Quantitative evaluation & comparison of world-maps

[RogerOwens]

First, because most of theses comparison-quantities described below refer to scale, then something should be said about scale.

Scale is distance on the map for a given distance on the Earth. ...such as inches on the map per mile on the Earth (On a map,that's usually written as its more convenient reciprocal, miles/inch.)

But scale is defined for a point on the map. Just as a car's speed is defined for an instant, and is called "instantaneous speed", so scale on a map is defined at a point.

In mathematical terms, then, at any particular point on the map, and in any particular direction from that point, scale is the limit of (say) inches on the map per mile on the Earth, as those two distances approach zero.

...just as an instantaneous speed in meters per second is the limit of an object's meters per second as the duration in seconds approaches zero.

In general, scale in various different directions at a particular point differ from eachother.

Of course you can divide a finite (measurable, non-infinitessimal) distance by a finite duration, and then you get a car's average speed over that duration. You could do the same thing with miles on the Earth and inches on the map, to give you the average scale along some path on the map. But, when said without that "average" qualification, scale refers to scale at some point on the map.

On a globe, and if the Earth is assumed perfectly spherical, the scale is constant everywhere on the globe. That can't be true of any flat map. On any flat map, the scale is different at different points on the map.

That's to be expected, since to place a globe's surface onto a flat table, of course there would have to be much deformation of that surface...expansion, compression, and maybe shear.

Some maps are specially designed so that, at every point, the scale is the same in every direction at any particular point on the map. Such maps are called conformal. Actually, the first such map (Stereographic) wasn't deigned to be have that property, but was discovered centuries later to have it.

But, in general, at at least nearly all points on non-conformal maps, the scale at any point is different in different directions.

Some maps are designed so that the areas of all regions on the map are in the correct proportions to eachother. They're called equal-area maps. Achieving the equal-area property results in an increase the amount by which scale differs in different directions at points on the map.

Some maps are neither equal-area nor conformal, and some of those are intended to attempt a compromise between those two properties.

I define three reference distances:
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X-reference-distance: The map’s largest extent in its X-dimension
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Y-reference-distance: The map’s largest extent in its Y-dimension
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Mean-reference-distance: The geometric-mean of the above two reference-distances.
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Relevant reference-distance, for a given projection and a given display-space, means:
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1. X-reference distance, if, in that display-space, the size of a map on that projection is limited by the display-space’s available space in the map’s X-dimension
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2. Y-reference-distance, if, in that display-space, the size of a map on that projection is limited by the display-space’s available space in the map’s Y-dimension.
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3. Mean-reference-distance, if neither of the above two conditions obtains. (Maybe it isn’t known where or in what orientation the map will be mounted. Or maybe it will be sharing a wall with various smaller cards and posters, so that its area is all that’s relevant to its fit there.)
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When I say speak of a map’s “min scale”, I refer to the smallest scale on that map (out to lat 71.2, the lat of North Cape), divided by the relevant reference-distance.
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When I speak of a map’s “min/max” scale, I refer to the smallest scale on that map (out to North Cape), divided by the largest scale on that map.
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“Point-min-scale” refers to the smallest scale at some particular point on the map.
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A map’s “av point-min-scale” is the average of the point-min-scale, over all of the map’s points, divided by the relevant reference-distance.
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“Point-min/max scale” refers to the smallest scale at some particular point on the map, divided by the largest scale at that same point.
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A map’s “av point-min/max scale” refers to the average of the point-min/max scale, over all of the map’s points.
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A map’s “av scale” refers to the square-root of the map’s area, divided by the relevant reference-distance.
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(Of course if the relevant reference-distance is the mean reference-distance, then the map’s av scale equals the square-root of its space-efficiency.)
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A map’s “space-efficiency” is the % of its circumscribing rectangle that that map fills.
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(I acknowledge that the above “av scale” isn’t necessarily really the arithmetic-mean, over all of the map’s points, of the map’s scales in all directions at each point. …but it’s easier to determine.)
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One more measure that I sometimes us for equal-area maps, because it’s easily estimated via a map that shows Tissot-ellipses:
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Tissot-ellipse-estimated min-scale:
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The square-root of: (The min/max dimensions of the thinnest Tissot-ellipse that the map shows, multiplied by the map’s area). …with that result divided by the relevant reference-distance.
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If the relevant reference-distance is the mean–reference-distance, then then the above expression is equal to the square root of (min/max dimensions of the thinnest Tissot-ellipse that the map shows, multiplied by the map’s space-efficiency).
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If desired, here is a more complete definition of scale:

On the flat surface of a map. From a point on that surface, there are displacements along that surface in every direction along that surface from the point. Furthermore, any displacement along that surface from that point is in a direction from that point, on that surface .
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A displacement is a vector. By one description, a vector has a magnitude and a direction. The magnitude of a displacement is called a "distance".
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From some particular point on the map, I'll refer to a particular displacement along the map-surface, and call it the "map-displacement. It corresponds to a displacement on the Earth, which I'll call the "Earth-displacement).
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The direction of the map-disiplacement is the "Earth-direction". The direction, on the Earth's surface, of the displacement on the Earth that the Map-Displacement corresponds to is the "Earth-direction".
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The "map-distance" is the magnitude of the map-displacement. The "Earth-distance" is the magnitude of the Earth-displacement.
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At that point on the map, and for a particular map-displacement and its corresponding Earth-displacement, and their directions, the map-direction and the Earth-direction:
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I define the "scale" at that point, in that map-direction, as the limit of the map-distance for a given Earth-distance, as the map-distance approaches zero.
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That could be expressed by the llmit of (map-distance divided by the Earth-distance), as the map-distance approaches zero.
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As I said, from any point on a map there are displacements along the map-surface in every direction from that point along that surface. And any displacement from that point, along that surface, is in a direction on that surface from that point.
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So obviously there's a scale in every direction along the map surface from every point on the map.
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Among those there are a max and min scale at that point.
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Among the min scales at all of the points on the map, the smallest of those is the min scale for the whole map.
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Michael Ossipoff
38 M
September 9th
2210 UTC

[daan]

“Point-min/max scale” refers to the smallest scale at some particular point on the map, divided by the largest scale at that same point.

Which is everywhere 1 for a conformal map.

A map’s “av point-min/max scale” refers to the average of the point-min/max scale, over all of the map’s points.

Which is 1 for a conformal map, and the closer to conformal the map is, the closer to 1 the value will be.

More generally, little of all that applies to “world map” in any generic sense. Indeed, it all really only applies to pseudocylindric maps in equatorial aspect—and even then, the “av min/max” stuff above is heavily biased against equal-area maps as being the least conformal.

— daan

[quadibloc]

One could also note that space efficiency is heavily biased in favor of cylindrical projections. But if an incomplete list of desiderata favors some projections over others, then adding the ones that are obviously missing will balance things out.

So one can add to the list a measure of how area varies from place to place. And certainly it's useful to have a list of numerical measures for the properties of a map.

If I were inclined to criticize something, instead, I think the more pertinent question is: even if I know which of the desirable properties of a map are more important to me, is there really a way in which the numerical ratings of different projections can be put to use to help me choose a projection? Can they be weighted in a useful way?

Suppose that I want a map that must be equal-area. I want the distortion of shapes to be low, consistent with that, but I also don't want the projection to be interrupted too many times. That is a very common example of what some people want. But judging which projection satisfies those conditions best is difficult. An uninterrupted Wagner VII? A MacBryde-Thomas Flat Polar Quartic with only a few interruptions? A Strebe Asymmetric? That is best left as a subjective judgment, rather than being settled on a point score.

EDIT: Upon reflection, instead of criticizing the technique for what it can't do, even if that is a better and more fundamental criticism, it is better and more useful to praise it for what it can do, and make use of that.

Yes, it isn't adequate as a way to pick out the "best" projection for a job. But there are a great many projections in existence. So having numerical scores for the various desirable attributes of a projection could be helpful in sorting through all these choices. One might, for example, have a system suggest additional projections useful for one's purpose that one had not known of, or at least had not considered. So this is where its usefulness lies.

[Atarimaster]

But there are a great many projections in existence. So having numerical scores for the various desirable attributes of a projection could be helpful in sorting through all these choices.

A while ago, I suggested a feature like this (see No. 3).

One might, for example, have a system suggest additional projections useful for one's purpose that one had not known of, or at least had not considered. So this is where its usefulness lies.

Did you check the “map projection selection tree” in the Geocart manual? It is helpful, altough it of course doesn’t list all the projections that are available in Geocart.
(The Geocart manual is available for free at the download page.)
It would be nice if you could select certain purposes and/or properties (and maybe, the area which is to be mapped, like in the Projection Wizard), and you get a list of appropriate projections, including the numerical scores you mention.

Whoever, I guess a “decision tool” like this would be a hell of work, I doubt that the benefits of it justify the expenditure.

Kind regards,
Tobias

[daan]

Upon reflection, instead of criticizing the technique for what it can't do, even if that is a better and more fundamental criticism, it is better and more useful to praise it for what it can do, and make use of that.

Broadly, I agree. For me, there is more to it.

1. Does this thing do what its promoter claims? (Arno Peters on the Gall orthographic: Fail)
2. How effectively does it to what it is claimed? (Arno Peters on the Gall orthographic: Poorly)
3. How does it compare to others? (Gall orthographic: Not useless; not usually desirable)

The business of concocting and promoting things without examining prior art to ascertain (1), (2), and (3), as well as how to talk about the thing, seems self-absorbed and disrespectful of other people’s time when it happens persistently or is argued forcefully. In extreme cases we have people embarked on a quest to convince people that the earth is flat, with some success. Violating (1) or(3), as well as misrepresenting (2), is miseducation. Miseducation, even if not recognized as such by its purveyor, particularly rankles me.

None of those violations necessarily mean that an idea is meritless when properly characterized, contextualized, and qualified. The Gall–Peters projection does have its occasional use. For example.

— daan

[RogerOwens]

Daan & Quadibloc—
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Thanks for your replies. They arrived late last night, and I can reply in more detail later today.

It’s true that my comparison system can’t be called universally-applicable until it expresses the currently-used comparisons in its own consistent language. …things like area variation in compromise-maps, and the difference between angles on the map and the angles on the Earth that they portray.
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Min/max scale and point min/max scale cover what’s usually compared in conformal-maps, even if in a different form.
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I claim that my comparison-system applies perfectly well to maps such as Briesemeister. More about that when I write again.
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Weighting the various measures of merit is too subjective to write into a comparison-system. The individual must choose what seems important. A comparison system can only provide some measures for comparison.
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More later…
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Michael Ossipoff

[daan]

“Point-min/max scale” refers to the smallest scale at some particular point on the map, divided by the largest scale at that same point.

This metric seems unmotivated. It is nothing other than an expression of angular error. Its relationship to ω is:
 Let c = asin(ω/2). [ should be sin, not asin, noticed by RogerOwens ]
 Let P = b/a, corresponding to Ossipoff’s “min/max scale”.
  Then c = (1 – P)/(1 + P).
Now, compared to, ω, the measure in common use, what characteristics does it have? Well, the graph makes it clear:

P as a function of ω; grid space = ½

Omega vs P.png (4.72 KiB) Viewed 3164 times

The graph’s x-axis is ω. The red curve is where P is for a value of x. Does that transformation help me visualize the distortion any better? Not really. With ω, I get a measure of 0 where there is no distortion—as expected. With P, the measure is 1. And then they kind of just look like an inverse scaled representation of each other, with a little curvature. Meanwhile, ω tells me the maximum deviation from orthogonal any projected axis at the point will reach. P doesn’t tell me anything specific about the geometry of the distortion. It also doesn’t tell me about the minimum and maximum scales because their separate contributions have been lost by combined into a single metric, just like ω.

So, no particular benefit. Ignoring ω and proposing P seems contrary.

— daan

[RogerOwens]

Dan & Quadibloc--
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Typo:
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I said: “Min/max scale and point min/max scale cover what’s usually compared in conformal-maps, even if in a different form.”
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Instead of “point min/max scale”, which, as you said, is 1.0 everywhere in all conformal maps, I meant to say “point scale”. (…something with meaning only on a conformal map).
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Point-min/max scale” refers to the smallest scale at some particular point on the map, divided by the largest scale at that same point.

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Which is everywhere 1 for a conformal map.

