A scale bar is right in one place

A representative fraction is a number, and a number can be dismissed as technical. A scale bar is a picture of a distance, drawn on the map, and it invites exactly one use: hold something against it, then hold that against the map, and read off a distance.

On a world map the invitation is false almost everywhere on the sheet.

The measurement #

The bar’s length on the page is fixed. What it represents is the page length divided by the projection’s scale factor at the point where it is read, so the ground distance a fixed length of paper spans is

$$d(\varphi) = d_0 \cdot \frac{k(\varphi_0)}{k(\varphi)}$$

with $k$ the scale factor along the direction the bar is held. There is no modelling in that; it is the definition of a scale factor, applied twice.

On Mercator, with the bar drawn for the equator, the readings run 1,000 · 966 · 866 · 707 · 500 · 259 kilometres at 0°, 15°, 30°, 45°, 60° and 75°. The bar is exact where it was drawn — 0.0 per cent, asserted to under $10^{-6}$, because a measurement that had drifted would fail on the one row whose answer is known in advance — and it is 74 per cent short at the top of the sheet.

The direction it is held in #

A scale factor is a property of a point and a direction, so the same bar has two readings, and the pair distinguishes the two great families of projection more sharply than any description does.

On a conformal map the two readings are identical, because equality of the scale factor in every direction is what conformal means. The site’s gate demands it here to under $10^{-6}$ percentage points, and Mercator returns $4.6\times10^{-11}$.

On an equal-area map the two readings are reciprocal. Their product is exactly one — which is again the definition, since equal-area means $ab = 1$ — so the bar reads short along the parallel by precisely the factor it reads long along the meridian.

The consequence nobody draws #

The usual advice, when the Mercator-and-Greenland argument has run its course, is to use an equal-area projection. For area that is exactly right and is measured as such in the projection that shows true size.

For a scale bar it makes things worse in a specific and checkable way. On the conformal map a reader who knows the latitude can correct any measurement with one number. On the equal-area map the correction depends on the direction the measurement was taken in, so it is one number for a north–south distance, its reciprocal for an east–west one, and something between the two for anything diagonal.

At 60° north on Gall–Peters, the bar reads 50 per cent short east–west and 100 per cent long north–south. A distance measured along a diagonal is out by an amount that depends on the bearing, and no annotation in a map’s margin can express that.

This is not an argument against equal-area projections. It is purpose before property stated in the sharpest available form: the projection chosen for the honesty of its areas is, for the specific operation of reading a distance off a bar, the least correctable of the families.

The compromise projections, which do something odd #

A compromise projection is neither conformal nor equal-area and is chosen to keep both errors moderate — the subject of compromise projections. Measured as a scale bar, one of them does something the other two families cannot.

On Robinson the bar’s readings along a parallel run 1,000 · 976 · 902 · 789 · 626 · 384 kilometres, which is better than Mercator everywhere. Along a meridian they run 1,000 · 1,000 · 1,000 · 1,018 · 1,093 · 1,293 — flat to a tenth of a per cent as far as 30°, and within two per cent at 45°.

That is a real property rather than a coincidence: Robinson’s table of parallel spacings was built to keep the meridional scale nearly true, and the tabulated curve delivers it. So on that projection a north–south scale bar is a usable instrument over most of the inhabited world, and an east–west one is not.

The reading in between, which nothing in the margin can express #

Both directions have been measured; the directions between them are where the equal-area case stops being merely awkward.

The scale factor along a bearing $\theta$ from north, on a projection whose graticule is orthogonal, is $\sqrt{h^2\cos^2\theta + k^2\sin^2\theta}$ — the indicatrix, evaluated in that direction, which is what Tissot’s indicatrix is for. On Gall–Peters at 60° north, with $h = 0.7071$ and $k = 1.4142$, a bar drawn for a thousand equatorial kilometres reads

1,000 km due north · 756 at 30° · 632 at 45° · 555 at 60° · 500 due east.

Five different answers to one question, on one map, at one point, from one bar. A correction table would need a column for bearing; a bar in a margin has nowhere to put one.

On Mercator the same five readings are 500, 500, 500, 500 and 500. That is the practical content of conformality and it is worth stating in exactly those terms, because the property is usually explained as shapes are right locally, which sounds aesthetic. What it actually buys is that the correction has one column.

