The scale of a screen map is not one number

A paper map carries a scale in its margin: 1:25,000, or a bar with kilometres marked on it. The number is a ratio between a distance on the sheet and a distance on the ground, and on a sheet covering forty kilometres it is true everywhere on the sheet to a part in ten thousand.

A screen map carries the same kind of number and it is true along one line of latitude.

What a scale denominator is #

The representative fraction is a pure ratio: one unit on the map is $N$ units on the ground, whatever the units. That definition requires the map to have a physical size, which a printed sheet has and an image does not.

So turning a screen map’s resolution into a scale needs a convention for how large a pixel is. The one in general use is 0.28 millimetres, which is 90.7 dots per inch. Nothing measures it. It was chosen so that two programs asked for the same picture would report the same scale, and the number is now embedded in every specification that lets a layer appear between one scale and another.

With that convention, a zoom level’s scale denominator is

$$N(z, \varphi) = \frac{r(z)\cos\varphi}{0.00028}$$

which at level 12 is 136,495 on the equator and 68,765 at 60° north. The first of those is what a program prints.

Two things are wrong with the printed number, and they are different sizes #

The first is the projection’s scale factor, and it is enormous. Mercator’s is $\sec\varphi$: 1.15 at 30°, 1.41 at 45°, exactly 2 at 60°, 3.86 at 75°. Any map spanning more than a few degrees of latitude is at visibly different scales top and bottom, which is the whole content of why Mercator exists — the stretching is not a defect of the projection but the mechanism that makes it conformal.

The second is a few thousand parts per million and would be invisible next to the first if anyone ever looked at them together. The scheme applies spherical Mercator formulae to geodetic latitudes, so its scale factor is not $\sec\varphi$. Measured from the projection’s own derivatives against the WGS84 metric, the relative scale at 60° is 1.9850 rather than 2.0000 — a departure of 7,522 parts per million — and at 85° it reaches 9,949.

The control that makes the second one a measurement #

A departure of seven thousand parts per million from a textbook formula is exactly the size of thing that turns out to be an artefact of how the measurement was made. The site’s habit is to check that before printing it, and the check here is the same machinery run on a projection whose answer is known.

The spherical Mercator — the same forward map, measured against a spherical metric — reproduces $\sec\varphi$ to under $10^{-2}$ parts per million across the whole range. Same derivative code, same differencing, same Richardson extrapolation. So the difference between the two runs is not the machinery; it is the metric, which is to say the ellipsoid, which is to say the thing the projection ignored.

That is the two-sided form this site has used since foundation: an assertion that only checks the case expected to pass cannot distinguish a working measurement from one that has quietly stopped measuring. Here the passing case is the sphere and the failing case is the ellipsoid, and they differ by four orders of magnitude.

Which scale, in which direction #

On a conformal map the question what is the scale here has one answer, because the scale factor is the same in every direction — that is the definition. Mercator’s $h$ and $k$ are equal at every point, so a single number per latitude corrects any measurement taken off the map, in any direction.

That is not a small convenience and it is worth stating plainly, because this field spends most of its time on what the projection costs. On an equal-area projection the correction is two numbers and their product is one: the map reads short one way and long the other, and a scale bar is right in one place measures both.

The catch is that Web Mercator is not conformal — 0.3848° of maximum angular deformation, against a floor of $1.6\times10^{-6}$, which is the measurement Web Mercator is not conformal is built on. So even the one-number-per-point property is approximate here, at the fourth significant figure. It holds far better than any correction anyone applies, and it does not hold exactly.

Why the printed number survives #

A number that is wrong by a factor of two at 60° north would be expected to cause continuous trouble. It causes very little, and the reasons are worth setting out because they explain what the number is actually used for.

It is used as an ordering, not as a measurement. A specification that says a layer appears between 1:100,000 and 1:25,000 is choosing when detail arrives. The choice is a design decision about clutter, and clutter is a property of the picture — of how many labels are in view — which the projected resolution predicts rather better than the ground resolution does.

Nobody measures off a screen with a ruler. The operation a printed scale supports — dividers on paper — has no screen equivalent. What replaced it is a distance tool, which computes on the ellipsoid and never touches the projection at all.

And the error is invisible to comparison. Two features on the same screen at the same latitude are at the same scale, so anything a reader compares by eye is being compared fairly. The failure needs a comparison across latitudes, which is exactly the operation Mercator against Peters is about, and which people do far less often than the argument suggests.

Where it stops surviving #

The number stops being harmless the moment it is used as an input rather than as a label.

Print. A screen map exported at a stated scale for a printed page inherits the whole factor. A sheet produced at “1:10,000” for a site in Norway is a 1:5,000 sheet, and a length scaled off it with a ruler is out by half.

Ground sampling distance. A specification that requires imagery at half a metre per pixel, checked against a scheme’s published resolution table, is satisfied at the equator and over-satisfied everywhere else. That direction is safe. The reverse — a specification written for a high-latitude country and checked against the equatorial table — is not.

Anything with a tolerance. The tolerance decides the model is the practice field’s version of this, and it holds here: a correction that can be dropped depends on the size of the correction against a stated tolerance, and a factor of two in the scale moves that boundary by a factor of two.

The zoom level is itself a coarse quantisation of scale #

There is a third thing wrong with the printed number, and it is the one nobody complains about because it is not an error at all — it is a rounding, and it is larger than the projection’s error over most of the inhabited world.

The available scales are $1{:}559{,}082{,}264 / 2^z$: 1:136,495 at level 12, 1:68,247 at 13, 1:34,124 at 14. Nothing between them exists. A map wanted at 1:50,000 lands between the last two, 36 per cent away from one and 47 from the other, and whichever is chosen the picture is at a scale nobody asked for. The worst case is the geometric midpoint, where the nearest available level is $\sqrt2$ away — 41 per cent.

Set that against the projection’s own factor. $\sec\varphi$ reaches $\sqrt2$ at exactly 45°. So below 45° of latitude, the error from rounding a wanted scale to the nearest zoom level is larger than the error from the projection, and the second is the one that gets written about.

That comparison is not an argument that the projection’s factor is unimportant. It is a statement about which of the two a reader has ever noticed, and the answer is neither: the discretisation is invisible because a map is looked at rather than measured, and the projection factor is invisible for the same reason. Both become visible at the same moment, which is when somebody exports the picture and puts a ruler on it.

Continuous zoom removes the first of the two — a map scaled smoothly between levels can sit at any scale, at the cost of resampling tiles rather than serving them at their own resolution. It removes nothing at all from the second.

Drawn as equal cells of the graticule — 20° of longitude by 10° of latitude, the same ground everywhere but for the exact factor a cell’s latitude gives — the scheme’s own projection shows what the printed scale is claiming is one number. The cell at 70° north is drawn at 15.5 times the area of the one on the equator.

The device makes it worse, and this part is nobody’s fault #

The 0.28 mm pixel is a convention about a rendering pixel, not about a screen. A modern display has two, three or four physical pixels for each of those, and a browser may serve a tile of 512 pixels intended to be displayed at 256 units of layout.

So the same tile, on two devices, is the same number of layout pixels and a different number of millimetres. The printed 1:136,495 is therefore a statement about a hypothetical 0.28 mm pixel, at a hypothetical latitude, and neither of those hypotheticals is the reader’s.

This is not a criticism of the convention. The alternative is to have no comparable number at all, which is worse. It does mean that the representative fraction, which on paper is a measurable physical ratio, has become on a screen a label with two conventions and one latitude baked into it — and that saying so is more useful than pretending the number is what it looks like.

What an honest statement would be #

It takes three quantities rather than one, and all three are computable at the moment of drawing.

The zoom level, which is exact and is a property of the picture. The latitude of the view’s centre, which the program already knows. The ground resolution there, which is $r(z)\cos\varphi$ and takes one multiplication.

A caption reading 38.2 m/px, 23.8 m/px at 51.5° north carries more information than 1:136,495 and hides nothing. That it is not the convention is a historical accident: the representative fraction came from a world of printed sheets where the number was true across the whole sheet, and it survived into a medium where it is not.

What the same number does to an area #

A scale factor is a statement about lengths, and an area carries it twice. That is arithmetic, and it is the reason the numbers in this section are larger than any yet quoted.

At 60° north the linear scale is 2, so a square kilometre of ground occupies four times the paper it would on the equator. At 75° the linear factor is 3.86 and the areal one is 14.9; at the top of the frame, 85.05°, it is 132. Those are computed from the projection’s Jacobian rather than from $\sec^2\varphi$, and they agree with it — which is the check, since the areal factor is the determinant and the linear one is not.

The consequence for a printed scale is that the one number is wrong twice as fast for anything two-dimensional. A reader who has internalised that the map is “a bit stretched up there” and mentally halves a length at 60° will still be four times out on an area, and this is the mechanism behind the entire Mercator-and-Greenland argument. The projection that shows true size takes that apart on the projection’s own terms; here the point is narrower and about the label: a single denominator cannot describe a map on which areas and lengths are wrong by different powers of the same factor.

Measured against three other projections the same way, the scheme’s own is the steepest of them, which is the price of the conformality that makes a pyramid pleasant to drag; an equal-area member is flat at one by construction and would make the printed denominator nearly honest at the cost of the dragging.

Scale is a derivative #

The deeper point, and the one that connects this to the rest of the site, is that a map does not have a scale. It has a scale at a point, in a direction, and that quantity is a derivative of the projection — which is what measuring instead of naming sets out and what every figure on this site computes.

A representative fraction is a summary of that derivative under an assumption: that it does not vary much over the sheet. For a printed 1:25,000 sheet the assumption is excellent, and the scale factor across it varies by a few parts per million, which is why the scale factor of a line has to integrate rather than sample when it wants millimetres. For a map of the world the assumption is not approximately true; it is false by a factor of a hundred and thirty between the equator and the top of the frame.

The screen map is the case where a convention built for the first situation was carried unchanged into the second. Nothing was measured wrongly. What happened is that a summary statistic outlived the assumption that made it a summary of anything.

What made the convention portable, and what that cost #

The representative fraction survived the move from paper to screen for a reason worth naming, because the reason is what made it so hard to notice that it should not have.

It is a single number, and single numbers travel. A scale denominator fits in a caption, a menu, a metadata field, a printed legend, an API parameter and a conversation. Everything about a summary statistic that makes it useful also makes it portable, and portability is what decides which quantities survive a change of medium.

The assumption it summarises does not travel with it. The scale factor varies little across this sheet is a property of a particular map at a particular size, established by whoever drew it, and it is not part of the number. So the fraction moved to the new medium and the condition for its meaningfulness stayed behind — which is the same shape as a transformation’s sign convention and as a rate stored beside a geometry, arriving in a different currency.

And on paper the assumption was so nearly always true that stating it would have looked pedantic. A few parts per million across a topographic sheet is beneath any use the number is put to, so a century of practice built up in which the fraction genuinely was the whole truth, and nobody had occasion to write down why.

The screen broke the assumption without changing the number’s appearance. A zoom level’s scale denominator looks exactly like a sheet’s; it is presented in the same place, in the same format, and to the same purpose. There is nothing on the interface to indicate that the quantity has changed from a property of the map into a property of one latitude on it.

Which is the general hazard with a summary that becomes an interface. It keeps working long after it stops summarising, because a number cannot report the failure of its own preconditions — and the only way to find out is to compute the thing it was summarising and compare, which is what this rung does.

What this rung settles #

The printed scale of a screen map is true at one latitude, wrong by $\sec\varphi$ everywhere else, and wrong by a further few thousand parts per million because the projection is not the projection it is named after. The first factor reaches two by 60° and 11.5 by 85°. The second is small enough to ignore for every use the number is actually put to, and is measured here rather than assumed small — with the spherical Mercator as the control that makes the measurement mean something.

The next rung takes the same failure to the one place a reader can see it directly: the scale bar, which is a picture of a distance and is therefore checkable by eye, and which fails differently on a conformal map and on an equal-area one.