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Sure, but these comparisons aren’t for choosing between equal-area and conformal maps—That choice is made before one chooses a projection.
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I don’t mean for those comparisons to be used in a weighted aggregation of various merits into one rating.
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If some of my comparison-measures seem biased in against equal-area maps, that’s because I’m particularly interested in comparing equal-area maps, so it’s their faults that interest me most. I don’t choose between conformal maps because there’s only one that I like. But yes, of course, as I said, if I want to say that my comparison-system is universally-applicable, then it should also express the comparisons that others make but I don’t make. …and compare maps that I don’t care for, such as compromise-maps.
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Such things as min-scale, av point-min-scale, and av-scale are relevant to all kinds of maps.
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Sure, space-efficiency can be said to be biased in favor of cylindricals, but only in the sense that races are biased in favor of fast-runners. Space-efficiency measures for a kind of genuine merit, and it also sometimes figures in the determination of min-scale or av scale.
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Sure, I said nothing about a measure of departure from equal-area, but, for a conformal map, min/max scale or its reciprocal, or the square of that reciprocal is what is typically compared, isn’t it? Maybe sometimes point-scale or its square in various regions of the map is shown.. So my comparison-system includes a comparison that tests and compares conformal maps by the usual standards for them.
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Admittedly my comparison-system doesn’t have a measure for area-misportrayal in maps that aren’t conformal or equal-area, but I don’t like those maps anyway, and so it didn’t occur to me. But yes, not measuring that for those maps could be called a biased lack of universality. I admit my bias in favor of equal-area and conformal maps, but a comparison-system limited by my biases can’t be called universally-applicable.
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So yes, a universally-applicable comparison system must deal, in its own language, with area inequality on compromise-maps, and the amount by which angles on the map differ from the angles on the Earth that they portray. …things, so far, missing from my comparison-system.
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A map’s “av point-min/max scale” refers to the average of the point-min/max scale, over all of the map’s points.

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Which is 1 for a conformal map, and the closer to conformal the map is, the closer to 1 the value will be.

…thereby rewarding an equal-area map that doesn’t hurt point-min/max-scale too badly. Low min-max scale, especially at the center of the map, is what disqualifies Tobler CEA. In general, of course low min-max scale is the bane of CEA. That’s why I like the idea of improving CEA, with CEA-Stack. It seems likely that there’s a combination of parameters for it that will give CEA-Stack acceptable min/max scale, while keeping a min-scale that’s better than any other cylindrical or pseudocylindrical equal-area world-map.
(I was disappointed when I found-out about Optimal CEA-Stack’s low min/max scale, but there’s probably a parameters-compromise that will fix that, and still have prizewinning min-scale.)
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More generally, little of all that applies to “world map” in any generic sense.

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Well, all of it is intended to. All world-maps have imperfect min/max scale, min-scale, and av-point-min-scale. All of those things sometimes need to be tested. All nonconformal maps have imperfect av point-min/max scale.
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Indeed, it all really only applies to pseudocylindric maps in equatorial aspect

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Well, even if the graticule is displaced to make it oblique with respect to the projection-equator & central-meridian, the projection & central-meridian are still there. Also, “X & Y” could be replaced by “short & long dimensions”. A map with curved meridians, even if its graticule is then tipped with respect to the original equator & central-meridian, still has short & long dimensions. …and, in fact, still has projection equator and central-meridian (as opposed to graticule equator and the graticule meridians all of which are now all curved).
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My comparison system doesn’t need anything more than that there be a map-dimension that is the one that’s critical to fit in a given display-space. Depending on what orientation you want to mount the map in, and depending on whether the display-space’s space-limitation is horizontal or vertical, either the map’s Y-dimension (parallel to the projection central-meridian) or its X-dimension will be the one that limits the size of the map that you can mount in that display-space.
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Whether you call it “NS & EW”, or “projection X & Y”, or “long and short dimensions”, if anything, there’s always a dimension of the map that’s fit-critical in a given display situation. The map’s measure in that dimension is the “relevant reference-distance”.
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As I mentioned, if you aren’t particular about the mounting-orientation, then you can just mount the map with its shortest dimension parallel to the display-space’s space-limitation. Then, whichever map-dimension is the shorter one (usually the projection NS or Y dimension) will be the one whose distance is the relevant reference-distance.
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Anyway, that system applies perfectly-well to such maps as Briesemeister, and so I don’t think that there’s a universal-applicability problem, other than that the system doesn’t measure area-variation in nonconformal maps.
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—and even then, the “av min/max” stuff above is heavily biased against equal-area maps as being the least conformal.

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…because equal-area maps are what I most want to compare, making their faults more of interest to me. But, as I was saying, min/max scale, min scale, av point-min-scale, and av scale are of interest for all maps.
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if an incomplete list of desiderata favors some projections over others, then adding the ones that are obviously missing will balance things out.

