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The van der Grinten projection is a compromise map projection that is neither equal-area nor conformal.

History

In 1904, the projection was the first of four proposed by Alphons J. van der Grinten.

In 1922, the projection was made famous when the National Geographic Society adopted it as their reference map of the world.

In 1988, 66 years later, it was supplanted by the Robinson projection.

Strengths and weaknesses

Unlike perspective projections, is an arbitrary geometric construction on the plane.

Areas of a fixed size at a distance from the equator look smaller than they do on a Mercator map but larger than a globe. It projects the entire Earth into a circle, although the polar regions are subject to extreme distortion.

Geometric construction

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

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

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

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

Should it occur that φ = 0, then

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

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.

See also

• List of map projections
• Robinson projection (successor)

References

1. ^ Flattening the Earth: Two Thousand Years of Map Projections, John P. Snyder, 1993, pp.258-262, ISBN 0-226-76747-7.
2. Map Projections - A Working Manual, USGS Professional Paper 1395, John P. Snyder, 1987, pp.239-242.

Bibliography

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

External links

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