdot \mathit{\theta }\right)=\mathit{\pi }\cdot \mathrm{sin}\left(\mathit{\phi }\right)$ $\mathit{x}=\frac{\mathit{\lambda }\cdot \mathrm{cos}\left(\mathit{\theta }\right)}{2\cdot \mathit{\pi }}$ $\mathit{y}=\frac{\mathrm{sin}\left(\mathit{\theta }\right)}{4}$

## Robinson

$\begin{array}{ccc}\mathit{Latitude}& \mathit{XScale}& \mathit{YScale}\\ 90°& 0.5322& 1.0000\\ 85°& 0.5722& 0.9761\\ 80°& 0.6213& 0.9394\\ 75°& 0.6732& 0.8936\\ 70°& 0.7186& 0.8435\\ 65°& 0.7597& 0.7903\\ 60°& 0.7986& 0.7346\\ 55°& 0.8350& 0.6769\\ 50°& 0.8679& 0.6176\\ 45°& 0.8962& 0.5571\\ 40°& 0.9216& 0.4958\\ 35°& 0.9427& 0.4340\\ 30°& 0.9600& 0.3720\\ 25°& 0.9730& 0.3100\\ 20°& 0.9822& 0.2480\\ 15°& 0.9900& 0.1860\\ 10°& 0.9954& 0.1240\\ 5°& 0.9986& 0.0620\\ 0°& 1.0000& 0.0000\\ -5°& 0.9986& -0.0620\\ -10°& 0.9954& -0.1240\\ -15°& 0.9900& -0.1860\\ -20°& 0.9822& -0.2480\\ -25°& 0.9730& -0.3100\\ -30°& 0.9600& -0.3720\\ -35°& 0.9427& -0.4340\\ -40°& 0.9216& -0.4958\\ -45°& 0.8962& -0.5571\\ -50°& 0.8679& -0.6176\\ -55°& 0.8350& -0.6769\\ -60°& 0.7986& -0.7346\\ -65°& 0.7597& -0.7903\\ -70°& 0.7186& -0.8435\\ -75°& 0.6732& -0.8936\\ -80°& 0.6213& -0.9394\\ -85°& 0.5722& -0.9761\\ -90°& 0.5322& -1.0000\end{array}$ $\mathit{x}=\frac{\mathit{XScale}\cdot \mathit{\lambda }}{2\cdot \mathit{\pi }}$ $\mathit{y}=\frac{13523\cdot \mathit{YScale}}{16974\cdot \mathit{\pi }}$

## NaturalEarth

$\begin{array}{ccc}\mathit{Latitude}& \mathit{XScale}& \mathit{YScale}\\ 90°& 0.5630& 1.0000\\ 85°& 0.6270& 0.9761\\ 80°& 0.6754& 0.9394\\ 75°& 0.7160& 0.8936\\ 70°& 0.7525& 0.8435\\ 65°& 0.7874& 0.7903\\ 60°& 0.8196& 0.7346\\ 55°& 0.8492& 0.6769\\ 50°& 0.8763& 0.6176\\ 45°& 0.9006& 0.5571\\ 40°& 0.9222& 0.4958\\ 35°& 0.9409& 0.4340\\ 30°& 0.9570& 0.3720\\ 25°& 0.9703& 0.3100\\ 20°& 0.9811& 0.2480\\ 15°& 0.9894& 0.1860\\ 10°& 0.9953& 0.1240\\ 5°& 0.9988& 0.0620\\ 0°& 1.0000& 0.0000\\ -5°& 0.9988& -0.0620\\ -10°& 0.9953& -0.1240\\ -15°& 0.9894& -0.1860\\ -20°& 0.9811& -0.2480\\ -25°& 0.9703& -0.3100\\ -30°& 0.9570& -0.3720\\ -35°& 0.9409& -0.4340\\ -40°& 0.9222& -0.4958\\ -45°& 0.9006& -0.5571\\ -50°& 0.8763& -0.6176\\ -55°& 0.8492& -0.6769\\ -60°& 0.8196& -0.7346\\ -65°& 0.7874& -0.7903\\ -70°& 0.7525& -0.8435\\ -75°& 0.7160& -0.8936\\ -80°& 0.6754& -0.9394\\ -85°& 0.6270& -0.9761\\ -90°& 0.5630& -1.0000\end{array}$ $\mathit{x}=\frac{\mathit{XScale}\cdot \mathit{\lambda }}{2\cdot \mathit{\pi }}$ $\mathit{y}=\frac{13\cdot \mathit{YScale}}{50}$