Yes, in its present form, my comparison-system is biased, by not measuring area-distortion in compromise projections, and not mentioning difference between angles on the map and the angles on the Earth that they represent.
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So one can add to the list a measure of how area varies from place to place.

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Yes, and now that you’ve mentioned it, there are 4 measures of areal-accuracy that I’ll add, along with the application, to point-min-scale and point-max-scale, of that formula that you posted for angular-error.
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And certainly it's useful to have a list of numerical measures for the properties of a map.

[/quote][/quote]
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Exactly. Too few measures are used now. There are applications for which additional measures are needed, and that’s why I propose my comparison-system, with its various measures for comparison.
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Here are the 4 areal-accuracy measures that I suggest:
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1. min point-min*max-scale/max point-min*max scale. That’s the map’s most reduced area divided by the most exaggerated area.
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2. area divided by what it should be (what it would be on an equal-area map of the same area). …either shown in regions over the map, or stated as that quantity’s minimum throughout the map.
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At any point, there are a point-min-scale and a point-max-scale. Say they’re expressed in the convenient scale of cm on the map per radian on the Earth. So their product has units of square cm per square radian on the Earth.

Say that there's a very tiny circle on the Earth with an area of A square radians, where of course A is a very small number.

What will be the area of its portrayal on the map, in square cm?:

A * point min*max scale

What area should it have? What area would it have if the map were equal-area?:

A * (Area of map in square cm)/(Area of the Earth in square radians)

= A * (Area of map in square cm)/(4 pi)

So, the area of the portrayal of that circle on the map, divided by what it should be (would be if the map were equal-area) is:

A * point min*max scale, divided by
A * (Area of map in square cm)/(4 pi)

= point min*max scale * 4 pi, divided by (Area of map in square cm)

At least for now, I'll call that quantity "Is/Should". I don't claim that that's the best possible name for it.

One could state the minimum, over all of the points on the map, of Is/Should. That’s one of the areal-accuracy measures I propose,.

3. …or one could have a display-map showing the value of Is/Should in various regions of the map.

4. Similarly to some of my other comparisons, one could divide the map’s lowest value for point-min*max scale by the square of the relevant reference-distance.
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Those are the 4 measures of areal-accuracy that I propose.
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Michael Ossipoff
38 Tu
September 9th
2357 UTC

[RogerOwens]

First, I want to emphasize that I don’t claim that angular-error shouldn’t be used if someone likes it as a measure for comparison. Obviously it isn’t for me to say what someone else should like for comparison of the merit of projections.
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First I proposed some comparison-measures that aren’t currently used, while also referring to a few that maybe are currently in use. I didn’t mean that whatever I didn’t mention shouldn’t be used.
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Then, wanting to propose a complete and universally applicable and useful comparison-system, I realized that that means that I should explicitly declare that I propose, as part of that comparison-system proposal, that measures of areal-accuracy for all projections, and the widely-used angular-distortion measure, are part of that proposed system.
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So, in case I haven’t said it well enough yet: I hereby declare that the comparison-system that I propose includes the 4 generally-applicable areal-accuracy measures that I’ve defined, and also the widely-used measure called angular-distortion, calculated from point-min/max scale in the manner described in recent postings here by Quadibloc & daan.
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I personally don’t use angular-error in comparisons. And the only maps on which I’d make areal-comparisons are the conformal-projection (because compromise-projections don’t interest me), and I don’t choose between conformal projections, because there’s only one of them that I like.
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But I make the above-stated declaration because obviously a complete and universally applicable and useful comparison-system shouldn’t leave out widely-used and widely-preferred comparison-measures, or evaluation of widely-published kinds of projections.
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Having emphasized that, I can discuss the relative appeal or justification for point-min/max-scale vs omega, though I declare that omega part of the comparison-system that I propose.
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RogerOwens wrote:
“Point-min/max scale” refers to the smallest scale at some particular point on the map, divided by the largest scale at that same point.

This metric seems unmotivated.

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It’s motivated by this:





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Peters & Tobler CEA look as if they were made of wax, and someone forgot to turn on the air-conditioner.
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It is nothing other than an expression of angular error.

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Angular-error is calculated from it, but it’s primarily an expression of the ratio of two scales.
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Its relationship to ω is:
 Let c = asin(ω/2).
 Let P = b/a, corresponding to Ossipoff’s “min/max scale”.
  Then c = (1 – P)/(1 + P).

