Definition of lenticular projections

[Milo]

Tobias has already coined the term teleophasic for projections of this shape,

and teleophasic categories, which already exist (even if teleophasic is relatively new and little used)

A nitpick, but according to Wikipedia's articles on mitosis, the telophase stage does not necessarily involve any actual cleavage furrow (the first example picture on the former page is oval, while the latter page hedges its bets with "cell division may or may not occur at this time depending on the organism"). The more proper term is apparently cytokinesis, but that doesn't sound as cool (and map projections do not, in general, move). I guess we can just assume we're talking about one of the organisms that does start cytokinesis before telophase finishes, whichever those are...

And that's for eukaryotes. Prokaryotes follow a similar-looking but different-under-the-hood mechanism that seemingly doesn't even have names for its stages.

Whatever you call it, the category includes the Dietrich-Kitada projection, which is currently classified as lenticular. An argument could of course be made that it should be reclassified...

For that matter, one could also argue that the Hammer projection looks unlike the "core" lenticular projections because it has convex rather than concave poles. Indeed, you acknowledge that the Hammer projection looks worse than the Mollweide projection despite not even liking pseudocylindrical projections in general, which suggests that concave poles are essential for lenticular to be able to do their thing. (Usually that means pole lines, but it's technically possible to contrive a way to have concave curves at the poles even when the poles themselves are points.)

If we're talking about only projections with concave pole lines and convex interruption meridians, then I think of those as "axehead projections" because of their shape.

All the non-conformal projections that I can think of that blow the polar regions up to massive proportions also have either a circular or a teleophasic boundary.

Conformal projections by necessity can never have a pole line (the Mercator projection gets around this because, unlike most cylindrical projections, it technically doesn't have a pole line, since it's pushed out to infinity), and once you contract the poles to points, there are only so many shapes the map can take. Almost-but-not-quite conformal projections would logically imitate this aesthetic, even if they don't strictly have to (but probably need to at least come close anyway, so you might as well go all the way).

[PeteD]

For that matter, one could also argue that the Hammer projection looks unlike the "core" lenticular projections because it has convex rather than concave poles. Indeed, you acknowledge that the Hammer projection looks worse than the Mollweide projection despite not even liking pseudocylindrical projections in general, which suggests that concave poles are essential for lenticular to be able to do their thing.

Apart from the Hammer (and related projections like the Briesemeister and the Eckert-Greifendorff), there are several other lenticular projections that don't have concave poles:

A4
Aïtoff
Canters W21 and W32
Ciric I
Dedistort
Gott elliptical
Györffy D–F (if we allow increasing parallel spacing)
Kramer VII
Strebe–Hammer
Strebe–Kavrayskiy V
Strebe–Mollweide
Strebe–sinusoidal
Winkel Tripel

So I don't think concave poles are essential for lenticular projections to "do their thing".

[daan]

Maybe the answer here is another category for projections that do not fit the “constant or decreasing spacing” rule.

If we're moving projections out of the lenticular category, I feel that moving them into the globular and teleophasic categories, which already exist (even if teleophasic is relatively new and little used), would be preferable to inventing a new category.

I agree; I don’t necessarily mean defining a new category. However:

All the non-conformal projections that I can think of that blow the polar regions up to massive proportions also have either a circular or a teleophasic boundary.

I know we don’t usually pay much attention to the polar regions, which is part of why pole lines even exist. Being sensitive to it myself, Ginzburg VI and Györffy this and that reach inflation figures toward the poles that I think of as “blown up to massive proportions”. Obviously there is no dividing line, and I’m not suggesting that a polar inflation metric ought to determine categorization, but the non-increasing parallel spacing criterion largely handles my concerns there. If parallel spacing increases toward the poles, the effect is going to be to blow them up unless you resort to Christmas-tree ornament contortions that nobody wants in a real map projection. Due to their other constraints, pseudocylindrical projections already naturally avoid non-decreasing parallel spacing, and so I simply formalized that requirement in conceiving of lenticular projections as pseudocylindrical projections generalized by relaxing the straight parallel requirement.

Cheers,
— daan

[daan]

Addressing just the polyhedral thing in this posting:

Depending on interrupting along one meridian is already arbitrary.

Somewhat, but it's something that I think matters because it's such an obvious distinguishing feature from many other noteworthy projections, such as azimuthal projections (which are interrupted at only a point) and polyhedral projections (which are interrupted at far more than a meridian).
…
I would say that aesthetically, Eisenlohr has more in common with Dietrich-Kitada, and circular Lagrange with circular Hammer, than they do with other "miscellaneous" projections, such as polyhedral ones (not currently a separate category on your site - in fact, a dodecahedron is your icon for miscellaneous projections as a whole), or the armadillo projection (okay, that's just a weird one, even by miscellaneous standards).

This is part of what I was hinting at about the definition of projection. I wouldn’t consider polyhedral to be a category of projections at all. It is what I would call an arrangement. I call it that because the projection mathematics and the bounding shape are independent of each other in polyhedral projections. Interrupting along the outer meridian in equatorial aspect is also an arrangement decision (you can always arbitrarily slice up a projection and scatter the pieces), but it also happens to be the mathematically natural arrangement for several categories of projections. (Not going to explain “natural” rigorously unless someone wants to force me to be pedantic.)

