Quantitative evaluation & comparison of world-maps

[RogerOwens]

I searched for some of the comparisom-scheme names you listed. Nothing.

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I thought the whole world as on the same Internet. My Internet gives:
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Airy-Kavrayskiy
Jordan-Kavrayskiy
Goldberg-Gott

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Searching those names doesn't lead to concise specifications of map-projection comparison or rating schemes.
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But don't worry about it.
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i once found a complete description of how Gott & Goldberg actually rate maps, but I'm not finding it in current searches.
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But, to answer your question about how my comparions-quantities relate to the 3 comparison-schemes you named in the above quote, I don't claim any such relation. But I guess it could be said that my comparison-criteria relate to those by being much more briefly, concisely and simply defined.
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For example, when I defined av-scale, I emphasized that I don't claim that it's really the arithmetical-mean of the scales in all directions at every point on the map. But it's a lot more easily-determined.
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As I said, i don't claim that my comparison-quantities are better, or that they should replace other ones. I merely propose them as additions.
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I've heard of Gott & whatever, but their comparison-scheme is a weighted aggregation of several comparison-quantities (which, it seems to me, were largely RMS global-aggregations of various -point-metrics).

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Nothing about that description is remotely correct.

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Maybe you mean that it isn't really true to say that flexion and skewness are point-limit quantities.
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But if you believe that G&G didn't use RMS, or that they didn't aggregate, with weighting, various comparison-quantities into a single rating-number, then you're mistaken.
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Michael Ossipoff
41 Su
October 6th
2231 UTC

[RogerOwens]

I should just add, in answer to your question about how my comparison-quantities compare to others:

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In addition to being briefly-defined, and simpler, less complicated than some, my favorite comparison-quantities have division by a reference-distance (or its square, in the case of map-area).

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If other proposed or used comparison-quantities have that, I haven't heard of it.

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Again, I'm not claiming that my comparison-quantities are better or should replace other ones. I just like them as an addition to the already-used comparison-quantities that I've heard of.

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Michael Ossipoff
41 Su
October 6th
2354 UTC

[daan]

I've heard of Gott & whatever, but their comparison-scheme is a weighted aggregation of several comparison-quantities (which, it seems to me, were largely RMS global-aggregations of various -point-metrics).

Nothing about that description is remotely correct.

Maybe you mean that it isn't really true to say that flexion and skewness are point-limit quantities.

I meant no such thing because I have no idea what “point-limit quantities” are because that’s not a term that any text I know of recognizes.

But if you believe that G&G didn't use RMS, or that they didn't aggregate, with weighting, various comparison-quantities into a single rating-number, then you're mistaken.

The Goldberg-Gott metrics are flexion and skew. Those measures don’t aggregate, weight, bring in “various ‘comparison quantities’” and they don’t combine them into a single metric.

They integrate flexion and skew each independently across the entire map, but not by RMS, not by weighting, not by “aggregating ‘various’ quantities”…

In their paper, they extend Laskowksi’s scheme to include flexion and skew “as a simple example of how our results may be combined with previous studies of map projections.” I presume this is what you are talking about. It’s not their scheme. You’re mistaken.

— daan

[quadibloc]

i once found a complete description of how Gott & Goldberg actually rate maps, but I'm not finding it in current searches.

Their paper "Flexion and Skewness in Map Projections of the Earth" contains this description, and it is still online.

[RogerOwens]

Maybe you mean that it isn't really true to say that flexion and skewness are point-limit quantities.

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I meant no such thing because I have no idea what “point-limit quantities” are because that’s not a term that any text I know of recognizes.

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Alright, then you didn't mean that flexion and skewness are not quantities consisting of limits evaluated at a point.
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In that case I have no idea what you meant, and I'm not going to continue trying to find out.
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Anyway, I'm not interested in debating the matter, or the matter of what G&G did.
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Michael Ossipoff
42 M
October 7th
1425 UTC

[RogerOwens]

i once found a complete description of how Gott & Goldberg actually rate maps, but I'm not finding it in current searches.

Their paper "Flexion and Skewness in Map Projections of the Earth" contains this description, and it is still online.

In an article of theirs, I clicked on "our paper", and received a screen at which I could pay to buy their paper.

When I clicked that link some years ago, the link led directly to their paper. This time it led to a payment-screen.

What G&G did or didn't do isn't sufficiently of interest to me to justify paying money to find out.

Michael Ossipoff

[RogerOwens]

The Goldberg-Gott metrics are flexion and skew. Those measures don’t aggregate, weight, bring in “various ‘comparison quantities’” and they don’t combine them into a single metric.

They integrate flexion and skew each independently across the entire map, but not by RMS, not by weighting, not by “aggregating ‘various’ quantities”…

I looked at their paper some years ago. It seems to me that they used a number of metrics of map-accuracy, including flexion and skewness, and some more familiar ones. Unless I'm mistaken, they evaluated each metric at many places on the map (but didn't sum over all the map's points). And it seems to me that they said that they used RMS to aggregate each metric over those many evaluations.

That's just what it seems to me that they said. As i told Quadibloc, when I clicked on "our paper" in their article this time, it led to a payment-screen.

Because the matter of how they aggregated all the various metrics they used into a single rating-number isn't at all important to me, then I'll just defer to whatever you say about it and take your word for it.

Michael Ossipoff
42 M
October 7th
1439 UTC

[RogerOwens]

As I've been saying, attempts to combine several different metrics into one rating-number don't interest me at all.

Michael Ossipoff
42 M
October 7th
1443 UTC

[RogerOwens]

Daan seems to be claiming that G&G didn't use RMS to globally-aggregate even any one of their various metrics, and that they didn't combine those separate metric-aggregations into one number via weighting.

That's surprising, and seems to contradict the G&G paper. The G&G articles that I've looked at gave numerical ratings of some projections. One can only wonder how daan thinks G&G got those single-number ratings if not by a weighted combination of various metrics.

Michael Ossipoff
42 M
October 7th
1850 UTC

[daan]

Daan seems to be claiming that G&G didn't use RMS to globally-aggregate even any one of their various metrics, and that they didn't combine those separate metric-aggregations into one number via weighting.

That's surprising, and seems to contradict the G&G paper. The G&G articles that I've looked at gave numerical ratings of some projections. One can only wonder how daan thinks G&G got those single-number ratings if not by a weighted combination of various metrics.

In their paper, they extend Laskowksi’s scheme to include flexion and skew “as a simple example of how our results may be combined with previous studies of map projections.” I presume this is what you are talking about. It’s not their scheme. You’re mistaken.

What I wrote flatly contradicts how you are characterizing what I wrote. I state, right there, that they gave an example of incorporating their results into Laskowksi’s scheme. Laskowksi’s scheme does aggregate metrics. Goldberg and Gott do not endorse their particular example of it; they flatly say that everything about it is debatable, and that its merely an example of what might be done.

The Goldberg-Gott measures are flexion and skew, not Laskowksi’s method of aggregation. Laskowksi’s scheme is Laskowksi’s scheme. Oddly.

You have categorically stated, several times now, that you are not interested in this topic of aggregated measures, and specifically stated this with respect to Goldberg-Gott. Why are you mischaracterizing what I wrote in order to talk about something you don’t want to talk about?

— daan