Why Mercator for the Web? Isn’t the Mercator bad?

General discussion of map projections.

Why Mercator for the Web? Isn’t the Mercator bad?

Postby daan » Thu Mar 15, 2012 6:56 pm

As you may know, Google Maps uses the Mercator projection. So do other Web mapping services, such as Bing Maps and MapQuest. Over the years I’ve encountered antipathy toward the use of the Web Mercator from map projection people. I know of two distinct schools of opposition. One school, consisting of cartographic folks and map aficionados, thinks the Mercator projection is “bad”: The projection misrepresents relative sizes across the globe and cannot even show the poles, they are so inflated. The other school, consisting of geodesy folks, thinks mapping services have corrupted the Mercator projection, whether by using the wrong formulæ for it or by using the wrong coordinate system for it.

The cartographic objection

Let’s first examine the objections from cartographers with some historical background. Gerardus Mercator presented his new map projection on a famous and influential world map in 1569. He had somehow figured out, probably through trial and error, how to space the parallels on a rectangular map so that sailing courses that do not change direction (rhumbs) could be represented on a map as straight lines. It turns out the parallel spacing needs to increase more and more the higher the latitude, with the unfortunate side-effect of blowing things up way out of proportion the further away from the equator you look. The poles are way off at infinity, so mapmakers choose some latitude at which to cut off the arctic regions.

Sailors don’t care about the poles, where there’s no sailing to be done. They also don’t care about proportions. They care about getting where they’re going. For them the drawbacks to the Mercator were hardly relevant.

Unfortunately, map publishers in the 19th century embraced the Mercator projection for general-purpose world maps, not just sailing maps. There aren’t any records about why they did this, but we can make some inferences. For one thing, if publishers were already drafting on the Mercator for sailing maps, they wouldn’t want to go to the expense of drafting on other projections just because the market was different. For another, the enterprise of sailing was so vital and prestigious that we can infer the accessories to it were prestigious as well, and that include the maps. Finally, after growing up with maps on a particular projection, we can infer the public preferred them that way, fueling the cycle. It is also true that the Mercator was convenient for the European market in that the inflated areas also happened to be the ones they knew the most about and had the most place names to stuff into: Europe.

It would be a mistake to think that the Mercator was all people saw. Maps were portrayed in many projections even in the Mercator-prone era. And just as the Mercator climaxed in usage in the mid 19th century, it was already being criticized for its inappropriate use in general reference maps. A very long succession of cartographers and amateurs promoted alternatives, with varying degrees of success, culminating in the sordid Peters affair of the 1980s.

In 1974, amateur historian Arno Peters presented his “new” projection, the purpose being to right the wrongs of the Mercator and restore the oppressed people of the world to their rightful cartographic stature. In short, his thesis was that cartographers had been deliberately promoting a Euro-centric view of the world for centuries by creating maps on the Mercator, inflating the apparent size of the mid-latitude European imperial homelands. Peters equated this size disproportion to establishing dominance and oppressing the undeveloped world, which happens to lie largely in the tropics. He claimed this is so because representing a country as disproportionately large makes it seem more important than it is. While mean-spirited and unsupported by historical fact or psychological experiment, Peters’s thesis is not easily disproved. It’s essentially a faith-based ideology. The Peters claims did not end there, however. He invented all kinds of virtues for “his” projection that are clearly wrong, some being simply impossible. He misled his disciples into thinking his projection was the first or only equal-area projection. And in a sad bit of irony, his projection was not even novel; it had been presented in 1855 by Scottish clergyman James Gall as an alternative to… the Mercator.

Despite being just one in a long line of map projection crusaders, Peters found sympathetic ears. Most of the European colonies had attained independence by then. Academia was immersed in postmodernism. Social activism abounded. The time was ripe, and Peters made inroads. His successes, while modest, galled cartographers. They had been trying to steer publishers away from the Mercator for a hundred and fifty years by then. It stung to be accused of conspiracy, and it certainly seemed unfair for Peters to be attracting favorable attention on the basis of claims undemonstrable if not flatly wrong. The American Cartographic Association mounted a counter-campaign of map projection education and recommended against using any rectangular map projection for general-purpose world maps. Cartographers needed to be perfectly clear that they “hated” the Mercator projection, too, and always had. This was important in demonstrating the Peters thesis to be a straw man.

