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van der Grinten Projection

\begin{figure}\begin{center}\BoxedEPSF{maps/vdgr.epsf scaled 500}\end{center}\end{figure}

A Map Projection given by the transformation


$\displaystyle x$ $\textstyle =$ $\displaystyle \mathop{\rm sgn}\nolimits (\lambda-\lambda_0){\pi[A(G-P^2)+\sqrt{A^2(G-P^2)^2-(P^2+A^2)(G^2-P^2)}\,]\over P^2+A^2}$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle \mathop{\rm sgn}\nolimits (\phi){\pi\vert PQ-A\sqrt{(A^2+1)(P^2+A^2)-Q^2}\over P^2+A^2},$ (2)

where
$\displaystyle A$ $\textstyle =$ $\displaystyle {1\over 2}\left\vert{{\pi\over\lambda-\lambda_0}-{\lambda-\lambda_0\over\pi}}\right\vert$ (3)
$\displaystyle G$ $\textstyle =$ $\displaystyle {\cos\theta\over\sin\theta+\cos\theta-1}$ (4)
$\displaystyle P$ $\textstyle =$ $\displaystyle G\left({{2\over\sin\theta}-1}\right)$ (5)
$\displaystyle \theta$ $\textstyle =$ $\displaystyle \sin^{-1}\left\vert{2\phi\over\pi}\right\vert$ (6)
$\displaystyle Q$ $\textstyle =$ $\displaystyle A^2+G.$ (7)

The inverse Formulas are


$\displaystyle \phi$ $\textstyle =$ $\displaystyle \mathop{\rm sgn}\nolimits (y)\pi\left[{-m_1\cos(\theta_1+{\textstyle{1\over 3}}\pi)-{c_2\over 3c_3}}\right]$ (8)
$\displaystyle \lambda$ $\textstyle =$ $\displaystyle {\pi\vert X^2+Y^2-1+\sqrt{1+2(X^2-Y^2)+(X^2+Y^2)^2}\vert\over 2X}+\lambda_0,$  
      (9)

where

$\displaystyle X$ $\textstyle =$ $\displaystyle {x\over\pi}$ (10)
$\displaystyle Y$ $\textstyle =$ $\displaystyle {y\over\pi}$ (11)
$\displaystyle c_1$ $\textstyle =$ $\displaystyle -\vert Y\vert(1+X^2+Y^2)$ (12)
$\displaystyle c_2$ $\textstyle =$ $\displaystyle c_1-2Y^2+X^2$ (13)
$\displaystyle c_3$ $\textstyle =$ $\displaystyle -2c_1+1+2Y^2+(X^2+Y^2)^2$ (14)
$\displaystyle d$ $\textstyle =$ $\displaystyle {Y^2\over c_3}+{1\over 27}\left({{2{c_2}^3\over {c_3}^3}-{9c_1c_2\over{c_3}^2}}\right)$ (15)
$\displaystyle a_1$ $\textstyle =$ $\displaystyle {1\over c_3}\left({c_1-{{c_2}^2\over 3c_3}}\right)$ (16)
$\displaystyle m_1$ $\textstyle =$ $\displaystyle 2\sqrt{-{\textstyle{1\over 3}}a_1}$ (17)
$\displaystyle \theta_1$ $\textstyle =$ $\displaystyle {\textstyle{1\over 3}} \cos^{-1}\left({3d\over a_1m_1}\right).$ (18)


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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© 1996-9 Eric W. Weisstein
1999-05-26