van der Grinten Projection

A Map Projection given by the transformation

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

$$y = \mathop{\rm sgn}\nolimits (\phi){\pi\vert PQ-A\sqrt{(A^2+1)(P^2+A^2)-Q^2}\over P^2+A^2},$$ (2)

where

$$A = {1\over 2}\left\vert{{\pi\over\lambda-\lambda_0}-{\lambda-\lambda_0\over\pi}}\right\vert$$ (3)

$$G = {\cos\theta\over\sin\theta+\cos\theta-1}$$ (4)

$$P = G\left({{2\over\sin\theta}-1}\right)$$ (5)

$$\theta = \sin^{-1}\left\vert{2\phi\over\pi}\right\vert$$ (6)

$$Q = A^2+G.$$ (7)

The inverse Formulas are

$$\phi = \mathop{\rm sgn}\nolimits (y)\pi\left[{-m_1\cos(\theta_1+{\textstyle{1\over 3}}\pi)-{c_2\over 3c_3}}\right]$$ (8)

$$\lambda = {\pi\vert X^2+Y^2-1+\sqrt{1+2(X^2-Y^2)+(X^2+Y^2)^2}\vert\over 2X}+\lambda_0,$$ (9)

where

$$X = {x\over\pi}$$ (10)

$$Y = {y\over\pi}$$ (11)

$$c_1 = -\vert Y\vert(1+X^2+Y^2)$$ (12)

$$c_2 = c_1-2Y^2+X^2$$ (13)

$$c_3 = -2c_1+1+2Y^2+(X^2+Y^2)^2$$ (14)

$$d = {Y^2\over c_3}+{1\over 27}\left({{2{c_2}^3\over {c_3}^3}-{9c_1c_2\over{c_3}^2}}\right)$$ (15)

$$a_1 = {1\over c_3}\left({c_1-{{c_2}^2\over 3c_3}}\right)$$ (16)

$$m_1 = 2\sqrt{-{\textstyle{1\over 3}}a_1}$$ (17)

$$\theta_1 = {\textstyle{1\over 3}} \cos^{-1}\left({3d\over a_1m_1}\right).$$ (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.