My own opinion is that American cartographers overreacted. Understandably, they think maps are important, and so they implicitly accept Peters’s hypothesis that how a map looks is critical, even if they reject his solution. In that way, Peters won even though his map never took over because cartographers’ legitimizing that belief in the power of maps opens the door to Peters-style quacks and charlatans. Yet when you look objectively at the impact a map projection has on the beliefs and world view of the general populace, you really don’t come up with much. Maps are only a tiny part of most people’s lives. They’re exposed to many projections anyway—as they should be. They take cues from a map in many ways, not just relative sizes. And mostly, they really don’t care.

That brings us to the present. We live amongst a generation of mapmakers trained in the truism that Mercator is not just inappropriate, but evil. You would be hard-pressed to find one who would admit the Mercator is good for anything but sailing charts.

But is that true? You could point to the irony of straight rhumbs on the Mercator. Google has no use for straight rhumbs, so the one thing it’s good for is what Google isn’t using it for. But no, it turns out the truism is not true. The Mercator is good for more than just straight rhumbs. Let’s look at some other properties.

First, it is conformal. This means that, if you look at a small part of the map, no matter where you look, it will look right. That is in stark contrast to any equal-area map, where you will find large swaths of the map horribly bent up.

Secondly, north is always the same direction—in this case, upward. No matter which piece of the map you look at, you know which way north is, and it’s always upward. That’s only true on cylindric maps. Not only is north always up on Mercator, but every direction from north to south and in between heads off at the right angle. That’s only true on conformal maps. The combination of cylindric and conformal equals… Mercator.

What is important in a projection to a Web street-mapping service?
    • Easily calculated.
    • One projection of the entire world so that the map can be panned and zoomed without having to change the projection and without favoring any one part.
    • Suitable for large-scale maps—that is, maps of small areas like cities or blocks within cities.
    • Directions are always the same regardless of what portion you look at because you won’t necessarily have a graticule to inform you.

What is not important in a projection to a Web street-mapping service?
    • Showing large areas with “fidelity” (which is not possible anyway).
    • Historical conventions in map projection usage.
    • Being equal-area, or even being perfectly conformal, as long as it’s “practically” conformal.

With that list of requirements and anti-requirements, there is no alternative to the Mercator. It exactly fits the bill. If Web mapping providers tried to use an equal-area projection in order to be—what? Politically correct?—then they have two choices: They can favor some locations over other locations, showing the favored locations with lower or no distortion; or else they are obliged to draw equal-area maps with the projection specific to the location. The latter option would consume far more energy and compute power because they can’t reuse the same map’s rendering for other people’s requests. Yet the latter option STILL favors this location over that location when zooming out because an equal-area map always has a preferred locale of presentation. The computer can infer that you are interested in the center of the map and therefore draw the center with less distortion, but when you zoomed out, you implicitly stated you were interested in more than just the center. The computer has no way to give you an equal-area presentation in which your entire zoomed-out region can be examined in its particulars without wrecking directional fidelity of some regions. In Peters’s quixotic map, that preferred locale was… Europe and the mid latitudes in general. Such lovely irony.

A conformal map, by contrast, does not favor one region over another. Any region is locally correct. You may decide that “larger” is better than “smaller” and therefore that the map has “favored” one region over another, but your decision is arbitrary. Conformal maps have a problem with relative presentation across wide areas, meaning, they cause problems in comparing sizes of different regions to each other, but equal-area maps have a problem with absolute presentation even across small areas. That is, even if you do not compare different regions to each other, you will find equal-area maps to be grossly distorted locally across most of their territory. That’s a grave problem for a service like Google Maps, where the purpose is local.

My criticisms of equal-area maps as an alternative to the Mercator in Web street-mapping services apply also to “compromise maps”, those which are neither equal-area nor conformal. While compromise maps reduce the severity of the problems, they remain far from eliminating them. Conformal maps eliminate them.

Yes, you could use some other conformal map. But Mercator is the only one that yields north-up everywhere. Standardization is important in an environment where you are likely to look at a plethora of maps of many locales. Other conformal maps either come with similar problems as Mercator (infinite expanse) or much greater computational complexity.

The geodesy objection

I first ran into this objection on the proj4 mailing list. While much was said, including invective directed toward Google engineering, my own interpretation is that it amounts to much ado about nothing.

One charge: Google and their ilk aren’t really using a Mercator.

Guilty as charged: The Web Mercator as deployed by Google Maps et al is not actually a Mercator. It’s close, but not quite. The reason is this: The earth is not perfectly spherical. It is closer to an ellipsoid. Coordinates as measured on the earth are referenced against the ellipsoid, not against a sphere. But Google maps and the others use the spherical version of the Mercator, not the ellipsoidal version, even though they use coordinates referenced against the ellipsoid. So in the view of geodesists, these mapping services either ought to use formulæ adapted for the ellipsoid, or they ought to use coordinates adapted for the sphere.

Why does the mixing up of the two matter? In my view, it doesn’t. Hence much ado about nothing. Using the “wrong” formulæ means that the resulting map is not quite conformal. Yet for the purposes of service Google and others provide, the projection only needs to be conformal “enough”, not perfectly conformal. It is conformal enough if people don’t notice the deviation from conformal when they use the maps. Because the deviation from conformal is only a few percent, in fact people will not notice. So, in the quest for perfectly conformal, engineers could have opted to use the much more complicated ellipsoidal formulæ for the ellipsoidal Mercator; or, they could bow to practicality and opt to use the spherical Mercator, which is what they did.

How much practicality? These Web map services render hundreds of millions of maps per day. It is not just the services who have to worry about this, though; thousands of other sites mash up their geodata onto these same services. They, too, have to deal with the projection. The difference in complexity between the spherical and ellipsoidal versions of the Mercator could add up to tens of millions of dollars per year (back-of-envelope calculation) in electrical costs, plus the environmental impact. And remember, those thousands of programmers outside of Google doing Web mash-ups are working in a domain they typically know little about. Hence their odds of making mistakes soars with the complexity. The cost of those mistakes across many thousands of Web developers could easily reach millions of dollars. All in order to achieve—theoretical purity?

Other objections were tendered, such as:
    • Lack of precedent (But anything new has that problem);
    • Fostering confusion (How? This is all documented, and indeed, given that any old Web developer needs to implement this, wouldn’t simpler formulæ foster less confusion?);
    • Violating established practices (But this is the Web, where there were no practices, not geodesy);
    • Reinventing the wheel (But the existing wheels were the wrong size)

The “established practices” and “lacks precedent” arguments don’t really hold in any case. We’ve always had a “Web Mercator” and a “Web-any-thing-else” projection. Small-scale projections assume a sphere. There’s no reason to bother with ellipsoids for maps at scales smaller than about 1:1,000,000. Yet survey data is always referred to ellipsoids. When a small-scale map goes into preparation, nobody bothers to transform the ellipsoidal coordinates to spherical-datum coordinates before projecting to the sphere because the difference would be imperceptible. If Google and the like are “guilty” of violating precedent, that violation is more one of degree rather than it is some startling new development. They represent (perhaps) the first widespread use of the spherical Mercator for large scale maps.

In summary

The reason we have so many map projections is because none serves every need. Match the projection to the need. In this case, the need is street maps that demand little calculation and that do not deform according to position within the frame or location on the earth’s surface.

I think the Web Mercator is a fine choice for its use. I would have made the same decision myself, and still would. All things considered, I can’t think of a better way to solve the problems that needed to be solved.

— daan Strebe
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Re: Why Mercator for the Web? Isn’t the Mercator bad?

Postby Piotr » Fri Dec 22, 2017 9:07 am

"In Peters’s quixotic map, that preferred locale was… Europe and the mid latitudes in general. Such lovely irony."

Correction:

"In Peters's Gall-Peters map, that free of angular distortion locale was... Europe and the mid latitudes in general. At least it isn't like Mercator which has a preferred locale as Europe (think of size distortion for "preferring", not angular distortion)."
My favorite projection is Winkel Tripel.

