At the time when the best known of the Van der Grinten projections, actually the Van der Grinten I projection, but usually just known as the Van der Grinten projection, was announced, three other projections were also described. While two of them are not of much interest, the Van der Grinten IV projection is an interesting conventional projection which does not share the strong bias towards shape and against area of its more famous companion.
The construction of this projection is illustrated in the diagram below: note that the arcs are drawn through a point on the central meridian (which is divided equally) and a circle, not the boundary of the map.
The red lines satisfy the equation
y lat --------- = ----------- x + 180 270 - lat
and the circle they intersect satisfies the equation
2 2 2 x + y = 90
so the location of the points on the circle that determine the arcs of the parallel is found by means of a quadratic equation, derived by substituting for y:
Denoting the ratio above by q, as follows:
lat q = ----------- 270 - lat
we substitute as follows:
2 2 2 2 x + q (x + 180) = 90
obtaining the quadratic equation:
2 2 2 2 (q + 1) x + 360 q x + (32400 q - 8100) = 0
which can then be solved by the usual methods.
The globular projection, if extended to cover the whole world, instead of being used in its usual format to cover only a hemisphere,
has the same outline as the Van der Grinten IV projection, but the less rapid expansion of the parallels in the Van der Grinten IV clearly makes it a more attractive projection.
The Van der Grinten IV projection is relatively obscure, but a subroutine for drawing it is included in the program GRASS, and I have seen some climate maps drawn with this projection in one German-language atlas.
Although an attractive projection, still there is quite a bit of shear and thus it is likely that Winkel's Tripel, for example, would normally be preferred to this projection. However, it occurred to me that if I allowed a degree of tilt towards the conformal, but not as much as in the Van der Grinten I, quite an attractive projection could be obtained.
Here, the central meridian is divided as in the Stereographic, with the parallels then derived from the central meridian position, and the meridian positions suitably scaled. Thus, this projection is conformal at the central meridian, and can be mated with the stereographic.
However, a less extreme modification, which preserves the uniform scale of the central meridian, is also possible. Instead of projecting the lines from the point on the edge of the map 180 degrees away from the central meridian, increasing that distance to 360 degrees produces a generally more pleasing projection, although with more shear in some places:
However, looking at this has led me to the conclusion that a better projection would be found somewhere in between that and the original Van der Grinten IV projection, so I tried again with a distance of 270 degrees, and obtained the following:
which I find to be quite reasonable.
Since this projection seems to do quite well, here is my standard oblique case (which places southern Africa near the South Pole in order to avoid interrupting Antarctica, which is why you likely haven't seen this kind of oblique projection before) for comparison and for a new view of its distortions, even though conventional projections aren't usually used in oblique or transverse cases, since that defeats the primary purpose of their design, making them easy to draw: