A new map metric: resolution-efficiency

[daan]

Of course there is some abiguity in what projections are considered "analogous to" each other. There doesn't (currently!) exist any formal theory of "dual projections", so it's more a matter of looking similar in some aspect that we happen to care about.

Definitions:
• “aphylactic” = neither equal-area nor conformal.
• “antisocol” = the gradient path from a point on an isocol (the path of greatest change in distortion).
• “conformant projection” = a projection having a conformal dual.

My well defined (formal?) proposal, which I partially talk about here, is that duals are:
• projections whose isocols (level curves of angular deformation and flation) are everywhere aligned; and
• each projection of the dual has a power relationship γ of the Tissot axes a to b that is constant within that projection: a^γ = b. [Updated because I confused Milo with my original verbiage.]

What “aligned” means is that, for every isocol on conformant projection A, there is an isocol on conformant B having the same preimage on the spheroid; and that, within an aphylactic A or B, if a point has both angular deformation and flation, then the isocol of angular deformation through that point will coincide with the isocol of flation through that point (of course both on the plane and in the preimage).

This implies:

1. All conformal projections have one equal-area and infinite aphylactic duals.
2. Many equal-area and aphylactic projections have no conformal duals (= they are not conformant).
3. For conformal projections, γ = 1; for equal-area, γ = –1; for equidistant, γ = 0.

This does not imply:

1. That the shape of the outer boundary is the same for A and a dual of A.
2. That the dual of A is contiguous even if A is contiguous.
3. That A’s dual’s isocols are planar scalings of A’s isocols (in case you are tempted by that thought).
4. That the range of γ is restricted to [–1…1], although very few useful projections would occur outside that range.

Notice that this proposes a rigorous definition for equidistant projections, which I believe has been lacking heretofore.

With regard to (d), the gnomonic projection has γ = 2.

— daan

[Milo]

My well defined (formal?) proposal, which I partially talk about here, is that duals are:

An interesting link, but I don't see the word "dual" anywhere on that page?

• the power relationship γ of the Tissot axes a to b is constant: a^γ = b.

Umm, this is a property of a single projection, not a relationship between two projections. Clearly many projections have nonconstant γ, therefore their dual would also be expected to. You would need some way to relate the γ at a specific point on the dual to the γ at the same point on the original (such as stating that they are each other's additive inverses).

3. For conformal projections, γ = 1; for equal-area, γ = –1; for equidistant, γ = 0.

Notice that this proposes a rigorous definition for equidistant projections, which I believe has been lacking heretofore.

This definition is essentially equivalent to my "constant-resolution" property, mentioned in the first post of this thread.

Note, however, that the two-point equidistant projection does not have this property, and is equidistant in a different sense. Likewise for the lateral equidistant and Hammer retroazimuthal projections.

With regard to (d), the gnomonic projection has γ = 2.

Which makes sense, because the gnomonic projection can be thought of as "beyond" the stereographic projection, in terms of having even worse area distortion than the stereographic projection.

By the same logic, we would expect the orthographic projection to lie "beyond" the azimuthal equal-area projection, since it has even worse angle distortion (and the actual functions involved also follow a simple relationship: tan(r) ≥ 2 tan(r/2) ≥ r ≥ 2 sin(r/2) ≥ sin(r)). The gnomonic and orthographic projections also have the same "same resolution-efficiency" property with each other as the stereographic with the azimuthal equal-area.

In actuality, attempting to find a value of γ for the orthographic (as log(b)/log(a)) gives a value of infinity, in a way that does not clarify whether it is positive or negative infinity, instead of the "gnomonic-dual" value of −2.

EDIT: No, wait a moment, you made a mistake. The value for the gnomonic projection isn't +2, it's +0.5. That's even worse, because it wrongly gives the impression that the gnomonic projection is a compromise projection (lies between a conformal and equal-area projection).

I tried using the same principle once myself to attempt to formalize the spectrum of compromise projections (etc.). I knew it went wrong somewhere.

[daan]

My well defined (formal?) proposal, which I partially talk about here, is that duals are:

An interesting link, but I don't see the word "dual" anywhere on that page?

I don’t use the term in that old posting; I merely express a unique relationship between conformal projections and a restricted set of equal-area projections. That restricted set would be the duals.

• the power relationship γ of the Tissot axes a to b is constant: a^γ = b.

Umm, this is a property of a single projection, not a relationship between two projections. Clearly many projections have nonconstant γ, therefore their dual would also be expected to.

What I expressed there is a constraint that a conformant projection must satisfy, not a relationship between projections.

Clearly many projections have nonconstant γ, therefore their dual would also be expected to.

Such projections are not conformant and do not have duals of the kind that I am referring to. The cardinality of conformal projections is smaller than the cardinality of equal-area projections, and therefore we cannot expect arbitrary projections to have conformal duals, which means they do not have duals by my usage of the term.

3. For conformal projections, γ = 1; for equal-area, γ = –1; for equidistant, γ = 0.

Notice that this proposes a rigorous definition for equidistant projections, which I believe has been lacking heretofore.

This definition is essentially equivalent to my "constant-resolution" property, mentioned in the first post of this thread.

Nearly. You do permit a constant stretching in the direction in which the equidistant property is realized, which mine does not. My definition has another constraint: isocols coincident with its conformal dual. I’m not sure my second constraint is reasonable. Just a proposal.

Note, however, that the two-point equidistant projection does not have this property, and is equidistant in a different sense. Likewise for the lateral equidistant, though that sees less real use.

Right. The term “equidistant” has been applied to several projections, including, for example, Bonne/Werner (because distances from the north pole are correct). I think these usages expand the meaning in arbitrary directions that aren’t really helpful. The ”two-point equidistant” is itself an unfair appropriation; any normally recognized equidistant projection is equidistant from two-points. You just don’t get to specify the two points arbitrarily.

By the same logic, we would expect the orthographic projection to lie "beyond" the azimuthal equal-area projection…

Well… by the definition I gave, γ could never be greater than 1, since greater than that would exchange the major and minor axes. The gnomonic would actually be –2, not 2, by the definition I gave. Sorry about that. Your point about the orthographic, and my comment in the previous sentence, shows that the definition I gave for γ is incomplete. You’ll notice that I defined antisocol but never used it in my posting. The reason I mentioned antisocol is that the real definition includes the orientation of the major axis against the antisocol, a fact that I failed to recall explicitly as I was bashing out words. In a conformant projection, one of the two axes is oriented along the isocol, and the other along the antisocol. The usual formulas given for calculating semimajor and semiminor axes are likewise broken in that they give reversed results for major and minor axes when the major axis lies along the antisocol (that is, when spacing of parallels accelerates more than the stereographic in polar azimuthals, for example). I’ll post more later when I get time.

 — daan

[Milo]

I don’t use the term in that old posting; I merely express a unique relationship between conformal projections and a restricted set of equal-area projections. That restricted set would be the duals.

Ah, I found the paragraph you're refering to. It's near the bottom: "What this implies is that there is an equal-area analog to any conformal map such that the equal-area map has isocols coinciding with the isocols of the conformal map that is optimal for the same region."

What I expressed there is a constraint that a conformant projection must satisfy, not a relationship between projections.

What you meant, maybe. It's not what you said.

You prefaced your two conditions with "duals are:", not "conformant projections are:", and the first condition was definitely in terms of comparing two projections rather than defining a conformant projection (explicitly confirmed in your clarification in the following paragraph, where you explicitly state that "aligned" is comparing the isocols of two different projections, not comparing the isocols of one projection to some other property of that same projection), so your second condition which is about a single projection doesn't fit as written.

...It also doesn't work because all normalized equal-area projections satisfy the condition that γ = −1 (and therefore constant), contradicting your claim that not all equal-area projections are conformant.

This definition is essentially equivalent to my "constant-resolution" property, mentioned in the first post of this thread.

Nearly. You do permit a constant stretching in the direction in which the equidistant property is realized, which mine does not.

Hmm?

My definition is that b has the same value everywhere, while your definition is that b is 1 everywhere. Clearly, rescaling so that whatever constant value you have becomes 1 is trivial, and does not meaningfully change the projection. Although I suppose you're right that your definition would technically exclude formulations for which this rescaling has not been applied.

(The same vagary is also applicable to equal-area projections. Technically, any projection with constant flation everywhere, even if that flation is not 1, would meet the literal definition of "equal-area", but in practice it's pretty much always convenient to renormalize so the constant is 1. At least for the purposes of analyzing the mathematics, even though most world maps aren't life-size.)

My definition has another constraint: isocols coincident with its conformal dual. I’m not sure my second constraint is reasonable. Just a proposal.

Ah, okay. Yes, that narrows things down. Or... not, since you defined duals as projections with the same isocols, so if a conformal dual exists at all, then its isocols will be coincident.

Though if you're trying for that kind of thing, it would seem more natural to use a definition that works in the projection itself, rather than needing to reference an as-yet-uncomputed dual. Such as requiring that the projection has zero torsion (in the sense you define in your linked post: Tissot major axes are parallel to isocols). It is not immediately obvious that this is the same as having a conformal dual. Or, well, that would require you to clean up your definition of what a conformal dual is before deciding.

Then again, as you point out, equal-area projections are more diverse than just the ones with zero torsion / conformal duals / whatever, so why shouldn't the same be true for equidistant projections?

Anyway, if both equal-area and equidistant projections can have conformal duals, but conformal projections are in short supply, then either duality isn't guaranteed in the other direction either (some conformal projections have equal-area duals, and some conformal projections have equidistant duals, but not both), or conformal projections can have more than one dual, in which case, well, calling it a "dual" is a misnomer. When things are called "duals" in mathematics, it's generally meant to be a self-inverse operation: the dual of a dual is the original thing again. Under the logic where I fancied conformal and (some?) equal-area projections being each others' duals, based on them having the same resolution-efficiency, equidistant / constant-resolution projections would be the self-dual ones.

Right. The term “equidistant” has been applied to several projections, including, for example, Bonne/Werner (because distances from the north pole are correct). I think these usages expand the meaning in arbitrary directions that aren’t really helpful.

You're right that this precludes defining a consistent criterion of "equidistance" which is satisfied by all equidistant projections.

Nonetheless, these projections (well, at least the two-point equidistant one) are clearly useful and their equidistant property is relevant in the sense that you will often use them for applications where you want to measure distances. One of my examples was even one that you named "equidistant" yourself.

In general being equidistant along some paths is not a particularly useful property, unless there is a reason why those specific paths (such as meridians) interest you more than others.

This is why I avoided trying to define "equidistant" and instead used the term "constant-resolution".

The ”two-point equidistant” is itself an unfair appropriation; any normally recognized equidistant projection is equidistant from two-points. You just don’t get to specify the two points arbitrarily.

Unless your two points are "a point and its antipode", I don't see what you mean here. (On the sphere, knowing the distance from one point will automatically tell you the distance from its antipode just because of how spherical geometry works. This does not necessarily extend to the projected antipodes, though, so the two-point equidistant projection is still not a four-point equidistant projection.)

Well… by the definition I gave, γ could never be greater than 1, since greater than that would exchange the major and minor axes. The gnomonic would actually be –2, not 2, by the definition I gave. Sorry about that.

Note my edit. If the axes are sorted so a is always the semimajor (larger) and b is always the semiminor (smaller) axis, then the proper value for the gnomonic projection would be 0.5, not 2 or −2. I've computed this multiple times, I'm pretty sure it's right.

It's still technically possible for γ to be greater than 1, but this requires a semimajor axis which is smaller than 1, and a semiminor axis which is even more smaller than 1. While clearly possible to construct as an artificial example, this is never seen in any practical projection.

