The van der Grinten projection is a compromise

map projection In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longit ...

, 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.''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp. 258–262, .

History

Alphons J. van der Grinten invented the projection in 1898 and received US patent #751,226 for it and three others in 1904.''A Bibliography of Map Projections'', John P. Snyder and Harry Steward, 1989, p. 94, US Geological Survey Bulletin 1856. The

National Geographic Society 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, an ...

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
 x &= \pm \pi \frac, \\
 y &= \pm \pi \frac,
\end

where ''x'' takes the sign of , ''y'' takes the sign of ''φ'', and

\begin
 A &= \frac \left,  \frac - \frac \, \\
 G &= \frac, \\
 P &= G \left(\frac - 1\right), \\
 \theta &= \arcsin \left, \frac\, \\
 Q &= A^2 + G.
\end

If ''φ'' = 0, then

\begin
 x &= (\lambda - \lambda_0), \\
 y &= 0.
\end

Similarly, if ''λ'' = ''λ'' or ''φ'' = ±/2, then

\begin
 x &= 0, \\
 y &= \pm \pi \tan \frac.
\end

In all cases, ''φ'' is the

latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...

, ''λ'' is the

longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek lette ...

, and ''λ'' 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."van der Grinten IV"

References

Bibliography

External links