Some more information

[Atarimaster]

Hello,

here are three suggestions for Geocart, all of them refer to information of a selected projection, like you get from the information palette.
I don’t know if they’re really useful/interesting enough to put work into it, much less how much work it’d be to integrate them into Geocart…

1.)
I came up with this idea shorty after daan posted a very useful hint in another thread:

In Geocart you can always stretch or shrink a projection in one or both dimensions by any arbitrary amount. That capability lets you set a chosen parallel to have whatever scale factor you want. You can even figure out what values you should use by noting in the info palette the k scale factor when you have the pointer over the chosen parallel. Multiplying the width of the map by the reciprocal of that value will result in that parallel having constant scale.

Well, it’s near impossible to retrieve information for a specific point on the map.
For example, if I’d like to know the k-value for 30°N, I always end up with something like 29°59´43.438˝N, and when I move up the mouse pointer as carefully as I can, I’m at 30°0´1.191˝N.
Of course, this doesn’t make a bit of a difference for all practical purposes. Whether I multiply the width of the map by the reciprocal of 0.9458315 or 0.9458785 – well, I just tried, and even on a map which is three and a half meters in width, there isn’t a single pixel difference.

However, my suggestion is a new menu entry, something like "Show information for…", which prompts you to enter latitude/longitude values. Then a new window will be opened which looks like the information palette, but showing the values for this specific point.

As I’ve said, this is of little practical use, and the only reason that I’m suggesting this at all is that I guess it’d be fairly easy to implement, since Geocart already is capable of retrieving information about a specific point on the map, and the only difference is that this time, the point is determined by a numerical input rather than the position of the mouse pointer.
If my guess is wrong and it would require a greater amount of work: Forget about it. In my eyes it’s really more of a »nice to have« feature.


2.)
Another feature of that kind: Something that might be called »pixel scale«. What I mean is that in the information palette (the one that is already there, not the one I just proposed) it is shown how many square miles/kilometers are covered by the single pixel at the mouse pointer position.
Probably this only makes sense in the »Actual pixel of Selected Object« view mode, because otherwise, you’d have to deal with the fact that the screen pixel at that position is either interpolated by several actual pixels, or only a part of an actual pixel. So I’d be fine if that bit of information only appears in this view mode and is automatically dismissed at any other zoom stage.


3.)
This one is, in my opinion, the most interesting one because it provides information that might be of some real use. Unfortunately, it’s also the one that probably would require the most work. (Again, I’m just guessing here.)

I really like the »Distortion Tables« in Jenny’s & Patterson’s FlexProjector.
They look like this:

distortion-tables.png (22.88 KiB) Viewed 15806 times

They show mean values for scale, areal and angular distortions (all of them both for the entire globe and continental areas only) and Capek’s Q-index.
While I’m aware that values of that kind are only one criterion in finding an appropriate map for a given purpose, they might prove a valuable help especially when you’re working with one of the many projections that can be changed by certain parameters in Geocart.
While Geocart already provides a great deal of information for single points on the map, it’s those mean values that I miss a bit.



And finally, I’ve got a question – which is, admittedly, kind of another suggestion…
Are there any plans on adding Canters’s low-error polyconics to Geocart?
I especially like the *take a deep breath* low-error polyconic projection with twofold symmetry and equally spaced parallels and the low-error polyconic projection with twofold symmetry, equally spaced parallels and correct ratio of the axis.
Since they haven’t been added already, I speculate that they’re examples of those projections that you recently mentioned, which aren’t numerically stable everywhere.
Well, either that or you haven’t figured out so far how to fit those blasted names into Geocart’s menu…

Kind regards,
Tobias

[daan]

Wow. Thanks, Tobias. I like all of these suggestions. As you surmise, the third is the most work.

You’ll find Canters’s EU-optimized projection in Geocart 3.1. I might add others, if specific requests are made. He presents a •lot• of projections in his “Small Scale Map Projection Design” book, and it’s just not clear who might care about them.

— daan

[Atarimaster]

Ooops, I just realized that I never replied here anymore.
I was certain that I did, that’s why I wrote (in the "flawed cartometry" thread) that "I still think that [a list of Canters projections] would make a great addition to Geocart".

[Atarimaster]

Hello,

about two and a half year ago, I talked about my cartometry to calculate Capek’s Q; and a few weeks ago, I mentioned Peter Denner who helped me writing scripts for d3.

Recently, I started tinkering with the cartometry again, refined and extended it…
And it occurred to me that Q has some serious drawbacks. I know that all the comparison schemes have drawbacks and merits, but…
I already mentioned at some point that the van Leeuwen projection which has a large amount of angular deformation, gets an almost unbeatable Q-value. Nearly as good is the Gall-Peters, although on this projection, almost everything between the tropics has a max. angular deformation > 30°. And the Q value for the Györffy E^[1] – while I can’t get it using my cartometry, you can see by the contour plots provided by Györffy that it would be determined by the flation value K_max only, not all taking into account that it has a huge area with angular deformation below 20°.

