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 4648 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 Jesuit 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).

[quadibloc]

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).

I thought this matter was raised earlier in the thread. The perspective cylindrical projection with a projection point at 3/4 of the Earth's diameter is the one that's almost identical to the Mercator; it, and the one with the projection point at the full distance of the Earth's diameter away, like the Stereographic, but without the stretching of either the Gall Stereographic or the BSAM, are due to Braun. Mine, wilth a point at 7/8 of the Earth's diameter, halfway between them, is original as far as I know.

[daan]

At least I read about him creating another perspective cylindrical that was called the Pseudo-Mercator(the same name applied to Web-Mercator).

Slight rant: I don’t know why people refer to it as “pseudoMercator”. There is nothing “pseudo” about it. It’s just the spherical Mercator with a particular set of units and a well defined boundary.

— daan

[mapnerd2022]

Yes, I agree. It's still a genuine Mercator. It's just that while the Web Mercator's formulas are for the spherical form of the Mercator, geographical coordinates are required to be in the WGS 84 ellipsoidal datum, which makes it only very slightly not conformal. Which is very hard to notice.

[Milo]

I assume that it's "pseudo" in the sense that if the underlying data is from an ellipsoid rather than a sphere, it's not truly conformal, only approximately so. This can be a problem if you were actually counting on the Mercator's property of being conformal.

Mind you, this is fairly easy to fix if you're putting in the effort. You just need to convert your data to conformal latitude, instead of the geodetic latitude that geographical coordinates are usually given in. I can understand why you might not bother with this if you're just using a free toy like G.Projector, but it's really lazy if professional mapmakers making commercial web applications can't be bothered to put in the effort to do this properly, especially after people have been mocking "Web Mercator" for years.

Note that of all of the forms of auxiliary latitude given on that page, geodetic latitude is actually easily the worst for mapmaking, not giving optimal results for any type of map projection whatsoever, and there is absolutely no reason to use it if you have a choice in the matter. Despite that, geodetic latitude is the accepted standard for geographical coordinates, and is the format that equirectangular-source-images-for-reprojection usually come in (even though it's still not advisable to actually use even for equirectangular projections), presumably because it's the kind of latitude that's easiest to measure using traditional celestial navigation techniques.

[daan]

I assume that it's "pseudo" in the sense that if the underlying data is from an ellipsoid rather than a sphere, it's not truly conformal, only approximately so. This can be a problem if you were actually counting on the Mercator's property of being conformal.

I’ve heard this argument before, but it makes no sense to me. Any spherical projection is “pseudo” by that reasoning, since nobody surveys against a spherical datum.

While I understand the special pleading that Web Mercator is used in more local contexts than most other world projections, that does not change the fact that it’s a spherically conformal projection with development identical to the “not-pseudo” form.

The only argument I’ve ever heard against Web Mercator specifically that made any sense to me is that, in geodetic contexts, people might get confused thinking that they are dealing with coordinates projected against the ellipsoidal form and therefore get mismatches. But that doesn’t really make sense to me, either: Nobody uses the ellipsoidal equatorial Mercator for geodetic purposes. Possibly some equatorial countries do, but they’re not the ones making this complaint.

It’s a straw man.

— daan