In the previous post we explored compass directions on a toroidal world. Now let's take a look at the broader question of map-making.
The simplest projection is an equirectangular projection, which just uses longitude and latitude as x- and y-coordinates. Since I've chosen a torus with a 4-to-1 ratio of major radius to minor radius, the outer equator has a circumference five times that of a meridian. Maps of our torus will tend to be quite long compared to maps of Earth. Here's a 5-to-1 equirectangular graticule showing 45° increments, centered on the outer equator:
If we hadn't chosen the directional conventions we did in the previous post, map-makers would have pushed us there for sanity's sake. Besides the confusion of swapping east and west when crossing a polar rim, choosing north to be topward would require map-makers to include separate compass roses at different latitudes:
If forced into such a convention, they might simplify a bit by centering on the bottom polar rim, so that "north" means "away from the middle of the map":
But given the sensibilities of "north" meaning a constant direction along a meridian, one compass rose is sufficient. The challenge remains, however, that while north is always up, up on the map is not always toward the top polar rim, so perhaps some regions do require something like a compass rose, to help navigators orient themselves in terms of the planetary axis.
The inner equator of a torus is shorter than the outer equator, so an equirectangular projection has to stretch it. To get a better idea of the relative sizes of different areas, we can project different latitudes to their relative lengths:
Denizens of the inner equator might not appreciate being relegated to the edges of the map, so a map centered on them could also be drawn:
Or a polar-centric map:
Whichever center is chosen, the inner equator is distorted less than than the polar points on equator-centric maps of Earth. Here's a Tissot indicatrix of the proportional-latitude projection shown above:
Unfortunately, longitudes far from the central meridian are badly distorted. We can reduce this distortion by not shrinking the inner equator all the way down its actual proportions:
A small improvement. Inner latitudes are only slightly stretched, but middle latitudes far from the central meridian are quite twisted. Let's try tearing the map into halves, creating a toroidal version of the Nicolosi globular projection:
This projection doesn't stretch the inner equator, and it gives us two central meridians with minimal distortion, but it doesn't do any better at reducing side distortion than the projection above it. In fact, it creates four sides with that much distortion.
For mapping a torus, our best option may where we began, the equirectangular projection:
The inner equator is stretched, but not that much. A fatter torus would require more expansion, but a 4-to-1 ratio of radii only stretches the interior to 5/3 its actual size, 67% bigger, which is hardly a great deformity. Trying to minimize such already minor distortion doesn't seem worth it, especially given the twisting distortions we encountered in the obvious alternatives.
If you have ideas for other ways to map a torus's surface to a flat plane, please email me.