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{{Short description|Compromise map projection}}
{{lowercase|van der Grinten}}
]]] ]
] of deformation]]
The '''van der Grinten projection''' is a compromise ], which means that it is neither ] nor ]. Unlike perspective projections, the van der Grinten projection is an arbitrary geometric construction on the plane. Van der Grinten projects the entire Earth into a circle. It largely preserves the familiar shapes of the ] while modestly reducing Mercator's distortion. Polar regions are subject to extreme distortion. Lines of longitude converge to points at the poles.<ref name="snyder">''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp. 258–262, {{ISBN|0-226-76747-7}}.</ref>


==History==
The '''van der Grinten projection''' is neither ] nor ]. It fits the entire Earth into a circular shape, though the polar regions are subject to extreme distortion. The projection was proposed by Alphons J. van der Grinten in 1904, and, unlike most projections, is an arbitrary geometric construction on the plane. It was made famous when the ] adopted it as their reference map of the world from 1922 until 1988.
] invented the projection in 1898 and received US patent #751,226 for it and three others in 1904.<ref name="bibliography">''A Bibliography of Map Projections'', John P. Snyder and Harry Steward, 1989, p. 94, US Geological Survey Bulletin 1856.</ref> The ] 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 ].<ref name="snyder" />

== Geometric construction==
The geometric construction given by van der Grinten can be written algebraically:<ref> {{Webarchive|url=https://web.archive.org/web/20100701103721/http://pubs.er.usgs.gov/usgspubs/pp/pp1395 |date=2010-07-01 }}, ] Professional Paper 1395, John P. Snyder, 1987, pp. 239–242.</ref>

<math display="block">\begin{align}
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{P Q - A \sqrt{(A^2 + 1) (P^2 + A^2) - Q^2}}{P^2 + A^2},
\end{align}</math>

where ''x'' takes the sign of {{nowrap|''λ'' − ''λ''{{sub|0}}}}, ''y'' takes the sign of ''φ'', and

<math display="block">\begin{align}
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{align}</math>

If ''φ''&nbsp;=&nbsp;0, then

<math display="block">\begin{align}
x &= (\lambda - \lambda_0), \\
y &= 0.
\end{align}</math>

Similarly, if ''λ'' = ''λ''{{sub|0}} or ''φ'' = ±{{pi}}/2, then

<math display="block">\begin{align}
x &= 0, \\
y &= \pm \pi \tan \frac{\theta}{2}.
\end{align}</math>

In all cases, ''φ'' is the ], ''λ'' is the ], and ''λ''{{sub|0}} is the central meridian of the projection.

== Van der Grinten IV projection ==

The van der Grinten IV projection is a later polyconic map projection developed by Alphons J. van der Grinten.
The central meridian and equator are straight lines. All other meridians and parallels are arcs of circles.<ref>
.
</ref><ref>
.
p. 205.
</ref><ref>
.
</ref>

==See also==

*]
*] (successor)

==References==
{{reflist}}

==Bibliography==
* {{cite web|url=http://www.progonos.com/furuti/MapProj/Normal/ProjOth/projOth.html|title=Projections by Van der Grinten, and variations}}

{{Map projections}}

]


]


{{cartography-stub}} {{cartography-stub}}

Latest revision as of 10:45, 30 December 2023

Compromise map projection
Van der Grinten projection of the world
The Van der Grinten projection with Tissot's indicatrix of deformation

The van der Grinten projection is a compromise map projection, which means that it is neither equal-area nor conformal. Unlike perspective projections, the van der Grinten projection is an arbitrary geometric construction on the plane. Van der Grinten projects the entire Earth into a circle. It largely preserves the familiar shapes of the Mercator projection while modestly reducing Mercator's distortion. Polar regions are subject to extreme distortion. Lines of longitude converge to points at the poles.

History

Alphons J. van der Grinten invented the projection in 1898 and received US patent #751,226 for it and three others in 1904. 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.

Geometric construction

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

x = ± π A ( G P 2 ) + A 2 ( G P 2 ) 2 ( P 2 + A 2 ) ( G 2 P 2 ) P 2 + A 2 , y = ± π P Q A ( A 2 + 1 ) ( P 2 + A 2 ) Q 2 P 2 + A 2 , {\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 λλ0, y takes the sign of φ, and

A = 1 2 | π λ λ 0 λ λ 0 π | , G = cos θ sin θ + cos θ 1 , P = G ( 2 sin θ 1 ) , θ = arcsin | 2 φ π | , Q = A 2 + G . {\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

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

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

x = 0 , y = ± π tan θ 2 . {\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.

Van der Grinten IV projection

The van der Grinten IV projection is a later polyconic map projection developed by Alphons J. van der Grinten. The central meridian and equator are straight lines. All other meridians and parallels are arcs of circles.

See also

References

  1. ^ Flattening the Earth: Two Thousand Years of Map Projections, John P. Snyder, 1993, pp. 258–262, ISBN 0-226-76747-7.
  2. A Bibliography of Map Projections, John P. Snyder and Harry Steward, 1989, p. 94, US Geological Survey Bulletin 1856.
  3. Map Projections – A Working Manual Archived 2010-07-01 at the Wayback Machine, USGS Professional Paper 1395, John P. Snyder, 1987, pp. 239–242.
  4. "Van der Grinten IV Projection".
  5. "An Album of Map Projections". p. 205.
  6. "van der Grinten IV".

Bibliography

Map projection
By surface
Cylindrical
Mercator-conformal
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Pseudocylindrical
Equal-area
Conical
Pseudoconical
Azimuthal
(planar)
General perspective
Pseudoazimuthal
By metric
Conformal
Equal-area
Bonne
Bottomley
Cylindrical
Tobler hyperelliptical
Equidistant in
some aspect
Gnomonic
Loxodromic
Retroazimuthal
(Mecca or Qibla)
By construction
Compromise
Hybrid
Perspective
Planar
Polyhedral
See also


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