I had a long email conversation with Peter about things like that and my wish that Geocart would offer some kind of global distortion metric.
And he came up with the following idea:

Geocart already calculates areal distortion and angular deformation as a*b and 2*arcsin((a-b)/(a+b)), respectively. It would be straightforward to convert these to (log(a*b))^2 and (log(a/b))^2. These two local metrics could then be globally averaged separately, and the square roots of these global averages would give E_p and E_a: separate global metrics for areal distortion and angular deformation.
In many cases, these separate metrics are more useful than a combined metric. Since there isn't just one single correct weighting for combining areal distortion and angular deformation, I would suggest allowing the user to choose the weighting, with a default fifty–fifty value giving E_K = sqrt(1/2 * E_p^2 + 1/2 * E_a^2) = sqrt(2) * E_AK. In addition to providing separate global metrics for areal distortion and angular deformation, and allowing the user to choose their preferred weighting, this would have the further advantage of presumably being easier to implement than Jordan–Kavrayskiy, Canters or Goldberg–Gott.

This sounds quite reasonable to me, but of course I don’t know how easy/hard the implementation would be; or which drawbacks it would have.
So, what do you think?

Kind regards,
Tobias

[1] János Györffy: Minimum distortion pointed-polar projections for world maps by applying graticule transformation, page 14 of the PDF file (that’s printed page no. 236)

[daan]

Nice posting, Tobias.

It’s ironic that Geocart 2 had a global distortion metric, whereas I never put one into Geocart 3. Peter Denner’s comments are good. My feelings about permitting weightings are mixed, though: I don’t want people to create and publicize tables of distortion metrics that cannot be compared to others due to their personal choices. On the other hand, when studying a specific projection, I could see reasons to choose unequal weights.

One thing I will point out here is Kerkovits’s observation that finite distortion measures closely track integrated infinitesimals, ultimately based on Tissot’s treatment. Kerkovits’s follow-up paper reinforces this. Finite measures also inevitably bias the regions away from the boundaries. These are truths I haveI always suspected, so it was good to see an empirical treatment of the problem. Its results mean we can mostly ignore global or long-range measures (such as proposed by A.B. Peters, Tobler, Canters, Laskowski &c.), as redundant. That leaves, for me, the Airy–Kavraiskiy criterion as the most reasonable and useful metric.

An important aside from that study is that the Goldberg-Gott flexion/skewness metric is a serious outlier that is difficult to relate to the notions of size and deformation that most clearly express the effects of distortion.

Cheers,
— daan

[Atarimaster]

It’s ironic that Geocart 2 had a global distortion metric, whereas I never put one into Geocart 3.

Oh, I didn’t know that. Why did you drop it?

I don’t want people to create and publicize tables of distortion metrics that cannot be compared to others due to their personal choices.

A valid point.
It already is a nuisance that you find tables of distortion metrics in literature but they are rarely comparable because different metrics are used…

On the other hand, when studying a specific projection, I could see reasons to choose unequal weights.

How about that:
As Peter suggest, you provide separate global metrics for areal distortion and angular deformation and a fifty-fifty weighted combined metric. In the manual however, you provide the calculation to get the result with a different weighting. While this doesn’t prevent people from creating and publicizing tables with their own weighting, it maybe makes it less likely. Hopefully.

That leaves, for me, the Airy–Kavraiskiy criterion as the most reasonable and useful metric.

That’d be fine by me.

For me, there’s thing that I find much more interesting than the option to change the weighting:
Namely that the distortion metrics are given
a) for the entire globe and
b) continental areas only.

In many cases, e.g. on political maps, the oceans are depicted as a unicolored or slightly shaded area. Who cares about distortions there?
I realize that this could be a lot of work, because ideally, the “continental areas only” value also regards the current settings of
– the map center,
– the boundaries,
– affine transformations and
– the undistorted location.

Of course, people again might create and publicize data that cannot be compared to others, e.g. disregarding Antarctica (like Canters mostly did). But in my opinion, the usefulness of a distortion metric that takes the actual map in regard instead of the theoretical value that you could reach if you’d only choose the standard settings, outweighs the cons.

An important aside from that study is that the Goldberg-Gott flexion/skewness metric is a serious outlier that is difficult to relate to the notions of size and deformation that most clearly express the effects of distortion.

Good to know, thank you!

And oh, in order not to adorn myself with borrowed plumes:
In my posting above, I forgot to mention that the thing about the Györffy E and Capek’s Q was actually pointed out to me by Peter. I guess I would have seen that later by myself, namely when I used my cartometry on Laskowski’s Tri-Optimal… but of course, I don’t know that.

Kind regards,
Tobias