## Kavrayskiy VII

$\mathit{x}=\frac{\mathit{\lambda }\cdot \sqrt{1-3\cdot {\left(\frac{\mathit{\phi }}{\mathit{\pi }}\right)}^{2}}}{2\cdot \mathit{\pi }}$ $\mathit{y}=\frac{\mathit{\phi }\cdot \sqrt{3}}{3\cdot \mathit{\pi }}$

## Wagner VI

$\mathit{x}=\frac{\mathit{\lambda }\cdot \sqrt{1-3\cdot {\left(\frac{\mathit{\phi }}{\mathit{\pi }}\right)}^{2}}}{2\cdot \mathit{\pi }}$ $\mathit{y}=\frac{\mathit{\phi }}{2\cdot \mathit{\pi }}$

## Aitoff

$\mathit{\alpha }=\mathrm{arccos}\left(\mathrm{cos}\left(\mathit{\phi }\right)\cdot \mathrm{cos}\left(\frac{\mathit{\lambda }}{2}\right)\right)$ $\mathit{x}=\frac{\mathrm{cos}\left(\mathit{\phi }\right)\cdot \mathrm{sin}\left(\frac{\mathit{\lambda }}{2}\right)}{\mathrm{sinc}\left(\mathit{\alpha }\right)\cdot \mathit{\pi }}$ $\mathit{y}=\frac{\mathrm{sin}\left(\mathit{\phi }\right)}{\mathrm{sinc}\left(\mathit{\alpha }\right)\cdot 2\cdot \mathit{\pi }}$

## Hammer

$\mathit{x}=\frac{\mathrm{cos}\left(\mathit{\phi }\right)\cdot \mathrm{sin}\left(\frac{\mathit{\lambda }}{2}\right)}{2\cdot \sqrt{1+\mathrm{cos}\left(\mathit{\phi }\right)\cdot \mathrm{cos}\left(\frac{\mathit{\lambda }}{2}\right)}}$ $\mathit{y}=\frac{\mathrm{sin}\left(\mathit{\phi }\right)}{4\cdot \sqrt{1+\mathrm{cos}\left(\mathit{\phi }\right)\cdot \mathrm{cos}\left(\frac{\mathit{\lambda }}{2}\right)}}$

## Winkel I

$\mathit{x}=\frac{\mathit{\lambda }\cdot \left(\mathrm{cos}\left(\mathit{\phi }\right)\cdot \mathit{\pi }+2\right)}{2\cdot \mathit{\pi }\cdot \left(\mathit{\pi }+2\right)}$ $\mathit{y}=\frac{\mathit{\phi }}{\mathit{\pi }+2}$

## Winkel Tripel

$\mathit{\alpha }=\mathrm{arccos}\left(\mathrm{cos}\left(\mathit{\phi }\right)\cdot \mathrm{cos}\left(\frac{\mathit{\lambda }}{2}\right)\right)$ $\mathit{x}=\frac{\mathrm{sinc}\left(\mathit{\alpha }\right)\cdot \mathit{\lambda }+\mathrm{cos}\left(\mathit{\phi }\right)\cdot \mathrm{sin}\left(\frac{\mathit{\lambda }}{2}\right)\cdot \mathit{\pi }}{\mathrm{sinc}\left(\mathit{\alpha }\right)\cdot \mathit{\pi }\cdot \left(\mathit{\pi }+2\right)}$ $\mathit{y}=\frac{\mathrm{sin}\left(\mathit{\phi }\right)+\mathit{\phi }\cdot \mathrm{sinc}\left(\mathit{\alpha }\right)}{2\cdot \mathrm{sinc}\left(\mathit{\alpha }\right)\cdot \left(\mathit{\pi }+2\right)}$

