- Van der Grinten projection
The van der Grinten projection is neither equal-area nor conformal. It projects the entire Earth into a circle, though the polar regions are subject to extreme distortion. The projection was the first of four 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
National Geographic Society adopted it as their reference map of the world from 1922 until 1988 ["Flattening the Earth: Two Thousand Years of Map Projections", John P. Snyder, 1993, pp.258-262, ISBN 0-226-76747-7.] .The geometric construction given by van der Grinten can be written algebraically [ [http://pubs.er.usgs.gov/usgspubs/pp/pp1395 "Map Projections - A Working Manual"] , USGS Professional Paper 1395, John P. Snyder, 1987, pp.239-242] :
:x = frac {pm pi left(Aleft(G - P^2 ight) + sqrt {A^2 left(G - P^2 ight)^2 - left(P^2 + A^2 ight)left(G^2 - P^2 ight)} ight)} {P^2 + A^2},
:y = frac {pm pi left(P Q - A sqrt{left(A^2 + 1 ight)left(P^2 + A^2 ight) - Q^2} ight)} {P^2 + A^2}
where x, takes the sign of lambda - lambda_0,, y, takes the sign of phi, and
:A = frac {1} {2}|frac {pi} {lambda - lambda_0} - frac {lambda - lambda_0} {pi}|:G = frac {cos heta} {sin heta + cos heta - 1}:P = Gleft(frac {2} {sin heta} - 1 ight):heta = arcsin |frac {2 phi} {pi}|:Q = A^2 + G,
Should it occur that phi = 0,, then
:x = left(lambda - lambda_0 ight),:y = 0,
Similarly, if lambda = lambda_0, or phi = pm pi / 2,, then
:x = 0,:y = pm pi an { heta / 2 }
In all cases, phi, is the latitude, lambda, is the longitude, and lambda_0, is the central meridian of the projection.
Notes
References
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