The Other Twilight Projection

[quadibloc]

An example of a minimum-error perspective azimuthal projection was Clarke's Twilight projection.
G.Projector - and, I believe, Geocart - uses the formulas given in Snyder for it, and Snyder went back to original sources.
However, in an article on map projections in the Encyclopedia Britannica - which appeared in the eleventh edition, but also in revised form even in the fifteenth edition - which was written by Clarke himself, there's an illustration of the Twilight projection... which looks nothing like the Twilight projection as drawn by modern map-making software!
Well, I finally decided to take a serious look at the question. And my very first stab at the problem struck gold!
G.Projector offers the "Azimuthal Far-Side Projection", which lets one adjust the position of the light source. 1.707, for example, would give the La Hire projection.
Well, since that drawing had decreased scale towards the edges, giving an impression of curvature, somewhat like the Orthographic projection, I increased the distance... and my first choice, 3 Earth-radii, gave almost an exact match to the drawing!

[Milo]

Your link is broken. It seems like you're too used to HTML syntax.

Here's the correct version:


If the "Spherical Radius of 108°" caption was in the original drawing and means what I think it means, then that would indicate a farside perspective point of 3.236, or 1+√5. You can calculate it as "perspective_height = 1/cos(maximum_radius)", with positive values indicating nearside perspective and negative values indicating farside perspective. (This is for external perspective. For internal perspective, like the gnomonic projection, use "perspective_height = cos(maximum_radius)" instead, again with negative values indicating farside perspective, but also note that internal perspective projections need infinite space to depict the calculated radius, so the radius of a practical map would be less.)

...And why won't G.Projector let me generate an azimuthal farside perspective with an edge radius of greater than 90°? That's the entire point of farside perspectives. Apparently this was added in version 3.0.9. I refused to upgrade past version 2.x because changing the map controls to floating windows instead of part of the same window as the map seemed like a downgrade to me that would only make it harder to keep track of where my controls have run off to or which controls link to which map. I already often have far too many windows open on my desktop.

[quadibloc]

I refused to upgrade past version 2.x because changing the map controls to floating windows instead of part of the same window as the map seemed like a downgrade to me that would only make it harder to keep track of where my controls have run off to or which controls link to which map. I already often have far too many windows open on my desktop.

I'm too desperate to have support for more projections to be that fussy. So when I returned to updating my site, and found the new version of Map Designer Raster, I was overjoyed.
As for 3.26 being the correct value... if I understand you correctly, I suspect then that the border of the projection as shown would also be the ultimate limit of what could be drawn.

[Milo]

I'm too desperate to have support for more projections to be that fussy.

The new projections in version 3.x (or 2.x later than 2.5, which is what I have) seem to be fairly obscure ones, not anything that I'm interested in. If they added Eisenlohr, or better polyhedral projections, or the ability to make oblique aspects of arbitrary projections instead of only a small handful that have it explicitly coded, or Craig/Hammer retroazimuthal (currently there's just Littrow), or even Tobler hyperelliptical with exponents other than the default one, I might be tempted.

I guess the ones added in version 3.2.4 are mildly interesting? From what I can tell, they look like analogues to the Aitoff and Hammer projections, but based on different azimuthal steps. (And the pseudo-stereographic appears to be inferior to Lagrange, which is actually conformal. And the pseudo-orthographic isn't actually a true pseudocylindrical projection since it doesn't have equally-spaced meridians. But still, they make interesting showpieces, even if I'd never use them for a serious map.)

But 3.2.4 was also released more than two years after I lost interest in bothering to keep up.

Now that I look at it, I guess the SVG output option in 3.3.0 also catches my eye. Might be useful with shapefiles, though it's not much use if you're using raster input data to begin with. What I really wish G.Projector had is the ability to save PNG files with transparent background, instead of gray. Doesn't look like it's occurred to them to add that one yet.

As for 3.26 being the correct value... if I understand you correctly, I suspect then that the border of the projection as shown would also be the ultimate limit of what could be drawn.

Exactly. Like how the orthographic and gnomonic projection can only show half of the sphere (90° radius) no matter how much space you allocate, but the stereographic projection can show the entire sphere (minus one point) if given infinite space.

I'm just guessing that's what the author meant with the "108°" in the caption, but it seems likely given that it's pretty close to the value you determined experimentally. You'll notice that your image, while a relatively close match, still shows a meaningful mismatch, most visible in northwestern Africa.

I haven't actually seen how either the original paper or any modern reproductions attempt to define the Clarke Twilight projection, and don't care enough to try to look up any sources myself. G.Projector implements it as a special projection with no configurable parameters (except projection center), with no explanation in its projection table of how its constructed, despite some other cases of perspective projections being given as specific parameters of the general projection.

(Except it's 3.236, not 3.26.)

[quadibloc]

The new projections in version 3.x (or 2.x later than 2.5, which is what I have) seem to be fairly obscure ones, not anything that I'm interested in. If they added Eisenlohr, or better polyhedral projections, or the ability to make oblique aspects of arbitrary projections instead of only a small handful that have it explicitly coded,

Those are definitely improvements that I'd love to see as well.
Justin Kunimune's Map Designer Raster does have Eisenlohr, and also it now presents the conformal projection using Dixon elliptic functions in its most popular format, the Cahill butterfly, among other great additions.
While the latest G.Projector does lack making oblique aspects of arbitrary projections... it did allow oblique aspects of the equirectangular, so I had used a two-step process. However, one case when I did that is no longer necessary, as the Spilhaus world ocean aspect of August's conformal was explictly added (as a separate projection).

[Milo]

While the latest G.Projector does lack making oblique aspects of arbitrary projections... it did allow oblique aspects of the equirectangular, so I had used a two-step process.

Yeah, I can do that workaround easily, but it leads to sloppy graticules because they're also "reprojected". And it's just a hassle.