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van der Grinten Projection

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vanderGrintenProjection

The van der Grinten projection is a map projection given by the transformation

x=sgn(lambda-lambda_0)(pi[A(G-P^2)+sqrt(A^2(G-P^2)^2-(P^2+A^2)(G^2-P^2))])/(P^2+A^2)
(1)
y=sgn(phi)(pi|PQ-Asqrt((A^2+1)(P^2+A^2)-Q^2)|)/(P^2+A^2),
(2)

where

A=1/2|pi/(lambda-lambda_0)-(lambda-lambda_0)/pi|
(3)
theta=sin^(-1)|(2phi)/pi|
(4)
G=(costheta)/(sintheta+costheta-1)
(5)
P=G(2/(sintheta)-1)
(6)
Q=A^2+G.
(7)

The inverse formulas are

phi=sgn(y)pi[-m_1cos(theta_1+1/3pi)-(c_2)/(3c_3)]
(8)
lambda=(pi[X^2+Y^2-1+sqrt(1+2(X^2-Y^2)+(X^2+Y^2)^2)])/(2X)+lambda_0,
(9)

where

X=x/pi
(10)
Y=y/pi
(11)
c_1=-|Y|(1+X^2+Y^2)
(12)
c_2=c_1-2Y^2+X^2
(13)
c_3=-2c_1+1+2Y^2+(X^2+Y^2)^2
(14)
d=(Y^2)/(c_3)+1/(27)((2c_2^3)/(c_3^3)-(9c_1c_2)/(c_3^2))
(15)
a_1=1/(c_3)(c_1-(c_2^2)/(3c_3))
(16)
m_1=2sqrt(-1/3a_1)
(17)
theta_1=1/3cos^(-1)((3d)/(a_1m_1)).
(18)

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References

Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 239-242, 1987.

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van der Grinten Projection

Cite this as:

Weisstein, Eric W. "van der Grinten Projection." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/vanderGrintenProjection.html

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