Van der Grinten projection

Van der Grinten projection for the globe

The Van der Grinten projection is a compromise cartographic projection that is neither isometric nor equiangular. It projects the surface of the earth in a circle, the maximum distortion occurs in the regions of the poles. The projection was proposed by Alfons van der Grinten in 1904. It gained fame when the National Geographic Society in 1922 adopted it as a standard world map. As such, the projection existed until 1988 ^[1] .

The projection is expressed by the following formulas ^[2]

x={\frac {\pm \pi \left(A\left(G-P^{2}\right)+{\sqrt {A^{2}\left(G-P^{2}\right)^{2}-\left(P^{2}+A^{2}\right)\left(G^{2}-P^{2}\right)}}\right)}{P^{2}+A^{2}}}\,;

{\ displaystyle x = {\ frac {\ pm \ pi \ left (A \ left (GP ^ {2} \ right) + {\ sqrt {A ^ {2} \ left (GP ^ {2} \ right) ^ {2} - \ left (P ^ {2} + A ^ {2} \ right) \ left (G ^ {2} -P ^ {2} \ right)}} \ right)} {P ^ {2} + A ^ {2}}} \ ,;}

{\ displaystyle x = {\ frac {\ pm \ pi \ left (A \ left (GP ^ {2} \ right) + {\ sqrt {A ^ {2} \ left (GP ^ {2} \ right) ^ {2} - \ left (P ^ {2} + A ^ {2} \ right) \ left (G ^ {2} -P ^ {2} \ right)}} \ right)} {P ^ {2} + A ^ {2}}} \ ,;}

y={\frac {\pm \pi \left(PQ-A{\sqrt {\left(A^{2}+1\right)\left(P^{2}+A^{2}\right)-Q^{2}}}\right)}{P^{2}+A^{2}}},

{\ displaystyle y = {\ frac {\ pm \ pi \ left (PQ-A {\ sqrt {\ left (A ^ {2} +1 \ right) \ left (P ^ {2} + A ^ {2} \ right) -Q ^ {2}}} \ right)} {P ^ {2} + A ^ {2}}},}

{\ displaystyle y = {\ frac {\ pm \ pi \ left (PQ-A {\ sqrt {\ left (A ^ {2} +1 \ right) \ left (P ^ {2} + A ^ {2} \ right) -Q ^ {2}}} \ right)} {P ^ {2} + A ^ {2}}},}

Where $x$ ${\ displaystyle x}$ $x$ has the same sign as $\lambda -\lambda _{0}$ ${\ displaystyle \ lambda - \ lambda _ {0}}$ ${\ displaystyle \ lambda - \ lambda _ {0}}$ , but $y$ ${\ displaystyle y}$ $y$ - the same as $\phi$ ${\ displaystyle \ phi}$ $\ phi$ and

A={\frac {1}{2}}\left|{\frac {\pi }{\lambda -\lambda _{0}}}-{\frac {\lambda -\lambda _{0}}{\pi }}\right|;

{\ displaystyle A = {\ frac {1} {2}} \ left | {\ frac {\ pi} {\ lambda - \ lambda _ {0}}} - {\ frac {\ lambda - \ lambda _ {0 }} {\ pi}} \ right |;}

{\ displaystyle 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}};

{\ displaystyle G = {\ frac {\ cos \ theta} {\ sin \ theta + \ cos \ theta -1}};}

{\ displaystyle G = {\ frac {\ cos \ theta} {\ sin \ theta + \ cos \ theta -1}};}

P=G\left({\frac {2}{\sin \theta }}-1\right);

{\ displaystyle P = G \ left ({\ frac {2} {\ sin \ theta}} - 1 \ right);}

{\ displaystyle P = G \ left ({\ frac {2} {\ sin \ theta}} - 1 \ right);}

\theta =\arcsin \left|{\frac {2\phi }{\pi }}\right|;

{\ displaystyle \ theta = \ arcsin \ left | {\ frac {2 \ phi} {\ pi}} \ right |;}

{\ displaystyle \ theta = \ arcsin \ left | {\ frac {2 \ phi} {\ pi}} \ right |;}

Q=A^{2}+G\,.

{\ displaystyle Q = A ^ {2} + G \ ,.}

{\ displaystyle Q = A ^ {2} + G \ ,.}

If a $\phi =0$ ${\ displaystyle \ phi = 0}$ $\ phi = 0$ then

x=\left(\lambda -\lambda _{0}\right)\,;

{\ displaystyle x = \ left (\ lambda - \ lambda _ {0} \ right) \ ,;}

{\ displaystyle x = \ left (\ lambda - \ lambda _ {0} \ right) \ ,;}

y=0\,.

{\ displaystyle y = 0 \ ,.}

{\ displaystyle y = 0 \ ,.}

If a $\lambda =\lambda _{0}$ ${\ displaystyle \ lambda = \ lambda _ {0}}$ ${\ displaystyle \ lambda = \ lambda _ {0}}$ or $\phi =\pm \pi /2$ ${\ displaystyle \ phi = \ pm \ pi / 2}$ ${\ displaystyle \ phi = \ pm \ pi / 2}$ then

x=0\,;

{\ displaystyle x = 0 \ ,;}

{\ displaystyle x = 0 \ ,;}

y=\pm \pi \tan {\theta /2}.

{\ displaystyle y = \ pm \ pi \ tan {\ theta / 2}.}

{\ displaystyle y = \ pm \ pi \ tan {\ theta / 2}.}

In all formulas $\phi$ ${\ displaystyle \ phi}$ $\ phi$ - latitude $\lambda$ ${\ displaystyle \ lambda}$ $\ lambda$ - longitude $\lambda _{0}$ ${\ displaystyle \ lambda _ {0}}$ $\ lambda _ {0}$ - the central meridian of the projection.

Notes

↑ Flattening the Earth: Two Thousand Years of Map Projections , John P. Snyder, 1993, pp. 258-262, ISBN 0-226-76747-7 .
↑ Map Projections - A Working Manual , USGS Professional Paper 1395, John P. Snyder, 1987, pp. 239-242

Links

Projections by Van der Grinten, and variations (neopr.) . Date of treatment January 14, 2012. Archived May 18, 2012.
US Patent 751 226
US patent No. 751226 .

[1] Flattening the Earth: Two Thousand Years of Map Projections , John P. Snyder, 1993, pp. 258-262, ISBN 0-226-76747-7 .

[2] Map Projections - A Working Manual , USGS Professional Paper 1395, John P. Snyder, 1987, pp. 239-242

Van der Grinten projection

Notes

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