Mercator, Gall-Peters, Tobler's World in a square, Mollweide, Transverse Mercator, Hammer, Miller, Compact Miller, Sinusoidal, Equirectangular, Winkel Tripel, Foucaut, Lambert equal-area (cylindrical equal-area), Gall stereographic, Goode homolosine, Boggs eumorphic, Werner, Bonne, Loximuthal, Azimuthal Equidistant, Lambert azimuthal equal-area, Aitoff and Quartic (not in particular order) are also nice.
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Re: Why Mercator for the Web? Isn’t the Mercator bad?

Postby Atarimaster » Fri Dec 22, 2017 10:01 am

Piotr wrote:"In Peters's Gall-Peters map, that free of angular distortion locale was... Europe and the mid latitudes in general. At least it isn't like Mercator which has a preferred locale as Europe (think of size distortion for "preferring", not angular distortion)."


So, you’re saying that the term »preferred« can only be applied to areal distortion but in no way to angular distortion?
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Re: Why Mercator for the Web? Isn’t the Mercator bad?

Postby Piotr » Fri Dec 22, 2017 10:14 am

Atarimaster wrote:
Piotr wrote:"In Peters's Gall-Peters map, that free of angular distortion locale was... Europe and the mid latitudes in general. At least it isn't like Mercator which has a preferred locale as Europe (think of size distortion for "preferring", not angular distortion)."


So, you’re saying that the term "preferred" (oh please these angle quotes are confusing) can only be applied to areal distortion but in no way to angular distortion?

That's what Peters used it for apparently. I'm not using the term "preferred" to refer to map properties themselves, but Peters is.
My favorite projection is Winkel Tripel.

Mercator, Gall-Peters, Tobler's World in a square, Mollweide, Transverse Mercator, Hammer, Miller, Compact Miller, Sinusoidal, Equirectangular, Winkel Tripel, Foucaut, Lambert equal-area (cylindrical equal-area), Gall stereographic, Goode homolosine, Boggs eumorphic, Werner, Bonne, Loximuthal, Azimuthal Equidistant, Lambert azimuthal equal-area, Aitoff and Quartic (not in particular order) are also nice.
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Re: Why Mercator for the Web? Isn’t the Mercator bad?

Postby Atarimaster » Fri Dec 22, 2017 10:22 am

So your correction is merely a joke, namely stating how Peters would have put it?
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Re: Why Mercator for the Web? Isn’t the Mercator bad?

Postby Piotr » Fri Dec 22, 2017 1:13 pm

Atarimaster wrote:So your correction is merely a joke, namely stating how Peters would have put it?

I don't know! Stop diving into details in my brain!
My favorite projection is Winkel Tripel.

Mercator, Gall-Peters, Tobler's World in a square, Mollweide, Transverse Mercator, Hammer, Miller, Compact Miller, Sinusoidal, Equirectangular, Winkel Tripel, Foucaut, Lambert equal-area (cylindrical equal-area), Gall stereographic, Goode homolosine, Boggs eumorphic, Werner, Bonne, Loximuthal, Azimuthal Equidistant, Lambert azimuthal equal-area, Aitoff and Quartic (not in particular order) are also nice.
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Re: Why Mercator for the Web? Isn’t the Mercator bad?

Postby Atarimaster » Fri Aug 31, 2018 5:49 am

daan wrote:As you may know, Google Maps uses the Mercator projection.


I just found out:
Not anymore! When you zoom out, the flat map turns into a globe.
(Evidently, they changed that almost a month ago, but I didn’t notice that before.)

I’m not sure yet if this is the best solution, but it surely is noteworthy.
Kind regards,
Tobias
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Re: Why Mercator for the Web? Isn’t the Mercator bad?

Postby daan » Mon Sep 03, 2018 1:26 pm

Atarimaster wrote:
daan wrote:As you may know, Google Maps uses the Mercator projection.


I just found out:
Not anymore! When you zoom out, the flat map turns into a globe.
(Evidently, they changed that almost a month ago, but I didn’t notice that before.)

I’m not sure yet if this is the best solution, but it surely is noteworthy.

Indeed it is noteworthy. I noticed within a couple of days but wanted to think about this before posting anything. I have delivered a lecture several times over the past few years that addressed Google’s use of the Web Mercator as a major theme. I am scheduled to do this again in a few weeks, so I will gather my thoughts and post more later.

Cheers,
— daan
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