Definition of lenticular projections

[PeteD]

On the Map Projection Essentials page of this website, lenticular projections are defined as projections whose meridians and parallels:

1. curve towards the poles and
2. maintain or decrease their distances from each other going from the centre of the map to the edges.

What's the reasoning being the second criterion? Is it to exclude projections like van der Grinten and Lagrange? This would make sense because these projections don't seem to fit in with most lenticular projections. However, this criterion also excludes projections like Ginzburg VI, Györffy D-F and danseiji II, which I feel ought to be included. It seems particularly arbitrary to exclude Ginzburg VI and danseiji II since Ginzburg IV, V and IX and danseiji I and N are included. All these excluded projections are listed under lenticular projections on Tobias's website, and Ginzburg VI is also listed under lenticular projections on the Directory of Map Projections page of this website (where the other excluded projections aren't listed at all).

On the other hand, this definition includes the azimuthal equidistant and azimuthal equal-area projections in equatorial aspect, which, like van der Grinten and Lagrange, don't seem to fit in with most lenticular projections. Might it be better to specify that the aspect ratio be greater than 1 (and perhaps also that the projection be interrupted along (at least?) one meridian) instead? This would keep van der Grinten and Lagrange excluded, further exclude the azimuthal equidistant and azimuthal equal-area projections, and include Ginzburg VI, Györffy D-F and danseiji II.

[daan]

…lenticular projections are defined as projections whose meridians and parallels:

1. maintain or decrease their distances from each other going from the centre of the map to the edges.

What's the reasoning being the second criterion? Is it to exclude projections like van der Grinten and Lagrange?

Yes.

…this criterion also excludes projections like Ginzburg VI, Györffy D-F and danseiji II, which I feel ought to be included. It seems particularly arbitrary to exclude Ginzburg VI and danseiji II since Ginzburg IV, V and IX and danseiji I and N are included. All these excluded projections are listed under lenticular projections on Tobias's website, and Ginzburg VI is also listed under lenticular projections on the Directory of Map Projections page of this website (where the other excluded projections aren't listed at all).

Excellent points.

On the other hand, this definition includes the azimuthal equidistant and azimuthal equal-area projections in equatorial aspect, which, like van der Grinten and Lagrange, don't seem to fit in with most lenticular projections. Might it be better to specify that the aspect ratio be greater than 1 (and perhaps also that the projection be interrupted along (at least?) one meridian) instead? This would keep van der Grinten and Lagrange excluded, further exclude the azimuthal equidistant and azimuthal equal-area projections, and include Ginzburg VI, Györffy D-F and danseiji II.

Again, excellent points.

I aimed for more simplicity than my intent warranted, as you demonstrate. I like your proposal but would like to stew on it for awhile and possibly discuss more.

Many thanks.
— daan

[Milo]

Lenticular projections seem to be a less well-defined category than other common terms like "azimuthal" or "pseudocylindrical", and exactly what's included depends on who you ask. Some sources avoid using it entirely.

A category that I personally find more useful and seems to include all commonly-accepted lenticular projections is that of "meridian-interrupted projections" (or half-great-circle-interrupted, anyway, in oblique aspects). Though that's a little tricky because we ideally want pseudocylindrical projections to arise as a special limit case of lenticular projections (just as cylindrical projections are a special limit case of pseudocylindrical projections), yet those can be interrupted along more than one meridian as in the Goode homolosine! Though that one is still interrupted along a meridian as well.

Ah, you mentioned that already:

(and perhaps also that the projection be interrupted along (at least?) one meridian)

Perhaps you might also demand that (in normal aspect) the equator is a straight line, to exclude conic projections. I would not actually require symmetry across the equator, though, since while most lenticular projections have that, there are exceptions (such as the Strebe asymmetric).

The parallels curving towards the poles seems to be less a defining feature than it's something that just automatically tends to arise whenever you try to minimize distortion on a meridian-interrupted straight-equator projection without specifically demanding a particular shape for the other parallels (as pseudocylindrical projections do). The double Mollweide projection, while not exactly very practical outside of an amusing demonstration, still seems like it should be considered to form a continuum with the Mollweide, Hammer, and double Hammer projections, and you [Daan] identify the Hammer projection as lenticular (while the Mollweide is included under the "pseudocylindrical is a limit case of lenticular" principle).

