# Van der Grinten projection

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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. [1]

A map projection is a systematic transformation of the latitudes and longitudes of locations from the surface of a sphere or an ellipsoid into locations on a plane. Maps cannot be created without map projections. All map projections necessarily distort the surface in some fashion. There is no limit to the number of possible map projections.

In cartography, a conformal map projection is one in which any angle on Earth is preserved in the image of the projection, i.e. the projection is a conformal map in the mathematical sense.

The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because of its unique property of representing any course of constant bearing as a straight segment. Such a course, known as a rhumb or, mathematically, a loxodrome, is preferred by navigators because the ship can sail in a constant compass direction to reach its destination, eliminating difficult and error-prone course corrections. Linear scale is constant on the Mercator in every direction around any point, thus preserving the angles and the shapes of small objects and fulfilling the conditions of a conformal map projection. As a side effect, the Mercator projection inflates the size of objects away from the equator. This inflation starts infinitesimally but accelerates with latitude to reach infinite at the poles. So, for example, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator, such as Central Africa.

## History

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 National Geographic Society (NGS), headquartered in Washington, D.C., United States, is one of the largest non-profit scientific and educational organizations in the world. Founded in 1888, its interests include geography, archaeology, and natural science, the promotion of environmental and historical conservation, and the study of world culture and history. The National Geographic Society's logo is a yellow portrait frame—rectangular in shape—which appears on the margins surrounding the front covers of its magazines and as its television channel logo. Through National Geographic Partners, the Society operates the magazine, TV channels, a website, worldwide events, and other media operations.

The Robinson projection is a map projection of a world map which shows the entire world at once. It was specifically created in an attempt to find a good compromise to the problem of readily showing the whole globe as a flat image.

## Geometric construction

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 λλ0, 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.

In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earth's surface. Latitude is an angle which ranges from 0° at the Equator to 90° at the poles. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. On its own, the term latitude should be taken to be the geodetic latitude as defined below. Briefly, geodetic latitude at a point is the angle formed by the vector perpendicular to the ellipsoidal surface from that point, and the equatorial plane. Also defined are six auxiliary latitudes which are used in special applications.

Longitude, is a geographic coordinate that specifies the east–west position of a point on the Earth's surface, or the surface of a celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter lambda (λ). Meridians connect points with the same longitude. By convention, one of these, the Prime Meridian, which passes through the Royal Observatory, Greenwich, England, was allocated the position of 0° longitude. The longitude of other places is measured as the angle east or west from the Prime Meridian, ranging from 0° at the Prime Meridian to +180° eastward and −180° westward. Specifically, it is the angle between a plane through the Prime Meridian and a plane through both poles and the location in question.

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## 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, USGS Professional Paper 1395, John P. Snyder, 1987, pp. 239–242.