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Alright, P stands for point min/max scale.
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P doesn’t tell me anything specific about the geometry of the distortion.

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It tells us a lot about what a region will look like if, in that region, points predominantly have low point-min/max scale. In equal-area maps, it tells us by how much scale is shortened where point-min/max scale is low.
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So, no particular benefit.

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Point-min/max scale has the relevance and benefits that I mentioned.
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Ignoring ω and proposing P seems contrary.

[/quote]
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Yes I propose P, but I recognize that a lot of people value and use omega, and so I declare omega part of the comparison-system that I propose, though I don’t personally use it in comparisons.
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Ignore? Ignore this! :




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But let me answer about the motivation, justification, and need for P vs omega:
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Omega is calculated from P. As I understand it, first the min & max scales at a point are calculated, and then, from their ratio, omega is calculated. Omega is derived from the more fundamental quantity P.
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So, arguably P is the more natural, parsimonious and uncontrived of those two quantities. Omega seems just an unnecessary use of extra CPU time.
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2.
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P is more consistent with the rest of my comparison-system, which is based on scale. All of those comparison-ratings for all projections can be stated in terms of scale. To contrive a different expression for one of them seems unnecessary.
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Conformal projections are compared and evaluated mainly by the ratio of scale (or its square) in the map.
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That’s what min/max-scale is.
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Yes, in equal-area maps, the map’s largest and smallest scales are found at the same point, and in conformal map the max and min scales are always found in different parts of the map. You’ve made a distinction about that. But, whether the min and max scales are found at the same or different points, they’re still the map’s min and max scales. So, min/max scale is a comparison-measure that conformal maps have in common with equal-area maps. So it can be fairly said that the usual comparison-measure for conformal maps is nothing other than one of the comparison-measures that can be used among equal-area maps too.
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It’s simpler and more aesthetically-pleasing when there’s a common comparison-standard.
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3.
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Omega is the maximum departure, at a point, of an angle about that point on the map from the angle on the Earth that it portrays. But, looking at the map, even with Tissot-ellipses drawn, where should we look for that maximum? Who knows, without calculating it.
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In contradistinction, the scale-magnification, and the direction that has it, on a map is more obvious, noticeable, and often unmistakable and outright objectionable. Check out Tobler CEA again. It isn’t that I ignore omega, but P is what jumps up and slaps us in the face.
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So it’s what I compare.
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4.
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Yes, angular-error is important if you’re getting your sailing-direction from a map, when planning an ocean-voyage. But no one should get that direction by just measuring on a map other than Mercator. …and certainly not from a nonconformal map. …and least of all from an equal-area map. …but not from a compromise map either.
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Distances are relevant to travel time, and say something about travel-time, even if one only has a rough idea of scale. An approximate distance is useful for a first impression about a voyage. An approximate direction is likely to be a disaster at sea.
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So much for sailing-directions for sea-voyages. What about land-travel? Well, as at sea, approximate distance is of some use. Direction? You just follow the roads. You don’t need a compass or accurate direction from a map.
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Aside from road-trips, thematic-maps are the other instance in which someone might want to make a measurement on the map. Would it more likely be a direction, or a distance?
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I suggest that how far away from you a certain region is, with certain conditions portrayed on the thematic-map, is more of interest to you than in what direction it is. For one thing, as I said, you’d go there on roads. If you’re hiking cross-country, preferably use a conformal map, and certainly not an equal-area map. But certainly don’t use any kind of world-map for wilderness-hiking.
For the above reasons, I suggest that scale is more important than direction on a world-map.
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Michael Ossipoff
38 W
September 11th
2300 UTC

[daan]

I personally don’t use angular-error in comparisons.

Apparently you do. You just don’t use ω. You use a different, completely convertible metric that I abbreviate as P for this.

This metric seems unmotivated.

It’s motivated by this… Peters & Tobler CEA look as if they were made of wax, and someone forgot to turn on the air-conditioner.

My objection was to introducing P when ω already achieves the same thing. My objection was not to using (some measure of) angular error to prove how bad Gall–Peters is. &c. Straw man.

P doesn’t tell me anything specific about the geometry of the distortion.

It tells us a lot about what a region will look like if, in that region, points predominantly have low point-min/max scale. In equal-area maps, it tells us by how much scale is shortened where point-min/max scale is low.

