Map Projections

Plate Carrée (Equirectangular, φ1 = 0°)

x = \frac{λ}{2 \cdot π}

y = \frac{φ}{2 \cdot π}

Equirectangular, φ1 = 15°

x = \frac{λ}{2 \cdot π}

y = \frac{φ \cdot \sqrt{2} \cdot (\sqrt{3} - 1)}{2 \cdot π}

Equirectangular, φ1 = 30°

x = \frac{λ}{2 \cdot π}

y = \frac{φ \cdot \sqrt{3}}{3 \cdot π}

Plate Carrée (Equirectangular, φ1 = 45°)

x = \frac{λ}{2 \cdot π}

y = \frac{φ \cdot \sqrt{2}}{2 \cdot π}

Equirectangular, φ1 = 60°

x = \frac{λ}{2 \cdot π}

y = \frac{φ}{π}

Mercator

x = \frac{λ}{2 \cdot π}

y = \frac{\ln (\tan (\frac{π}{4} + \frac{φ}{2}))}{2 \cdot π}

Gall Stereographic

x = \frac{λ}{2 \cdot π}

y = \frac{\tan (\frac{φ}{2}) \cdot (\sqrt{2} + 1)}{2 \cdot π}

Miller

x = \frac{λ}{2 \cdot π}

y = \frac{5 \cdot \ln (\tan (\frac{π}{4} + \frac{2 \cdot φ}{5}))}{8 \cdot π}

Lambert (Cylindrical Equal Area, φ1 = 0°)

x = \frac{λ}{2 \cdot π}

y = \frac{\sin (φ)}{2 \cdot π}

Behrmann (Cylindrical Equal Area, φ1 = 30°

x = \frac{λ}{2 \cdot π}

y = \frac{2 \cdot \sin (φ)}{3 \cdot π}

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

x = \frac{λ}{2 \cdot π}

y = \frac{\sin (φ)}{{(\cos (\frac{5 \cdot π}{24}))}^{2} \cdot 2 \cdot π}

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

x = \frac{λ}{2 \cdot π}

y = \frac{\sin (φ)}{π}

Balthasart (Cylindrical Equal Area, φ1 = 50°)

x = \frac{λ}{2 \cdot π}

y = \frac{\sin (φ)}{{(\cos (\frac{5 \cdot π}{18}))}^{2} \cdot 2 \cdot π}

Sinusoidal

x = \frac{λ \cdot \cos (φ)}{2 \cdot π}

y = \frac{φ}{2 \cdot π}

Mollweide

2 \cdot θ + \sin (2 \cdot θ) = π \cdot \sin (φ)

x = \frac{λ \cdot \cos (θ)}{2 \cdot π}

y = \frac{\sin (θ)}{4}

Robinson

\begin{matrix} Latitude & XScale & 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{matrix}

x = \frac{XScale \cdot λ}{2 \cdot π}

y = \frac{13523 \cdot YScale}{16974 \cdot π}

NaturalEarth

\begin{matrix} Latitude & XScale & 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{matrix}

x = \frac{XScale \cdot λ}{2 \cdot π}

y = \frac{13 \cdot YScale}{50}

Kavrayskiy VII

x = \frac{λ \cdot \sqrt{1 - 3 \cdot {(\frac{φ}{π})}^{2}}}{2 \cdot π}

y = \frac{φ \cdot \sqrt{3}}{3 \cdot π}

Wagner VI

x = \frac{λ \cdot \sqrt{1 - 3 \cdot {(\frac{φ}{π})}^{2}}}{2 \cdot π}

y = \frac{φ}{2 \cdot π}

Aitoff

α = \arccos (\cos (φ) \cdot \cos (\frac{λ}{2}))

x = \frac{\cos (φ) \cdot \sin (\frac{λ}{2})}{sinc (α) \cdot π}

y = \frac{\sin (φ)}{sinc (α) \cdot 2 \cdot π}

Hammer

x = \frac{\cos (φ) \cdot \sin (\frac{λ}{2})}{2 \cdot \sqrt{1 + \cos (φ) \cdot \cos (\frac{λ}{2})}}

y = \frac{\sin (φ)}{4 \cdot \sqrt{1 + \cos (φ) \cdot \cos (\frac{λ}{2})}}

Winkel I

x = \frac{λ \cdot (\cos (φ) \cdot π + 2)}{2 \cdot π \cdot (π + 2)}

y = \frac{φ}{π + 2}

Winkel Tripel

α = \arccos (\cos (φ) \cdot \cos (\frac{λ}{2}))

x = \frac{sinc (α) \cdot λ + \cos (φ) \cdot \sin (\frac{λ}{2}) \cdot π}{sinc (α) \cdot π \cdot (π + 2)}

y = \frac{\sin (φ) + φ \cdot sinc (α)}{2 \cdot sinc (α) \cdot (π + 2)}

Van der Grinten

θ = \arcsin (| \frac{2 \cdot φ}{π} |)

A = \frac{| \frac{π}{λ} - \frac{λ}{π} |}{2}

G = \frac{\cos (θ)}{\sin (θ) + \cos (θ) - 1}

P = G \cdot (\frac{2}{\sin (θ)} - 1)

Q = A^{2} + G

x = {\begin{cases} \frac{λ}{2 \cdot π}, φ = 0 \\ 0, λ = 0 \lor φ = \pm \frac{π}{2} \\ \frac{sign (λ) \cdot | A \cdot (G - P^{2}) + \sqrt{A^{2} \cdot {(G - P^{2})}^{2} - (P^{2} + A^{2}) \cdot (G^{2} \cdot P^{2})} |}{2 \cdot (P^{2} + A^{2})}, otherwise \end{cases}

y = {\begin{cases} 0, φ = 0 \\ \frac{sign (φ) \cdot \tan (\frac{θ}{2})}{2}, λ = 0 \lor φ = \pm \frac{π}{2} \\ \frac{sign (φ) \cdot | P \cdot Q - A \cdot \sqrt{(A^{2} + 1) \cdot (P^{2} + A^{2}) - Q^{2}} |}{2 \cdot (P^{2} + A^{2})}, otherwise \end{cases}

Eckert II

x = \frac{λ \cdot \sqrt{4 - 3 \cdot \sin (| φ |)}}{4 \cdot π}

y = \frac{sgn (φ) \cdot (2 - \sqrt{4 - 3 \cdot \sin (| φ |)})}{4}

Eckert V

x = \frac{λ \cdot (1 + \cos (φ))}{4 \cdot π}

y = \frac{φ}{2 \cdot π}

Azimuthal Equidistant (Polar)

x = \sin (λ) \cdot (0.25 - \frac{φ}{2 \cdot π})

y = - \cos (λ) \cdot (0.25 - \frac{φ}{2 \cdot π})

Projections

Plate Carrée (Equirectangular, φ1 = 0°)

Equirectangular, φ1 = 15°

Equirectangular, φ1 = 30°

Plate Carrée (Equirectangular, φ1 = 45°)

Equirectangular, φ1 = 60°

Mercator

Gall Stereographic

Miller

Lambert (Cylindrical Equal Area, φ1 = 0°)

Behrmann (Cylindrical Equal Area, φ1 = 30°

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

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

Balthasart (Cylindrical Equal Area, φ1 = 50°)

Sinusoidal

Mollweide

Robinson

NaturalEarth

Kavrayskiy VII

Wagner VI

Aitoff

Hammer

Winkel I

Winkel Tripel

Van der Grinten

Eckert II

Eckert V

Azimuthal Equidistant (Polar)

Definitions and Notes