Also:

What's the reasoning being the second criterion? Is it to exclude projections like van der Grinten and Lagrange?

Yes.

Why? That seems to be arbitrarily excluding conformal/near-conformal projections, even though they otherwise look very much like lenticular projections, and it is unlikely that you could get a conformal meridian-interrupted projection without it looking somewhat like the Lagrange. Other generally-accepted projection categories like azimuthal and cylindrical are considered to be an orthogonal property to whether the projection is conformal/compromise/equal-area, with examples of each type, and I see no reason why lenticular should be different. (There is at least one category, pseudocylindrical, that incidentally excludes conformal projections - unless you count my little joke here, or the true-cylindrical Mercator projection - and I suppose lenticular projections are to some degree seen/intended as a "generalization" of pseudocylindrical projections, but still. It the math doesn't say lenticular projections can't be conformal for deep and important reasons, no reason to ban it arbitrarily.)

[PeteD]

... it is unlikely that you could get a conformal meridian-interrupted projection without it looking somewhat like the Lagrange.

While the elliptical conformal projection does look somewhat like the Lagrange, it also looks a lot more like other lenticular projections than the Lagrange does, on account of its aspect ratio. All of the 117 projections listed as being lenticular on Tobias's website have an aspect ratio between around 1.6 and around 2, so the van der Grinten and Lagrange projections would certainly stick out as outliers if added to that group.

The term "lenticular" presumably comes from the fact that pairs of parallels of the graticule of a lenticular projection approximate the shape of a lens, but some pairs of parallels of the graticules of the van der Grinten and Lagrange projections make a shape that's much thicker than a typical lens, so they're not very "lenticular" in the literal sense of the word.

Having said that, excluding projections based on their aspect ratio does seem a bit arbitrary, and I wouldn't be averse to a definition along the lines of "projections interrupted along (again, at least?) one meridian, wherein parallels other than the equator are (or can be?) curved in equatorial aspect". This would include Ginzburg VI, Györffy D-F and danseiji II while excluding azimuthal projections and would in my opinion be an improvement with respect to the current definition. I could live with the van der Grinten and Lagrange projections being included.

[Atarimaster]

First of all: Thank you for addressing this matter, PeteD!

Having said that, excluding projections based on their aspect ratio does seem a bit arbitrary

I agree, especially since this would exclude the circular Lagrange but not the one that Geocart renders by default.

and I wouldn't be averse to a definition along the lines of "projections interrupted along (again, at least?) one meridian, wherein parallels other than the equator are (or can be?) curved in equatorial aspect".

Of course, “along one meridian” refers to the equatorial aspect, too – is that clear in English? (I’m asking because it wouldn’t be in German.)
Apart from that, I like this definition.

I could live with the van der Grinten and Lagrange projections being included.

Me, too.
Although including the American Polyconic gives me a bit of a headache…

[daan]

I seem to be the lone voice unconcerned about “arbitrary”. I prefer consistency and rigor when I think it can be had, but this category of projections is just a bucket whose contents reflect our collective conventions and preferences in projections as designed to display this particular earth. Defining it by the graticule’s features in equatorial aspect is already arbitrary. Depending on interrupting along one meridian is already arbitrary.

I do not intend conformal, or conformal-like, projections to fit into this bucket, partly because doing that enlarges the bucket to the point where it gains significant overlaps with other buckets, but also because the term “lenticular” becomes less apt. The van der Grinten projections, for example, are already sometimes referred to as “globular”, which is a highly arbitrary, but still useful, category.

I agree, especially since this would exclude the circular Lagrange but not not the one that Geocart renders by default.

Not if the original criterion of equal or decreasing spacing of parallels is honored. In any case, the different parameterizations of a projection result in distinct projections, so I’m not so concerned that a “projection” might change categories based on parameterization. (Dare I start a thread on the fraught problem of the definition of “projection”?)

Cheers,
— daan

[daan]

There is also the question of purpose fulfilled by categorizing.

Some categories imply rigorous properties of some sort. By using this kind of category, we assert the presence of clear mathematical relationships between projections in the category. With those relationships, we can compare projections in the category quantitatively and meaningfully at least insofar as those relationships are concerned. The comparison becomes an exploration of the differences permitted within the categorization. Or, we can generate new projections within the category by changing the portions of its generating functions that do not affect the mathematical constraints defining the category.

