Van der Grinten projection

Geometric construction

The geometric construction given by van der Grinten can be written algebraically:

$${\displaystyle \begin{array}{cc}x& =\pm \pi \frac{A(G-P^2)+\sqrt{A^2(G-P^2{)}^2-(P^2+A^2)(G^2-P^2)}}{P^2+A^2},\\ y& =\pm \pi \frac{PQ-A\sqrt{(A^2+1)(P^2+A^2)-Q^2}}{P^2+A^2},\end{array}}$$

where x takes the sign of λ − λ₀, y takes the sign of φ, and

$${\displaystyle \begin{array}{cc}A& =\frac{1}{2}\left|\frac{\pi }{\lambda -{\lambda }_0}-\frac{\lambda -{\lambda }_0}{\pi }\right|,\\ G& =\frac{\cos \theta }{\sin \theta +\cos \theta -1},\\ P& =G\left(\frac{2}{\sin \theta }-1\right),\\ \theta & =\arcsin \left|\frac{2\phi }{\pi }\right|,\\ Q& =A^2+G.\end{array}}$$

If φ = 0, then

$${\displaystyle \begin{array}{cc}x& =(\lambda -{\lambda }_0),\\ y& =0.\end{array}}$$

Similarly, if λ = λ₀ or φ = ±π/2, then

$${\displaystyle \begin{array}{cc}x& =0,\\ y& =\pm \pi \tan \frac{\theta }{2}.\end{array}}$$

In all cases, φ is the latitude, λ is the longitude, and λ₀ is the central meridian of the projection.

Van der Grinten IV projection

The van der Grinten IV projection is a later polyconic map projection developed by Alphons J. van der Grinten. The central meridian and equator are straight lines. All other meridians and parallels are arcs of circles.