When I say that the decision to interrupt along one meridian is arbitrary, I do not mean that it is arbitrary for the projection mathematics. What I mean is that our geophysical circumstances have biased us toward projections that use mathematics that have this natural boundary. For example, if the earth had a single supercontinent (like it has in the past) concentrated entirely in the southern hemisphere, I don’t think pseudocylindric projections would be a serious thing at all. I have the same doubts if the earth’s rotational axis were tilted nearly to the plane of revolution, like Uranus’s.

Cheers,
— daan

[daan]

For example, if the earth had a single supercontinent (like it has in the past) concentrated entirely in the southern hemisphere, I don’t think pseudocylindric projections would be a serious thing at all. I have the same doubts if the earth’s rotational axis were tilted nearly to the plane of revolution, like Uranus’s.

In that last circumstance, I wonder if we would have adopted spherical coordinates at all. The reason I think that question is salient is because it seems to me that a lot of our cognition about projections is based on a mindset of spherical coordinates.

— daan

[Milo]

This is part of what I was hinting at about the definition of projection. I wouldn’t consider polyhedral to be a category of projections at all. It is what I would call an arrangement. I call it that because the projection mathematics and the bounding shape are independent of each other in polyhedral projections.

If you want to be pedantic, then what you really have is a polygonal projection, copied several times over in a polyhedral arrangement. With the exception of the gnomonic projection, though (which is really a special case), the projections you use to map a spherical polygon onto a Euclidean polygon are totally different from the ones you'd use for any other purpose. They're analogous in concept, though, to how azimuthal projections map circles onto circles, or cylindrical projections map a meridian-interrupted globe onto a rectangle. The source and destination shapes of the projection are absolutely essential characteristics.

For example, if the earth had a single supercontinent (like it has in the past) concentrated entirely in the southern hemisphere, I don’t think pseudocylindric projections would be a serious thing at all.

Would lenticular ones?

I have the same doubts if the earth’s rotational axis were tilted nearly to the plane of revolution, like Uranus’s.

Honestly, even the exact continental layout and axial tilt Earth has today would raise difficulties if only we weren't in an ice age, and Antarctica still had polar forests, like it did as recently as the Eocene.

Ultimately, every map projection needs to make a decision that some places don't matter as much, and that you can just use a second map projection on the occasions where you care about them anyway. If literally every significant landmass were inhabited by humans, where would you place the interruption? A single point in the South Pacific? The land of your ancient enemies who are surely all a bunch of backwards savages anyway?

At the same time, even if there are some interesting things near the poles, there is still a lot less surface area near the poles than in the tropics, simply because that's how spheres work. So people might still have an incentive to sacrifice the polar regions in their "main" maps, and just include insets to compensate.

Still, even if projections with pole-to-pole interruptions still got used, projections with pole lines would probably be less popular, because those distort the polar regions even worse than merely interrupting them does. You're never going to use something like Strebe 1995 or Wagner VII if you care about Antarctica or Greenland. Though those are also among the few places that look better in Hammer than Mollweide...

For example, if the earth had a single supercontinent (like it has in the past) concentrated entirely in the southern hemisphere, I don’t think pseudocylindric projections would be a serious thing at all.

In that last circumstance, I wonder if we would have adopted spherical coordinates at all. The reason I think that question is salient is because it seems to me that a lot of our cognition about projections is based on a mindset of spherical coordinates.

We might use colatitude (distance of parallel from the south pole) instead of latitude (distance of parallel from the equator), but transforming between the two is trivial (x ↦ 90° − x), so it wouldn't affect our thinking much. Using latitude measured from the equator certainly hasn't stopped us from being comfortable with polar-aspect azimuthal projections.

Aside from that detail, I think that a longitude-(co)latitude coordinate system is appropriate for the vast majority of planets and moons, even Uranus-like ones.

The one situation in which I think we might not use a longitude-latitude coordinate system is if we lived on a tidally-locked planet. In that case, it would be more natural to measure distance and azimuth from the substellar point, instead of distance and azimuth from the poles. This is in a way still a spherical coordinate system, but transverse to the usual kind.

I don't think there are really any serious alternatives to a spherical coordinate system of some sort. We live on a sphere. What else are we going to use?

A 4D sphere gives you more options, since there's hyperspherical coordinates vs Hopf coordinates, but we don't live on a 4D sphere. (If we did, I think Hopf coordinates would be a better match for how climate is likely to work, unless again the planet is tidally locked. If tidal locking is even a thing in a 4D universe...)

[PeteD]

Due to their other constraints, pseudocylindrical projections already naturally avoid non-decreasing parallel spacing, and so I simply formalized that requirement in conceiving of lenticular projections as pseudocylindrical projections generalized by relaxing the straight parallel requirement.

