Van der Grinten projection: Difference between revisions

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			{{Short description\|Compromise map projection}}
	]		]
	] of deformation]]		] 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.<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>		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==		==History==
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	== Geometric construction==		== Geometric construction==
	The geometric construction given by van der Grinten can be written algebraically:<ref>, ] Professional Paper 1395, John P. Snyder, 1987, pp. 239–242.</ref>		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>\begin{align}		<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}, \\		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},		y &= \pm \pi \frac{P Q - A \sqrt{(A^2 + 1) (P^2 + A^2) - Q^2}}{P^2 + A^2},
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	where ''x'' takes the sign of {{nowrap\|''λ'' − ''λ''{{sub\|0}}}}, ''y'' takes the sign of ''φ'', and		where ''x'' takes the sign of {{nowrap\|''λ'' − ''λ''{{sub\|0}}}}, ''y'' takes the sign of ''φ'', and

	: <math>\begin{align}		<math display="block">\begin{align}
	A &= \frac{1}{2} \left\| \frac{\pi}{\lambda - \lambda_0} - \frac{\lambda - \lambda_0}{\pi} \right\|, \\		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}, \\		G &= \frac{\cos \theta}{\sin \theta + \cos \theta - 1}, \\
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	If ''φ'' = 0, then		If ''φ'' = 0, then

	: <math>\begin{align}		<math display="block">\begin{align}
	x &= (\lambda - \lambda_0), \\		x &= (\lambda - \lambda_0), \\
	y &= 0.		y &= 0.
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	Similarly, if ''λ'' = ''λ''{{sub\|0}} or ''φ'' = ±{{pi}}/2, then		Similarly, if ''λ'' = ''λ''{{sub\|0}} or ''φ'' = ±{{pi}}/2, then

	: <math>\begin{align}		<math display="block">\begin{align}
	x &= 0, \\		x &= 0, \\
	y &= \pm \pi \tan \frac{\theta}{2}.		y &= \pm \pi \tan \frac{\theta}{2}.
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	In all cases, ''φ'' is the ], ''λ'' is the ], and ''λ''{{sub\|0}} is the central meridian of the projection.		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~~ ==		== 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 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>		The central meridian and equator are straight lines. All other meridians and parallels are arcs of circles.<ref>
	.		.
	</ref><ref>		</ref><ref>
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	* {{cite web\|url=http://www.progonos.com/furuti/MapProj/Normal/ProjOth/projOth.html\|title=Projections by Van der Grinten, and variations}}		* {{cite web\|url=http://www.progonos.com/furuti/MapProj/Normal/ProjOth/projOth.html\|title=Projections by Van der Grinten, and variations}}

		⚫	{{Map projections}}
	==External links==

⚫	{{Map ~~Projections~~}}

	]		]

Latest revision as of 10:45, 30 December 2023

Compromise map projection

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:

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

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

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

If φ = 0, then

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

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

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

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.

References

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

Bibliography

"Projections by Van der Grinten, and variations".

Map projection

By surface

Cylindrical

Mercator-conformal	Gauss–Krüger Transverse Mercator Oblique Mercator
Equal-area	Balthasart Behrmann Gall–Peters Hobo–Dyer Lambert Smyth equal-surface Trystan Edwards
Cassini Central Equirectangular Gall stereographic Gall isographic Miller Space-oblique Mercator Web Mercator

Pseudocylindrical

Equal-area	Collignon Eckert II Eckert IV Eckert VI Equal Earth Goode homolosine Mollweide Sinusoidal Tobler hyperelliptical
Kavrayskiy VII Wagner VI Winkel I and II

Conical

Pseudoconical

Azimuthal
(planar)

General perspective	Gnomonic Orthographic Stereographic
Equidistant Lambert equal-area

Bonne	Sinusoidal Werner
Bottomley	Sinusoidal Werner
Cylindrical	Balthasart Behrmann Gall–Peters Hobo–Dyer Lambert cylindrical equal-area Smyth equal-surface Trystan Edwards
Tobler hyperelliptical	Collignon Mollweide
Albers Briesemeister Eckert II Eckert IV Eckert VI Equal Earth Goode homolosine Hammer Lambert azimuthal equal-area Quadrilateralized spherical cube Strebe 1995