False. As I pointed out in different words, when you use the ratio b/a, you lose all information about a and b individually, with the exception of knowing whether or not b is 0. If, and only if, the map is equal-area, then will you still be able to compute a and b individually from it. Otherwise, no, you cannot claim anything about min/max because you have lost that information. You repeat this fallacy any number of times and ways in your posting.

Point-min/max scale has the relevance and benefits that I mentioned.

It provides no benefit over ω, and the benefits you claim beyond those of ω are false claims, with one exception that I note later.

Ignoring ω and proposing P seems contrary.

Ignore? Ignore this!

I was talking about ignoring ω, not ignoring distortion. I believe that was abundantly plain.

But let me answer about the motivation, justification, and need for P vs omega:
1. Omega is calculated from P. As I understand it, first the min & max scales at a point are calculated, and then, from their ratio, omega is calculated. Omega is derived from the more fundamental quantity P.

False. I cast ω in terms of P in order to show that they’re just simple transformations of each other, and so neither intrinsically contains any more or less information. Apparently you do not understand that I could cast P in terms of ω in the same way. In point of fact, no one uses P to compute ω; that would be pointless.

Omega seems just an unnecessary use of extra CPU time.

Computational expense is the one “benefit” of P, but paltry. Getting a and b in the general case is far more computational expense. As for “unnecessary”, given that P tells me nothing that ω does not, and ω tells me something more directly than P does, the claim is false.

P is more consistent with the rest of my comparison-system, which is based on scale. All of those comparison-ratings for all projections can be stated in terms of scale.

This is false. Once you divide b and a, you lose any relationship to scale. It make no difference if you used scale to compute P.

That’s what min/max-scale is.

One more time: It’s not. Minimum and maximum scale say everything about P, but P says nothing about minimum and maximum scale. Give that up.

Omega is the maximum departure, at a point, of an angle about that point on the map from the angle on the Earth that it portrays. But, looking at the map, even with Tissot-ellipses drawn, where should we look for that maximum? Who knows, without calculating it.

With some experience, it becomes obvious where those are just by looking, actually.

In contradistinction, the scale-magnification, and the direction that has it, on a map is more obvious, noticeable, and often unmistakable and outright objectionable. Check out Tobler CEA again. It isn’t that I ignore omega, but P is what jumps up and slaps us in the face.

One more time: It doesn’t. P tells you nothing about minimum and maximum scales at a point in the general case. Yes, it does in the case of equal-area projections, but you keep claiming things like “completely general”, which is exactly what this ad hoc stuff is not. Furthermore, do you even know how to manipulate P to tell you what the minimum and maximum scale is for a point on an equal-area maps And can you do that in your head?

And then look at conformal maps. P is everywhere 1.0. That tells you nothing about the actual minimum and maximum scales, so I don’t know what the relevance of this is:

So, min/max scale is a comparison-measure that conformal maps have in common with equal-area maps. So it can be fairly said that the usual comparison-measure for conformal maps is nothing other than one of the comparison-measures that can be used among equal-area maps too.

You seem to be persistently confusing the actual minimum and maximum scales with P.

Yes, angular-error is important if you’re getting your sailing-direction from a map, when planning an ocean-voyage. But no one should get that direction by just measuring on a map other than Mercator. …and certainly not from a nonconformal map. …and least of all from an equal-area map. …but not from a compromise map either.
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Distances are relevant to travel time, and say something about travel-time, even if one only has a rough idea of scale. An approximate distance is useful for a first impression about a voyage. An approximate direction is likely to be a disaster at sea.

This is a misunderstanding of the utility of ω, if that’s what you mean by angular error. Nobody uses it in the context of directions; the direction is not even part of ω. It’s used to express the amount of the greatest deviation of angle, not its direction.

I suggest that how far away from you a certain region is, with certain conditions portrayed on the thematic-map, is more of interest to you than in what direction it is. For one thing, as I said, you’d go there on roads. If you’re hiking cross-country, preferably use a conformal map, and certainly not an equal-area map. But certainly don’t use any kind of world-map for wilderness-hiking.
For the above reasons, I suggest that scale is more important than direction on a world-map.

Which has nothing to do with ω vs P because P has nothing to do with scale and ω has nothing to do with direction. P is just a variant of ω with no particular immediate utility, and since the two are completely convertible from each other, it has no extra utility no matter how much extra calculation you put into it.

— daan