Christmas tree decoration projections aside, there are several other pseudocylindrical projections with increasing parallel spacing:

Gall-Bomford
Hölzel
Times
Fahey
Baranyi I, II

and too many to list with constant parallel spacing, which also falls under "non-decreasing", so I don't see how excluding increasing parallel spacing (much less excluding non-decreasing parallel spacing) arises from generalizing by relaxing the straight parallel requirement of pseudocylindricals. In fact, the polar regions of the Gall-Bomford and Times projections are much worse than the polar regions of Ginzburg VI and Györffy D-F.

[daan]

This is part of what I was hinting at about the definition of projection. I wouldn’t consider polyhedral to be a category of projections at all. It is what I would call an arrangement. I call it that because the projection mathematics and the bounding shape are independent of each other in polyhedral projections.

If you want to be pedantic, then what you really have is a polygonal projection, copied several times over in a polyhedral arrangement. With the exception of the gnomonic projection, though (which is really a special case), the projections you use to map a spherical polygon onto a Euclidean polygon are totally different from the ones you'd use for any other purpose. They're analogous in concept, though, to how azimuthal projections map circles onto circles, or cylindrical projections map a meridian-interrupted globe onto a rectangle. The source and destination shapes of the projection are absolutely essential characteristics.

While what you say is true, the number of projections possible for a given polygonal facet is infinite, and a large subset of those can be mathematically continued beyond the face boundary without violating other constraints for a well-behaved map projection. Hence my observation that the mathematics and the boundary shape are independent. We just don’t spend time developing these projections because there isn’t much need. You could generalize the Chamberlin trimetric method, for example, to work on polygonal faces, and the utility of the result would go beyond the face: It would be a “low error” projection for regions the shape of the polygon. It’s just that no one has bothered (as far as I know).

For example, if the earth had a single supercontinent (like it has in the past) concentrated entirely in the southern hemisphere, I don’t think pseudocylindric projections would be a serious thing at all.

Would lenticular ones?

No.

I agree with your comments about earth-without-icecaps.

I don't think there are really any serious alternatives to a spherical coordinate system of some sort. We live on a sphere. What else are we going to use?

I’m not sure, but the conception of meridian is deeply rooted in the position of the sun in the sky and how it changes over the course of a day. On a tipped-over world, that relationship no longer holds — and indeed, a day and a year would be essentially the same thing [clarification: the point being that the concept of “day” becomes so distorted that its use as a time measurement surely wouldn’t have happened before modern survey methods]. Likewise, the conception of latitude is deeply rooted in the elevation of the sun, along with the effects that has on climate. Those effects would be upended with the axis. I don’t think we developed spherical coordinates with the intent of applying them to spheres, and then applied them to the earth because the earth was a sphere. I think we developed spherical coordinates for describing the earth, and then applied them to spheres more generally. On a planet tipped over onto its side, maybe we would have stayed with cartesian triplets to describe position. Or, if we really wanted a 2D system, maybe we would have started with something like a DGGS tessellation.

Cheers,
— daan

[daan]

Christmas tree decoration projections aside, there are several other pseudocylindrical projections with increasing parallel spacing:

Gall-Bomford
Hölzel
Times
Fahey
Baranyi I, II

and too many to list with constant parallel spacing, which also falls under "non-decreasing", so I don't see how excluding increasing parallel spacing (much less excluding non-decreasing parallel spacing) arises from generalizing by relaxing the straight parallel requirement of pseudocylindricals. In fact, the polar regions of the Gall-Bomford and Times projections are much worse than the polar regions of Ginzburg VI and Györffy D-F.

Sorry about “non-decreasing”: I meant “increasing”. I also should not have phrased what I wrote as to invite the interpretation that pseudocylindricals could not have increasing parallel spacing. Of course they can. I just mean to observe that the number that has been described, let alone used commonly, is tiny in comparison (with the implicit “there must be a reason for that” thought). When I considered the lenticular category, I formalized the constraint rather than leaving it to natural selection. I don’t think that’s unfair: In a pseudocylindric projection, you don’t get the parallel curvature to alleviate distortion toward the poles, so you might be tempted to stretch the parallels apart to get a more conformal look. With lenticulars, you have another degree of freedom for keeping the meridians and parallels closer to orthogonal. I think of that as the raison d’être for the category.

— daan

[daan]

Tobias has already coined the term teleophasic for projections of this shape, so danseiji I, II and N would join a ready-made group that also includes projections such as the Dietrich–Kitada and Canters W18, W20 and W23. This would result in globular, cordiform, lenticular and teleophasic forming four mutually exclusive groups of projections all defined by shape.

This makes sense to me with the caveat that I’m a little leery of group names that don’t describe anything about what goes on inside the boundary. There is some precedence for at least two other descriptors for teleophasic, such as epicycloidal (not to be taken literally, since most of them aren’t) and “apple-shaped”. (I see that Tobias acknowledges this second one his posting. Snyder also used this description.) I like “apple-shaped” less teleophasic because an apple doesn’t evoke symmetry to me; nor does the typical shape of a whole apple look a lot like these projections. That fact that not all teleophases look like what we’re going for here doesn’t bother me.

— daan