If you instead define a as the axis parallel to the isocol and b as the axis perpenducular to the antisocol (which can, of course, only be done when the Tissot ellipses are in fact oriented correctly), then the gnomonic projection does have a value of +2. All other values remain as stated. But with this, γ becomes undefined for many projections, even locally.

(I'm not entirely certain about your definition of "antisocol" as "the path of greatest change in distortion". Is it guaranteed that this path will always be perpendicular to the isocol? If not, then you clearly can't align Tissot ellipses to it.)

[daan]

What I expressed there is a constraint that a conformant projection must satisfy, not a relationship between projections.

What you meant, maybe. It's not what you said.

Apologies for the poor wording. I edited the original posting; hopefully it’s clearer now. I bungled several things.

...It also doesn't work because all normalized equal-area projections satisfy the condition that γ = −1 (and therefore constant), contradicting your claim that not all equal-area projections are conformant.

Both conditions must be fulfilled; hence the and.

This definition is essentially equivalent to my "constant-resolution" property, mentioned in the first post of this thread.

Nearly. You do permit a constant stretching in the direction in which the equidistant property is realized, which mine does not.

My definition is that b has the same value everywhere, while your definition is that b is 1 everywhere. Clearly, rescaling so that whatever constant value you have becomes 1 is trivial, and does not meaningfully change the projection.

Scaling in one direction, or radially, is trivial, but nothing about either of our definitions requires the Tissot minor axes to be oriented such that this is possible. Instead, the planar orientation of the axes can vary across the map, in which case it cannot be scaled in only one direction without changing the relationship of the major axes to the minor axes. That would meaningfully change the character of the map. Of course, without scaling the map, you could scale your reading of distance off of the map to compensate for the non-unitary semiminor axis, so it’s not a problem in that sense.

Though if you're trying for that kind of thing, it would seem more natural to use a definition that works in the projection itself, rather than needing to reference an as-yet-uncomputed dual. Such as requiring that the projection has zero torsion (in the sense you define in your linked post: Tissot major axes are parallel to isocols). It is not immediately obvious that this is the same as having a conformal dual.

Merely having zero torsion is an insufficient condition for having a conformal dual… at least, here is an example that makes the situation muddy: You can assign the nominal scale of a Mercator map to be whatever you want depending on what parallel you want to say has a scale factor of 1. Normally we just say it’s the equator. The only change is scaling based on an arbitrary nominal scale. Meanwhile, all of the cylindrical equal-area projections have zero torsion, but presumably only one of them is the dual of the Mercator. You could, possibly, say that the dual is the one that has no distortion along the parallel that you have assigned on the Mercator to have a scale factor of 1. But the very character of the equal-area map changes based on the standard parallel.

Then again, as you point out, equal-area projections are more diverse than just the ones with zero torsion / conformal duals / whatever, so why shouldn't the same be true for equidistant projections?

That’s a good argument, and I haven’t made any particular argument for why it should be constrained.

Anyway, if both equal-area and equidistant projections can have conformal duals, but conformal projections are in short supply, then either duality isn't guaranteed in the other direction either (some conformal projections have equal-area duals, and some conformal projections have equidistant duals, but not both), or conformal projections can have more than one dual, in which case, well, calling it a "dual" is a misnomer.

I think your question is an artifact of my poor explanation. I do not mean to say that a conformal projection has only one dual. Rather, it has one equal-area dual. It also has a lot of other duals, one for each γ. The dual is “per γ per conformal projection”, not “per conformal projection”.

Nonetheless, these projections (well, at least the two-point equidistant one) are clearly useful and their equidistant property is relevant in the sense that you will often use them for applications where you want to measure distances. One of my examples was even one that you named "equidistant" yourself.

I agree, but I also think I would not choose to name the “lateral equidistant” that a second time.

Unless your two points are "a point and its antipode", I don't see what you mean here.

That is what I mean.

