Most widely used projections

[Milo]

Simple scaling in the y direction is extremely easy to do on computers. I don't consider avoiding that to be a good reason to use Mercator, especially if you're already using other basic transformations like cropping, or switching to an azimuthal projection near the poles (actually more complicated computationally!). But yes, you do need to remember to do it.

[PeteD]

I guess the correct [pronunciation] is “ah-eetoff” – that’s what the diaeresis suggests in German, and Winkel must have suggested this spelling for a reason ... Even if the diaeresis originates in the french transliteration, I doubt that Winkel would have used it if the correct pronunciation was "eye-toff".

After a bit of googling, it seems you're right: the "aï " in "Aïtoff" comes from "аи" in Cyrillic, which in Russian is apparently pronounced as two syllables, as in "naïve". On the other hand, the "ay" in "Kavrayskiy" comes from "ай" in Cyrillic, which in Russian is apparently pronounced as one syllable, as in "eye".

[PeteD]

Simple scaling in the y direction is extremely easy to do on computers. I don't consider avoiding that to be a good reason to use Mercator, especially if you're already using other basic transformations like cropping, or switching to an azimuthal projection near the poles (actually more complicated computationally!).

So would you advocate using a latitude-adjusted equirectangular over the Mercator of Google Maps?

But yes, you do need to remember to do it.

And even with the Mercator, you still need to remember to adjust the length of the scale bar with latitude, so some adjustment for latitude is unavoidable.

[Milo]

So would you advocate using a latitude-adjusted equirectangular over the Mercator of Google Maps?

Yes.

And even with the Mercator, you still need to remember to adjust the length of the scale bar with latitude, so some adjustment for latitude is unavoidable.

"Scale" being one of those things that's intuitively-useful but not really mathematically accurate, since maps never have constant distance scale.

For an equal-area map, you could define "average scale" as the square root of flation. For an equidistant map, you could define "nominal scale" as the scale in the direction where equidistance holds (typically meridional), even if the scale in other directions is different. For a conformal map, scale is really not definable in any meaningful way, at least globally.

I will also correct myself by saying that strictly speaking, if you want to adjust an equirectangular projection's standard parallel while retaining the same nominal scale, you would want to rescale in the x direction, not the y direction. Of course, regional maps will usually be drawn at a larger scale than global ones, which may counteract this. You need to do some scaling regardless!

[PeteD]

For an equidistant map, you could define "nominal scale" as the scale in the direction where equidistance holds (typically meridional), even if the scale in other directions is different.

And this nominal scale would also match the scale along the standard parallel.

For a conformal map, scale is really not definable in any meaningful way, at least globally.

It is inhomogeneous but isotropic, the opposite of the scale for equal-area projections.

As I've mentioned before, most people use Google Maps zoomed in, either to plan routes or to look at local places of interest. In this case, scale varies little across the screen regardless of whether you use the Mercator or an equirectangular projection, but if you use the Mercator, you need to adjust the length of the scale bar as you scroll up and down, whereas if you use an equirectangular projection, you need to adjust the width of the projection as you scroll up and down.

[Milo]

For an equidistant map, you could define "nominal scale" as the scale in the direction where equidistance holds (typically meridional), even if the scale in other directions is different.

And this nominal scale would also match the scale along the standard parallel.

Sure, but I was talking in the more general sense, not just cylindrical maps, so you don't necessarily have a standard parallel. The azimuthal equidistant map doesn't, for example.

Meanwhile, for equal-area maps, at every point there exists some direction (two of 'em, in fact, or four if you count opposite directions on the same line) where average scale exactly holds, but usually these are arbitrary directions that you don't particularly care about. However, where the map is conformal (which almost all equal-area maps achieve somewhere, and, yes, cylindrical equal-area maps do so on their standard parallel), average scale is accurate in all directions.

if you use the Mercator, you need to adjust the length of the scale bar as you scroll up and down, whereas if you use an equirectangular projection, you need to adjust the width of the projection as you scroll up and down.

I doubt many people would actually scroll all the way from, say, Indonesia to Mongolia (and even less so from a specific city in Indonesia to a specific city in Mongolia). More likely if that interested you, you'd zoom out from Indonesia to a global map, then zoom back in to Mongolia.

Which raises the question: if someone did decide to scroll all the way from Indonesia to Mongolia, what would be the appropriate way to handle scale? Would you want scale to remain constant throughout your scroll, or would you want to keep the same Mercator projection and increase scale to match?

[PeteD]

I doubt many people would actually scroll all the way from, say, Indonesia to Mongolia (and even less so from a specific city in Indonesia to a specific city in Mongolia). More likely if that interested you, you'd zoom out from Indonesia to a global map, then zoom back in to Mongolia.

I agree that you wouldn't scroll from a city in Indonesia all the way to a city in Mongolia without zooming out and back in again, but you can scroll from a map showing the whole of Indonesia over to Mongolia easily enough without zooming – with my phone in portrait orientation (which is how most people hold their phone most of the time), I can do it in a single swipe.

When moving from a map showing the whole of one city to a nearby city, e.g. along the Boston–Washington Corridor, I think that most people would scroll/swipe rather than zoom out and back in again, although I admit that at this zoom level, the change in scale as you move around is negligible. On the other hand, if you move from a map showing the whole of the Boston–Washington Corridor over to, say, the Quebec City–Windsor Corridor, then the change in the length of the scale bar is clearly noticeable.

Sometimes I follow rivers or railway lines, and then you can't zoom out too much because the river or railway line would no longer be visible, but maybe that's just me.

I agree that if you've zoomed in to view a particular area of a city, then you'd zoom out at least to whole-city-level before scrolling/swiping to a different city.

Which raises the question: if someone did decide to scroll all the way from Indonesia to Mongolia, what would be the appropriate way to handle scale? Would you want scale to remain constant throughout your scroll, or would you want to keep the same Mercator projection and increase scale to match?

Good question. With an equirectangular projection, you'd want the former and would have to adjust the standard parallel in order to keep the E–W scale constant (for a given row of pixels on the screen, most likely halfway up – the E–W scale would inevitably vary between the top and bottom of the screen). On the other hand, with the Mercator projection, you could do it either way. Google Maps, of course, does the latter.

[daan]

Simple scaling in the y direction is extremely easy to do on computers. I don't consider avoiding that to be a good reason to use Mercator, especially if you're already using other basic transformations like cropping, or switching to an azimuthal projection near the poles (actually more complicated computationally!). But yes, you do need to remember to do it.

The problem is the need to rerender the raster tiles if you do anything at all to the projection, such as changing the standard parallel on a nonconformal projection. That’s extremely expensive at Google’s scale; would slow down delivery of maps; and would require arbitrary and constant generalization of features, especially problematic for text placement. By using Mercator tiles, all you need is a raster set for each zoom level, rendered once, along with whatever feature overlays you use, crafted and stored once. It’s a big deal.

— daan

[daan]

The problem is the need to rerender the raster tiles if you do anything at all to the projection, such as changing the standard parallel on a nonconformal projection. That’s extremely expensive at Google’s scale; would slow down delivery of maps; and would require arbitrary and constant generalization of features, especially problematic for text placement. By using Mercator tiles, all you need is a raster set for each zoom level, rendered once, along with whatever feature overlays you use, crafted and stored once. It’s a big deal.

I proposed a partial solution to this problem in this paper, section 2.2. Partial, because you still have to deal with the generalization problem. My solution frees you from rectangular projections; you could use whatever equal-area projection you like, in principle. Or not even equal-area.

— daan

[quadibloc]

Simple scaling in the y direction is extremely easy to do on computers. I don't consider avoiding that to be a good reason to use Mercator, especially if you're already using other basic transformations like cropping, or switching to an azimuthal projection near the poles (actually more complicated computationally!). But yes, you do need to remember to do it.

Well, it could be argued, I suppose, that changing the standard parallel as you move around would be disconcerting to people viewing Google Maps, so the fact that the Mercator can usually remain static - unless you include the North or South pole in your map, and then there's a switch to the azimuthal equidistant, I suppose - actually is an advantage.
They chose the simplest solution that would do the job, without necessarily thinking too much about it. I would say that, for their humble purposes, they got it right. But if they show a scale bar applicable to the Equator at all latitudes, that, of course, is a mistake.