South America in the Crosshairs

[quadibloc]

The Oxford World Atlas, at one point, used an oblique Mercator projection for a map of the Americas. This was truly conformal in its entirety, even if in some areas it was slightly inferior to the Bipolar Oblique Conic Conformal.

These days, though, due to the strong emphasis on making maps equal-area, except for the maps of the whole world, most atlases today use the Lambert Azimuthal Equal-Area projection for every single map.

This isn't necessarily a bad thing, as it lets maps be compared. Older atlases might use a hodge-podge of projections based on whatever is easiest to draw. So an atlas might show Europe and Canada on the simple conic projection, the United States on the polyconic (!), Russia on Bonne's projection, and so on and so forth.

But the Lambert Azimuthal Equal-Area is a good choice for an area equal in height and width, roughly like a circle or a square. Not all maps are of areas of such shapes.

So if one is preparing a map of an area that is twice as wide as it is long, why not use the central part of a Hammer-Aitoff projection?

And as a recent thread here showed, instead of squeezing longitude by 2 and then stretching the x-axis by 2, one could use other factors as well, thus creating equal-area projections suitable for areas with other elongations.

It occured to me that South America has a funny shape which lends itself well to the characteristics of the Sinusoidal projection, like this:



in case one is wondering, first I select the meridian of 32 degrees East longitude, tilt the globe forwards by 77 degrees, and assign the meridian to 95 degrees East on the map.

Africa also has an L-shape, but in its case, what to do is more simple and obvious.

[daan]

It occured to me that South America has a funny shape which lends itself well to the characteristics of the Sinusoidal projection, like this:

Yup. But your first proposal, an obliqued Hammer, would do better here because the region of low distortion is broader in the regions on which the continent lies.

You can tailor these regions even more without too much difficulty; see my An adaptable equal-area pseudoconic map projection. That is an Albers/Bonne equal-area hybrid in which you can specify all three “standard parallels” independently. They do not remain standard parallels, but they do control how the distortion gets distributed. Because of the Bonne’s lack of distortion along the central meridian, along with the three “standard parallels”, the technique is suitable for, among other shapes, “cross-shaped” regions.

— daan

[daan]

And as a recent thread here showed, instead of squeezing longitude by 2 and then stretching the x-axis by 2, one could use other factors as well, thus creating equal-area projections suitable for areas with other elongations.

In the equatorial Hammer scaled horizontally by √2/2 and vertically by √2, it is the poles that have low distortion, increasing rapidly away from them. Other scalings result in two patches of low distortion along the central meridian and away from the center, symmetric about the equator, with latitude of conformal point depending on the scale factors. These are ovoid in shape. If, for some reason, two disjoint, ovoid regions of low distortion along a great circle were wanted, this would work.

— daan

[quadibloc]

an Albers/Bonne equal-area hybrid

Ooh! Just as one can derive the Hammer-Wagner from the Hammer, one can apply the same sort of shrink to the Bonne projection, Yes, that would provide a great deal of flexibility.

On the one hand, a Hammer projection would do better for Peru and Uruguay, but I would have thought the Sinusoidal does very well for Argentina and Chile.