Some categories imply little more than an evocative appearance. “Globular” would be one. “Cordiform” would be another. Neither of these categories imply any particular mathematics, and therefore no particular mathematical relationships. That means we have no commonalities to ignore when comparing; we can only compare using the with more general tools applicable to any projection. You can also argue about whether a projection properly even belongs in the category, since boundaries for the category haven’t been defined. Basically, it’s an æsthetic judgment. It’s not much to go on, but at least I can rule out enormous chunks of projection space if you call a projection “cordiform”.

Then there are categories in between. “Lenticular”, as I defined it, does not imply any specific mathematical relationships between its members. On the other hand, it does imply some geometric traits when the constraint of aspect is honored. It is fairly distinct: You could reasonably resolve a debate over whether a projection belongs in the category almost every time; conversely, if it’s lenticular, it tells you it’s definitely not a lot of other things. Does that make it useful? I would say it describes a large chunk of world projections created since the advent of the twentieth century that would otherwise not fit into existing categories, and so I think it’s useful as originally described, but I do think PeteD’s original posting on the topic brings up some unwelcome members to the category, members that could be ejected with simple clarifications. Why are they unwelcome? Because they already belong to other rigorous categories. His posting also list some excluded projections that seem reasonable to include, ones that are similar to qualified members and that have no other category to live in.

Maybe the answer here is another category for projections that do not fit the “constant or decreasing spacing” rule. The reason this makes (a little) more sense to me than expanding the lenticular class is because increasing parallel spacing (in particular) implies a willingness to blow the polar regions up to massive proportions in much the way that conformal projections in equatorial aspect do, whereas I was distinctly thinking of lenticular as leaning more toward the equal-area camp.

Cheers,
— daan

[PeteD]

Of course, “along one meridian” refers to the equatorial aspect, too – is that clear in English? (I’m asking because it wouldn’t be in German.)

I suppose not. We could change this to "along half a great circle", but since it looks like this definition won't be adopted anyway, it's a moot point.

Perhaps if we can agree on which projections should be included and excluded, it'll make the job of agreeing on a definition a bit easier. Here's my proposal:

Projections to be excluded

1. Azimuthal projections

These could be excluded by:

• requiring an aspect ratio greater than 1, or
• requiring an interruption along (at least?) half a great circle.

2. Lagrange, van der Grinten I, II

These could be excluded by:

• requiring an aspect ratio greater than 1, or
• requiring constant or decreasing parallel spacing.

3. Eisenlohr, August epicycloidal

These could be excluded by:

• requiring constant or decreasing parallel spacing, or
• requiring that the projections do not extend vertically beyond the poles.

4. American polyconic, rectangular polyconic, van der Grinten IV

These could be excluded by:

• requiring that the projections do not extend vertically beyond the poles.

Projections to be included

Ginzburg VI, Györffy D–F

These will end up being excluded if we require constant or decreasing parallel spacing.

Teleophasic projections

Not one of the four different potential requirements that I've listed above can exclude all the projections that we want to exclude on its own. However, all these projections can be excluded using just two of these requirements:

• requiring an aspect ratio greater than 1, and
• requiring that the projections do not extend vertically beyond the poles.

Using these two requirements would also avoid excluding Ginzburg VI and Györffy D–F, which is good. It would, however, exclude danseiji II, but I could live with this as it would also exclude danseiji I and N, so these three projections would still be classified together.

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.

[Milo]

Me, too.
Although including the American Polyconic gives me a bit of a headache…

Looking at it, the American polyconic produces some rather questionable world maps, but then, I guess that's why it's called the American polyconic. It was never intended for whole-world maps. Still, if you do make one that covers the whole world, it definitely looks lenticular to me, having clear similarities to other lenticular projections even if it has worse distortion. Being "bad" has never been a disqualifier of other map projection categories, as the central cylindrical projection is still cylindrical,

This is a problem with all currently-proposed defintions of "lenticular", in that they largely talk about how the projection looks in normal-aspect world maps, and are dubious to apply to regional maps. To a degree this seems reasonable because in many cases you would be less inclined to use lenticular projections for regional maps in the first place (the smaller the region, the less it matters which projection you use, so there is more of an incentive to use mathematically-simple projections such as azimuthal or cylindrical, possibly in oblique aspect), however the example of the American polyconic proves that it does happen sometimes, so we need to have some way of classifying it.