## Van der Grinten

$\mathit{\theta }=\mathrm{arcsin}\left(|\frac{2\cdot \mathit{\phi }}{\mathit{\pi }}|\right)$ $\mathit{A}=\frac{|\frac{\mathit{\pi }}{\mathit{\lambda }}-\frac{\mathit{\lambda }}{\mathit{\pi }}|}{2}$ $\mathit{G}=\frac{\mathrm{cos}\left(\mathit{\theta }\right)}{\mathrm{sin}\left(\mathit{\theta }\right)+\mathrm{cos}\left(\mathit{\theta }\right)-1}$ $\mathit{P}=\mathit{G}\cdot \left(\frac{2}{\mathrm{sin}\left(\mathit{\theta }\right)}-1\right)$ $\mathit{Q}={\mathit{A}}^{2}+\mathit{G}$ $\mathit{x}=\left\{\begin{array}{l}\frac{\mathit{\lambda }}{2\cdot \mathit{\pi }},\mathit{\phi }=0\\ 0,\mathit{\lambda }=0\vee \mathit{\phi }=±\frac{\mathit{\pi }}{2}\\ \frac{\mathrm{sign}\left(\mathit{\lambda }\right)\cdot |\mathit{A}\cdot \left(\mathit{G}-{\mathit{P}}^{2}\right)+\sqrt{{\mathit{A}}^{2}\cdot {\left(\mathit{G}-{\mathit{P}}^{2}\right)}^{2}-\left({\mathit{P}}^{2}+{\mathit{A}}^{2}\right)\cdot \left({\mathit{G}}^{2}\cdot {\mathit{P}}^{2}\right)}|}{2\cdot \left({\mathit{P}}^{2}+{\mathit{A}}^{2}\right)},\text{otherwise}\end{array}$ $\mathit{y}=\left\{\begin{array}{l}0,\mathit{\phi }=0\\ \frac{\mathrm{sign}\left(\mathit{\phi }\right)\cdot \mathrm{tan}\left(\frac{\mathit{\theta }}{2}\right)}{2},\mathit{\lambda }=0\vee \mathit{\phi }=±\frac{\mathit{\pi }}{2}\\ \frac{\mathrm{sign}\left(\mathit{\phi }\right)\cdot |\mathit{P}\cdot \mathit{Q}-\mathit{A}\cdot \sqrt{\left({\mathit{A}}^{2}+1\right)\cdot \left({\mathit{P}}^{2}+{\mathit{A}}^{2}\right)-{\mathit{Q}}^{2}}|}{2\cdot \left({\mathit{P}}^{2}+{\mathit{A}}^{2}\right)},\text{otherwise}\end{array}$

## Eckert II

$\mathit{x}=\frac{\mathit{\lambda }\cdot \sqrt{4-3\cdot \mathrm{sin}\left(|\mathit{\phi }|\right)}}{4\cdot \mathit{\pi }}$ $\mathit{y}=\frac{\mathrm{sgn}\left(\mathit{\phi }\right)\cdot \left(2-\sqrt{4-3\cdot \mathrm{sin}\left(|\mathit{\phi }|\right)}\right)}{4}$

## Eckert V

$\mathit{x}=\frac{\mathit{\lambda }\cdot \left(1+\mathrm{cos}\left(\mathit{\phi }\right)\right)}{4\cdot \mathit{\pi }}$ $\mathit{y}=\frac{\mathit{\phi }}{2\cdot \mathit{\pi }}$

## Azimuthal Equidistant (Polar)

$\mathit{x}=\mathrm{sin}\left(\mathit{\lambda }\right)\cdot \left(0.25-\frac{\mathit{\phi }}{2\cdot \mathit{\pi }}\right)$ $\mathit{y}=-\mathrm{cos}\left(\mathit{\lambda }\right)\cdot \left(0.25-\frac{\mathit{\phi }}{2\cdot \mathit{\pi }}\right)$

## Definitions and Notes

• $\mathit{\phi }=\text{latitude in radians}$
• $\mathit{\lambda }=\text{longitude in radians}$
• $\mathrm{sinc}\left(\mathit{x}\right)=\left\{\begin{array}{l}1,\mathit{x}=0\\ \frac{\mathrm{sin}\left(\mathit{x}\right)}{\mathit{x}},\text{otherwise}\end{array}$
• The formulae listed here for the projections are scaled to assume a circumfrence of 1. The formulae may not match their traditional definitions.