The Seven-Eighths Perspective Cylindrical

[quadibloc]

Recently, I added the BSAM projection to my page about the Gall Stereographic projection. I noted that its reduced stretch was less objectionable.
It led me to thinking that another way to have less stretch but still have 40 degrees as the standard parallel, would be to start with a projection closer to the Mercator.
One way to have an intermediate between the stereographic and the Mercator would be to remember that the perspective projection that goes 3/4 of the diameter of the Earth below the tangent point closely approximates the Mercator. The Stereographic goes down a full diameter. So 7/8 is halfway in between.
It turns out that without a vertical stretch, leaving the standard parallel at the equator, this projection is close to the Miller Cylindrical.

[Milo]

I would describe this as a perspective projection with a "height" of −½, halfway between the gnomonic projection at 0 and the stereographic projection at −1. I can see where you'd get ¾ if you counted diameters instead of radii, but where do you get ⅞? Ah, got it, you're talking about two different projections. Okay then!

A graph shows that this projection stays very close to the Mercator projection until approximately 70° north, which is about as far north as anyone usually cares to look. So this projection looks similar to Mercator over most of the world, but, unlike the true Mercator, can depict the entire world in finite space.

mercapprox.png (13.48 KiB) Viewed 29 times

The aspect ratio of the projection (with standard parallel at the equator) is almost but not quite square: it's precisely π:3.

While the −½ value can be determined by just eyeballing the graphs, it can also be justified by looking at derivatives. The general formula for a perspective projection is:
p_h(x) = (h−1)/(h−cos(x))⋅sin(x)
Which produces the following derivatives:
p_h'(0) = 1
p_h''(0) = 0
p_h'''(0) = (2+h)/(1−h)
p_h''''(0) = 0
p_h'''''(0) = (h²+13h+16)/(1−h)²
The formula for the Mercator projection is:
m(x) = asinh(tan(x))
Which produces the following derivatives:
p_h'(0) = 1
p_h''(0) = 0
p_h'''(0) = 1
p_h''''(0) = 0
p_h'''''(0) = 5
Setting h=−½ will allow the third derivative, and by extension all of the first four derivatives, to match.

G.Projector appears to support general cylindrical perspective projections under the name "Solov'ev Perspective Cylindrical" (a name which I cannot find corroborated anywhere else), but only allows external farside projections (h≤−1), and has forced cropping when the aspect ratio would be more extreme than 2:1.

[quadibloc]

My drawing of the projection was at the bottom of my page on the Gall Stereographic, but there was a bug in the program, stretching it horizontally.
I have now corrected the bug so that I and everyone else can see correctly how it looks.

[mapnerd2022]

Recently, I added the BSAM projection to my page about the Gall Stereographic projection. I noted that its reduced stretch was less objectionable.
It led me to thinking that another way to have less stretch but still have 40 degrees as the standard parallel, would be to start with a projection closer to the Mercator.
One way to have an intermediate between the stereographic and the Mercator would be to remember that the perspective projection that goes 3/4 of the diameter of the Earth below the tangent point closely approximates the Mercator. The Stereographic goes down a full diameter. So 7/8 is halfway in between.
It turns out that without a vertical stretch, leaving the standard parallel at the equator, this projection is close to the Miller Cylindrical.

Didn't Carl Braun, a Jesuist priest, invent it along with his distortion-free equator version of the Cylindrical Stereographic back in 1867? At least I read about him creating another perspective cylindrical that was called the Pseudo-Mercator(the same name applied to Web-Mercator).