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). The interruption is also an immediate tradeoff in terms of map quality: make the interruption too small, and distortion will increase over the rest of the map because you have a harder time pushing it all off to the interruption, but make the interruption too large, and you'll cut through important landforms.

Furthermore, choosing the make maps which are interrupted at exactly one meridian is something that people seem to keep doing again and again, for all sorts of maps that are otherwise quite different. Whether it's arbitrary or not, it seems to be a choice that humans are intuitively drawn to, in a way that "monotonically-decreasing length of parallels" isn't. It's also a feature that a layman can actually identify a map as possessing or not possessing at a glance, unlike monotonically-decreasing length of parallels, which can fool even experienced mapmakers if they don't feel like sitting down and doing the math, as we saw in your correction of Atarimaster concerning the Lagrange projection. (I mean, you probably could design a map that looks close enough to meridian-interrupted to fool someone without actually being so, but I don't think any commonly-used map projections fit the bill.)

I do not intend conformal, or conformal-like, projections to fit into this bucket, partly because doing that enlarges the bucket to the point where it gains significant overlaps with other buckets,

Which buckets? You currently classify the Lagrange (both circular and not) and Eisenlohr projections as "miscellaneous", so they don't have a bucket.

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).

Both the Dietrich-Kitada and Hammer projections are identified by you as lenticular. (The circular Hammer projection isn't explicitly identified, but given it's a simple affine rescaling of an equal-area projection, it surely has the same properties as the standard 2:1 Hammer projection. And while we're at it, non-circular Lagrange looks quite similar to Eisenlohr and by extension Dietrich-Kitada.)

Also speaking of buckets, if you just want equal-area projections, then we already have a term for those: "equal-area projections". We don't need another one. Though right now, you seem to be trying to do something along the lines of defining a category which includes "compromise projections which are closer to equal-area, but not compromise projections which are closer to conformal"?

Does that make it useful? I would say it describes a large chunk of world projections created since the advent of the twentieth century that would otherwise not fit into existing categories, and so I think it’s useful as originally described, but I do think PeteD’s original posting on the topic brings up some unwelcome members to the category, members that could be ejected with simple clarifications. Why are they unwelcome? Because they already belong to other rigorous categories.

This is because the category you're trying to make just doesn't fit in with the other categories. It's trying to do something totally different.

There are (at least) three ways to categorize projections:
1. By construction method as it informs the basic structure of the map: azimuthal, cylindrical, conic, etc.
2. By aesthetic appearance, which, as you note, is too subjective to make for a particularly rigorous categorization system. (Often this is based more on the shape of the outer boundary of the map than on the shape of the graticules inside the map.)
3. By satisfied mathematical properties, which can be hard ("equal-area", "conformal") or soft ("has area distortion below a certain threshold over the majority of the map", "minimizes the Airy-Kavrayskiy criterion for compromise projections").

If you define most of your categories by method 1, and then have one that's defined by method 3, then of course you're going to have overlap and gaps. That doesn't mean that both methods aren't useful, but they should be considered orthogonal axes. Certainly you could have chosen to organize your projection list by equal-area, compromise-closer-to-equal-area, compromise-closer-to-conformal, conformal, worse-than-conformal (like gnomonic), and worse-than-equal-area (like orthographic) projections, but you didn't. You apparently care only about the compromise-closer-to-equal-area category to the exclusion of all others, which are instead classified by construction instead of properties.

[PeteD]

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. There's also more precedent for defining categories according to shape than according to properties along the lines of "constant or decreasing parallel spacing". Furthermore, moving projections into the globular and teleophasic categories seems to remove all the projections that we don't want to classify as lenticular, whereas only removing projections with increasing parallel spacing still leaves the azimuthal equidistant, azimuthal equal-area, American polyconic, rectangular polyconic and van der Grinten IV projections in the lenticular category.

The reason this makes (a little) more sense to me than expanding the lenticular class is because increasing parallel spacing (in particular) implies a willingness to blow the polar regions up to massive proportions in much the way that conformal projections in equatorial aspect do, whereas I was distinctly thinking of lenticular as leaning more toward the equal-area camp.

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.