Note my edit. If the axes are sorted so a is always the semimajor (larger) and b is always the semiminor (smaller) axis, then the proper value for the gnomonic projection would be 0.5, not 2 or −2. I've computed this multiple times, I'm pretty sure it's right.

This would be a place where I bungled my correction to my bungling. Of course it is either ½, with a = semimajor axis, or 2, with a = axis aligned to isocol.

It's still technically possible for γ to be greater than 1, but this requires a semimajor axis which is smaller than 1, and a semiminor axis which is even more smaller than 1. While clearly possible to construct as an artificial example, this is never seen in any practical projection.

This is the normalization problem for scale, which I did not address and would need to in order for the γ relationship to make sense in the general case.

If you instead define a as the axis parallel to the isocol and b as the axis perpenducular to the antisocol (which can, of course, only be done when the Tissot ellipses are in fact oriented correctly), then the gnomonic projection does have a value of +2. All other values remain as stated. But with this, γ becomes undefined for many projections, even locally.

It is intended to be undefined for most projections, and it is not intended as a local property. It is defined for any conformant projections as a condition for being conformant. It is an insufficient condition.

(I'm not entirely certain about your definition of "antisocol" as "the path of greatest change in distortion". Is it guaranteed that this path will always be perpendicular to the isocol? If not, then you clearly can't align Tissot ellipses to it.)

Am I grossly wrong here? Isn’t this why directional derivatives even work? The gradient is perpendicular to the direction of the level curve.

— daan

[Milo]

Scaling in one direction, or radially, is trivial, but nothing about either of our definitions requires the Tissot minor axes to be oriented such that this is possible.

Which is why I'm not scaling in one direction (which, while easy to do, would result in a qualitatively different projection - for example a one-directionally-scaled version of a conformal projection will not still be conformal). I'm talking about scaling in both directions.

Yes, this means that the semimajor axis will be scaled as well as the semiminor axis, but since neither of our definitions made any assertions of the value of the semimajor axis, only the semiminor axis, this is a non-issue.

Merely having zero torsion is an insufficient condition for having a conformal dual… at least, here is an example that makes the situation muddy: You can assign the nominal scale of a Mercator map to be whatever you want depending on what parallel you want to say has a scale factor of 1. Normally we just say it’s the equator. The only change is scaling based on an arbitrary nominal scale. Meanwhile, all of the cylindrical equal-area projections have zero torsion, but presumably only one of them is the dual of the Mercator. You could, possibly, say that the dual is the one that has no distortion along the parallel that you have assigned on the Mercator to have a scale factor of 1. But the very character of the equal-area map changes based on the standard parallel.

Perhaps the Mercator projection simply does have a different dual depending on how it's scaled, even though those scalings are for most purposes treated as "the same" projection.

Or perhaps it might be better to define duals on equivalence classes of projections, rather than individual projections. Then all equal-area projections, put together, can be considered the dual of the Mercator.

Then again, it's not unreasonable to suppose that the choice of standard parallel really is important. As I've noted, among the cylindrical equidistant projections, the plate carree (standard parallel at the equator) has properties of resolution-efficiency and constant-resolution not shared by other cylindrical equidistant projections.

I think your question is an artifact of my poor explanation. I do not mean to say that a conformal projection has only one dual. Rather, it has one equal-area dual. It also has a lot of other duals, one for each γ. The dual is “per γ per conformal projection”, not “per conformal projection”.

While you might be onto an interesting concept if it's refined some more, I still think that you should come up with a different name for it because this is not how "dual"s typically work in mathematics.

(I'm not entirely certain about your definition of "antisocol" as "the path of greatest change in distortion". Is it guaranteed that this path will always be perpendicular to the isocol? If not, then you clearly can't align Tissot ellipses to it.)

Am I grossly wrong here? Isn’t this why directional derivatives even work? The gradient is perpendicular to the direction of the level curve.

Maybe. It's not a field I've studied a lot.