The bar has to declare a direction as well as a latitude #

There is an ambiguity hiding between the two tables above, and it is worth drawing out because it doubles the number of declarations a bar owes its reader.

The meridian column reports 2,000 kilometres at 60° north and the bearing list reports 1,000 due north at the same point. Both are right, and they differ because they are two different bars. Gall–Peters at the equator has a parallel scale of 0.7071 and a meridian scale of 1.4142, so a page length that means a thousand kilometres east–west at the equator is not the page length that means a thousand kilometres north–south there — the two differ by a factor of two before the reader has moved anywhere.

So a scale bar on an equal-area map is under-specified by its latitude alone. It also needs the direction it was calibrated in, and a bar drawn without one is two instruments printed as though they were one. On a conformal map the question does not arise, which is the same single-column property in a second place.

The directional spread is the indicatrix’s axis ratio, already tabulated #

The five bearing readings have a closed form and it is one already in every projection table.

A bar of page length L read along a bearing θ from north returns L divided by the indicatrix’s radius in that direction, √(h²cos²θ + k²sin²θ). The largest and smallest readings are therefore at the two principal directions, and their ratio is

$$\frac{\text{longest reading}}{\text{shortest reading}} = \frac{k}{h} = \frac{a}{b},$$

which is the indicatrix’s own axis ratio — the quantity every distortion table on this site already reports, under the name that produces the angular deformation.

On Gall–Peters at 60° north that ratio is exactly 2, and the five measured readings — 1,000, 756, 632, 555, 500 — are √(0.5cos²θ + 2sin²θ) evaluated at the five bearings and divided into the bar. Nothing about them needs measuring once the two principal scales are known.

Which means the correctability of a scale bar is a published number. A projection whose worst axis ratio over a sheet is 1.05 has a bar whose directional ambiguity is five per cent and can reasonably be printed without a bearing; one whose ratio reaches 2 has a bar that means two different things depending on how it is held. The angular deformation figure at the end of this essay is exactly that ranking, and it is usually read as a statement about how shapes look.

It is also a statement about an instrument in the margin, and the second reading is the one a reader can act on: look up the projection’s angular deformation over the region, and that is how much the bar’s answer depends on which way it was held.

The bar on a screen moves, and is still wrong across its own picture #

A printed bar is drawn once for a whole sheet. A screen map’s bar is recomputed as the view moves, usually for the latitude at the centre of the viewport, which makes it correct there — and leaves it wrong at the top and bottom of the same screen by an amount that depends on the zoom.

A nine-hundred-pixel-tall window centred on 60° north spans 58.74° to 61.21° at level 8, over which the scale factor changes by 7.8 per cent. At level 10 the window spans two thirds of a degree and the spread falls to 1.9 per cent. At level 6 it spans ten degrees, from 54.7° to 64.6°, and the spread is 34.7 per cent — so a bar correct at the middle of that picture is a third out at the edge of it.

The pattern is the useful part: the error inside one view is a zoom-dependent quantity that vanishes as the reader zooms in, while the error of the printed scale denominator against a stated latitude does not. So the bar becomes an honest instrument at large scale for the same reason a walking map’s does — not because the projection improves, but because the sheet gets small.

No bar can be right everywhere, and the reason is the theorem #

The refusal built into this measurement is the strongest available on this site, because it is not about any of these projections.

A bar that read correctly at every point in every direction would be a map whose scale factor is one everywhere: $a = b = 1$, which makes the map an isometry from the sphere to the plane. Gauss’s Theorema Egregium forbids it — a sphere has $K = 1/R^2$ and a plane has zero, and curvature is intrinsic — which is what no map is faithful derives by computing the curvature two independent ways rather than citing it.

So the site’s gate asserts two things of every projection it draws a bar on: the bar is exact at the latitude it was drawn for, and it is wrong by more than one per cent somewhere else on the same sheet. The second would fail if a projection had appeared that beat the theorem, which is the point of asserting it rather than describing it.

A bar’s correctability follows the angular deformation exactly, because a single-number correction exists precisely where $\omega$ is zero. Mercator’s is at the noise floor by construction, the equal-area member’s is large, and the compromises sit between — so the ranking of the four projections by how well a margin can rescue their bars is a ranking already computed for another purpose.

What a sheet map gets away with #

None of this is an argument against the scale bar on a walking map, and the reason is a quantity rather than a distinction of kind.

A 1:25,000 sheet covering forty kilometres of ground sits inside a transverse Mercator zone whose scale factor varies across it by a few hundred parts per million — the subject of the scale factor of a line, where the variation has to be integrated rather than sampled to get millimetres out of it. Against a bar a reader is holding a thumb next to, a few hundred parts per million is a fraction of the thickness of the printed line.

So the bar is honest at that scale and dishonest at world scale, and the boundary between them is computable: it is the extent over which the scale factor’s variation stays below the precision of the reading. For a bar read to one per cent, that is a region about 800 kilometres across on a well-chosen projection, which is roughly the size of the sheet a bar has always been drawn on.

The whole scale variation across a single transverse Mercator zone is under 1,400 parts per million, against the 50 per cent a world map’s bar is out by at 60° north. That is a factor of three hundred and fifty between the two situations, and it is why the same instrument is trustworthy on one sheet and useless on the other.

Three honest alternatives, all of them used #

The problem has been recognised for as long as world maps have carried bars, and the responses divide into three.

Draw several bars, one per latitude band, which is what a number of nineteenth-century atlases did and which is exactly right and looks cluttered.

Draw no bar and let the graticule do it. A meridian is a scale bar with a known length — one degree of latitude is 110.6 to 111.7 kilometres, which is the ellipsoid’s whole variation — so a map showing its own graticule is carrying a distance reference that is correct by construction at every point. That it requires a reader to know the length of a degree is a real cost, and it is the one thing in this list that no projection can invalidate.

Put the bar’s own latitude in the caption, which is the cheapest and the least done: 1,000 km at 40° north is a complete statement, and a bar with no latitude attached is the same genre of published number as a resolution table with no latitude column.

What is actually being measured when a bar is read #

It is worth being precise about what a reader does with a bar, because the operation is not the one the bar supports.

The bar supports comparing a page length to a page length. What a reader wants is a ground distance, and the step between those is the projection’s scale factor, which the bar embodies at one point and pretends is constant.

That is the same substitution the whole of this field is about, in its simplest available form: a quantity that is a function of position, published as a constant, because at the scale the convention was invented for it very nearly was one. The tile scheme’s resolution table does it with metres per pixel. The representative fraction does it with a ratio. The scale bar does it with a picture — and the picture is the version a reader is most likely to trust, because it looks like a measurement rather than a claim.

What the bar shares with the other three instruments #

This field has now measured four things that report a scale, and they fail in the same way for the same reason, which is worth collecting before the ladder moves on.

The resolution table publishes metres per pixel with no latitude column. The representative fraction publishes a ratio with no latitude attached. The scale bar draws a distance with no latitude attached. And the zoom level itself is an integer standing in for a continuous quantity, as a screen map is a pyramid of tiles sets out.

All four are summaries of a derivative under the assumption that it is nearly constant. All four were invented in a setting where it was — a sheet, a survey, a zone — and carried unchanged into one where it is not. And in all four the honest form costs one extra symbol: the latitude the number is true at.

That is not a plea for better labelling. It is the observation that the measurement is available in every case and is one multiplication away, and that what stops it being printed is a convention rather than a difficulty. The scale bar is the version of this that a reader can check without any equipment at all — hold it against the map at two latitudes and the two answers differ — which is why it is the one worth drawing.

What a projection would have to be for the bar to work #

It is possible to say precisely what would be needed, which is a better way of showing the impossibility than restating the theorem.

A bar correct in every direction requires $a = b$: the map must be conformal. A bar correct at every latitude requires the scale factor to be the same everywhere: $k$ constant. Both together give $a = b = \text{const}$, and choosing the constant to be one at the standard line makes the map an isometry.

Give up the second and Mercator is the answer, with a correction of one number per latitude. Give up the first and an equal-area map is available, with a correction that has a bearing in it. Keep both and there is nothing at all. That is the trade-off is two lines reaching an instrument in a map’s margin, which is about as far from an abstract theorem as the argument gets.

Where this ladder goes next #

The two rungs so far have taken apart what a screen map says about its own scale. The next goes to the top of the world: the cut at 85.0511287798°, which is the one number in the scheme that everybody quotes and almost nobody derives, and which turns out to be forced by the data structure rather than chosen for the geography.