Van der Grinten projection

Alphons J. van der Grinten invented the projection in 1898 and received US patent #751,226 for it and three others in 1904.[2] The National Geographic Society adopted the projection for their reference maps of the world in 1922, raising its visibility and stimulating its adoption elsewhere. In 1988, National Geographic replaced the van der Grinten projection with the Robinson projection.[1]

The geometric construction given by van der Grinten can be written algebraically:[3]

${\displaystyle {\begin{aligned}x&=\pm \pi {\frac {A(G-P^{2})+{\sqrt {A^{2}(G-P^{2})^{2}-(P^{2}+A^{2})(G^{2}-P^{2})}}}{P^{2}+A^{2}}},\\y&=\pm \pi {\frac {PQ-A{\sqrt {(A^{2}+1)(P^{2}+A^{2})-Q^{2}}}}{P^{2}+A^{2}}},\end{aligned}}}$

where x takes the sign of λ − λ₀, y takes the sign of φ, and

${\displaystyle {\begin{aligned}A&={\frac {1}{2}}\left|{\frac {\pi }{\lambda -\lambda _{0}}}-{\frac {\lambda -\lambda _{0}}{\pi }}\right|,\\G&={\frac {\cos \theta }{\sin \theta +\cos \theta -1}},\\P&=G\left({\frac {2}{\sin \theta }}-1\right),\\\theta &=\arcsin \left|{\frac {2\varphi }{\pi }}\right|,\\Q&=A^{2}+G.\end{aligned}}}$

If φ = 0, then

${\displaystyle {\begin{aligned}x&=(\lambda -\lambda _{0}),\\y&=0.\end{aligned}}}$

Similarly, if λ = λ₀ or φ = ±π/2, then

${\displaystyle {\begin{aligned}x&=0,\\y&=\pm \pi \tan {\frac {\theta }{2}}.\end{aligned}}}$

In all cases, φ is the latitude, λ is the longitude, and λ₀ is the central meridian of the projection.

• List of map projections
• Robinson projection (successor)

• "Projections by Van der Grinten, and variations".