I looked it up on Wikipedia now, and it turns out you're right. (At least so long as the function is differentiable, which seems like a reasonable assumption.)

[daan]

Scaling in one direction, or radially, is trivial, but nothing about either of our definitions requires the Tissot minor axes to be oriented such that this is possible.

Which is why I'm not scaling in one direction (which, while easy to do, would result in a qualitatively different projection - for example a one-directionally-scaled version of a conformal projection will not still be conformal). I'm talking about scaling in both directions.

Got it. Thanks for the clarification.

While you might be onto an interesting concept if it's refined some more, I still think that you should come up with a different name for it because this is not how "dual"s typically work in mathematics.

Fair enough. I hadn’t used the term “dual” until you inspired me by mentioning duals in a context by which I thought you meant something like my analogs:

Of course there is some abiguity in what projections are considered "analogous to" each other. There doesn't (currently!) exist any formal theory of "dual projections", so it's more a matter of looking similar in some aspect that we happen to care about.

I’m not sure that my usage of “dual” here is mathematically objectionable, but I’m happy to abandon it if you think so. It seemed reasonable to me because the point of my exploration is to confirm or refute my hypothesis that conformal mapping concepts can be carried over into equivalent and aphylactic projections such that a given conformal map can be transformed into another mapping which can then itself be transformed back into a conformal map after operations that preserve duality. Each γ defines its own dual space.

Cheers,
— daan

[Milo]

Come to think of it, isocols are not even well-defined for general projections. Angle and area deformation, or Tissot semimajor and semiminor axes, can have isocols that in the general case are not equal to each other. Now, in the specific case of a projection with constant γ, isocols are well-defined, so it's not a huge dealbreaker... but if you define your γ in terms of the isocol and antisocol directions, rather than the Tissot semimajor and semiminor directions, then you get a bit of circular dependency in that you need to know where your isocols are to determine the value of γ, which you in turn need in order to unambiguously identify isocols. It's not quite intractable - I'm pretty sure you can reformulate the criteria in a way that doesn't rely on circular dependency - but it does require some careful mathematical juggling.

[PeteD]

I know I'm nearly a year late, but I have some observations about all this.

You can assign the nominal scale of a Mercator map to be whatever you want depending on what parallel you want to say has a scale factor of 1. Normally we just say it’s the equator. The only change is scaling based on an arbitrary nominal scale. Meanwhile, all of the cylindrical equal-area projections have zero torsion, but presumably only one of them is the dual of the Mercator. You could, possibly, say that the dual is the one that has no distortion along the parallel that you have assigned on the Mercator to have a scale factor of 1.

This suggestion makes goods sense to me. After all, the cylindrical equal-area projection that has no distortion along a given parallel has the same overall distortion value according to the (unweighted) Airy-Kavrayskiy criterion as the Mercator projection that has a scale factor of 1 along the same parallel.

Interestingly, while the azimuthal equal-area and stereographic projections are analogues according to daan's proposed definition and have the same resolution efficiency, the cylindrical equal-area and Mercator projections are also analogues according to daan's proposed definition but have different resolution efficiency values.

Well, resolution efficiency is zero for both the cylindrical equal-area and Mercator projections if the calculation is performed over the whole globe. What I mean is that if the calculation is limited to a given latitude, their resolution efficiency values are different, unlike for the azimuthal projections, where limiting the calculation to a given radius (which corresponds to latitude in polar aspect) still gives the same resolution efficiency for the azimuthal equal-area and stereographic projections.

Assuming my calculations are correct, the cylindrical projection with the same resolution efficiency as the Lambert cylindrical equal-area is given by

x = lambda
y = tan phi * sec phi

and is even worse than the central cylindrical projection:

analogue of Lambert CEA.png (72.66 KiB) Viewed 7907 times

On the other hand, the Airy-Kavrayskiy criterion gives the same distortion value to the cylindrical equal-area and Mercator projections but different distortion values to the azimuthal equal-area and stereographic projections.

[PeteD]

By the way, why do pictures now appear as links?