The polyconic projection, as shown here,
is not particularly suitable to a map of the world.
However, if one scales latitude and longitude both down to one-half their value, one obtains a map like the following:
which looks somewhat more promising.
In the polyconic projection, there are actually a number of different parameters that can be scaled independently. For example, let us begin by only changing one thing: let the radius of the arcs used for the parallels be determined by a latitude one-ninth of the actual latitude:
Because the parallels don't have the correct radii, I class this projection as a conventional projection, even though it is based on the Polyconic projection.
The result, with true scale along the parallels, resembles the sinusoidal, but the curved parallels do mean that it is not equal-area. Now, let us reduce shear by using a latitude that is 3/4ths of the actual latitude to determine the scale of longitude along the parallels:
At this point, we have quite an attractive-looking projection. It is most accurate at the equator, however, with stretch even at temperate latitudes.
Let us shift the intersections of the arcs for the parallels with the central meridian by stretching them, without otherwise stretching the projection, so that those arcs remain circles, by a factor of 1.218188, which will make 48.75 degrees latitude the standard parallel of the projection:
Finally, let's make our standard meridian 32.5 degrees East longitude the standard meridian, but have the projection asymmetrically disposed around the standard meridian, being primarily divided at 120 degrees west longitude, but allowed to extend beyond that range:
and one obtains an attempt at an approximation to the map projection often seen for maps of the world in Soviet atlases, as devised by one G. A. Ginzburg.
Of course, other combinations of parameters are possible. For example, if I allow the parallels to have more curvature, by scaling the latitude to 1/4th of its value for that purpose, and I also choose a shorter pole-line, by using 8/9ths of the latitude to determine the longitude scale along the parallels, I obtain the following:
This is interesting, but the pole-line is perhaps a bit too small. However, it is interesting to note that this projection does have a slight general resemblance to the Laskowski Tri-Optimal projection:
If we try basing the radii on 1/6th of the latitude, and the longitude scale on 5/6ths of the latitude, we obtain:
To remove the traces of Bonne's projection-style puckering near the pole line, we can return to basing the longitude scale on 3/4ths of the latitude, but continue to curve the parallels more strongly by basing the radii on 1/6th of the latitude, to obtain: