Scale distortion is the third failure

Every map carries a scale. It is printed in the margin as a ratio — 1:50,000 — or as a bar, and it is read as a property of the sheet.

It is a property of one line on the sheet. Everywhere else the scale is different, and the difference is a distortion that neither of the two measures this site tracks reports on its own.

What it is #

At a point, a projection has two principal scale factors $a$ and $b$: the largest and smallest amounts by which it stretches, over all directions. Angular deformation depends on their ratio and areal error on their product.

Scale distortion is the departure of $a$ and $b$ themselves from one — how much bigger or smaller the map is here than its nominal scale says.

A conformal projection has $a = b$, so it has no angular deformation at all, and $a$ still varies from place to place. Mercator’s is $\sec\varphi$: two at 60°, six at 80°, unbounded at the pole. The projection is exactly conformal throughout and the map is six times larger at 80° than at the equator, in every direction equally.

So scale distortion is present in projections that have no angular distortion, and it is the residue that conformality does not touch.

It is not a third independent quantity #

Being precise, because the essay’s title overstates slightly and the correction is the interesting part.

At a point, the projection’s derivative is a linear map, and up to rotations it is described by exactly two numbers. So there are two degrees of freedom and no more, and any pointwise measure whatsoever is a function of $a$ and $b$.

Given the angular deformation $\omega$ and the areal factor $A = ab$, the pair inverts: with $s = \sin(\omega/2)$,

$$a = \sqrt{A\,\frac{1+s}{1-s}}, \qquad b = \sqrt{A\,\frac{1-s}{1+s}}$$

So scale distortion is determined by the two quantities already tracked, and adding it adds no information.

What it adds is emphasis. The two standard measures are constructed to be scale-free — $\omega$ from a ratio, the areal factor normalised to one — precisely so that they describe the map rather than the paper it was printed on. That normalisation is right and it means neither of them says how large the map is anywhere, and a great many practical questions are exactly that.

Why every equidistance claim is qualified #

The clearest consequence, and it explains a piece of vocabulary that looks evasive and is not.

No projection preserves all distances — that would be the isometry the curvature forbids. So no projection is simply “equidistant”, and every one that carries the word in its name qualifies it:

• Azimuthal equidistant: distances from the centre are true, in every direction. Between any other pair of points they are not.
• Equidistant cylindrical: distances along the meridians are true, and along one chosen parallel.
• Equidistant conic: distances along the meridians and along two standard parallels.

Each is a statement about a one-parameter family of lines, not about the plane. That is the strongest form the property can take, and the reason is two lines of algebra: equidistance in all directions at a point means $a = b = 1$, which combined with anything else forces an isometry.

Which makes equidistance the odd one out among the three properties. Conformality and equal area hold at every point of a map; equidistance holds along curves, and the curves have to be named.

The azimuthal equidistant projection is the clean case. Every radius from its centre is at true scale in every direction, and the parallels are equally spaced circles because of it — while distances between two points neither of which is the centre are not preserved at all.

A printed scale can be measured against that. A UTM sheet’s nominal scale holds exactly on two lines 1.61° either side of the zone’s axis; between them the map is up to four hundred parts per million too small, and outside them up to 981 too large.

What the printed scale means #

The practical question, and the answer is more specific than most map users realise.

A representative fraction like 1:50,000 is the scale at the projection’s line or point of true scale — a standard parallel, a central meridian, or the centre of an azimuthal projection. Everywhere else, the true scale is that fraction multiplied by the local scale factor.

For a large-scale national grid the variation is small and stated. The British national grid’s scale factor runs from 0.9996 on the central meridian to about 1.0005 at the extremities, so a sheet nominally at 1:50,000 is between 1:50,020 and 1:49,975. That is under a part in two thousand, which is why the printed fraction is usable.

For a world map the variation is enormous and nobody prints a fraction at all. A Mercator world map has no meaningful representative fraction, and reputable ones print a scale bar valid at the equator with a note, or omit it entirely.

That difference in practice is a good indicator of when scale distortion matters: it matters exactly when a map has a printed scale worth trusting, which is to say at survey scales, which is to say in the regime where the forced distortion is small enough to manage.

Scale on a screen #

The printed representative fraction has a digital successor and it inherits the same problem in a form worth spelling out, because it is the version most people meet.

A web map’s zoom level fixes the number of pixels per degree of longitude, not per metre of ground. The ground distance one pixel covers at zoom $z$ is

$$\frac{156{,}543 \cos\varphi}{2^{z}}\ \text{metres}$$

so at zoom 15 a pixel is 4.8 metres at the equator, 2.4 metres at 60°, and 0.83 metres at 80°. The map does not change its zoom level as it is panned north; the ground scale changes underneath it.

That is Mercator’s $\sec\varphi$ arriving as an interface property. A scale bar on a slippy map is recomputed for the current latitude, which is the honest response, and a measurement tool that forgets to do it is wrong by a factor of two in Scandinavia.

It also explains an effect every user of such a map has noticed without naming: the apparent detail changes with latitude at a fixed zoom. A city at 60° is drawn at twice the ground resolution of one on the equator, which is why high-latitude cities look sparser at the same zoom — there is more map per metre and the same amount of data in it.

Measuring a distance on a map, correctly #

The operational form, since this is the failure that produces wrong numbers rather than wrong impressions.

To measure a ground distance on a projected map, the length has to be divided by the scale factor along the line being measured, at the place it is measured. For a conformal projection that is one number per point, since the scale is the same in every direction. For anything else it depends on the direction of the line as well, which is why conformality earns its keep on a chart: one ruler serves every bearing at a given latitude.

At survey scale the correction is small and standardised. A distance measured on a UTM grid is divided by the grid scale factor, which surveyors compute for the line’s mid-latitude and mid-longitude, and the correction is up to about a metre per kilometre.

At world scale there is no correction that works, because the scale factor varies by more across the line than the measurement’s precision. The only correct procedure is to leave the map entirely and compute the geodesic on the ellipsoid, which is what every piece of software does and what no ruler can.

The one theorem in the subject is about this quantity and not about either of the other two. Over a 30° region no conformal projection’s scale can vary by less than 7.2 per cent, and exactly one projection reaches that floor — which is a hard bound on the third failure and has no counterpart for the first or the second.

The bound nobody can beat #

Scale distortion is the quantity the one exact optimality result in this subject is about, which is a good reason to give it its own name.

Chebyshev’s criterion minimises the ratio of maximum to minimum scale over a region, among conformal projections. Not the angular deformation, which is zero for all of them; not the areal error, which is the square of the scale factor and therefore the same information. Scale variation is the quantity there is a theorem about.

For a cap of angular radius $\rho$, the least achievable ratio is $\sec^2(\rho/2)$: 76 parts per million at 1°, 7,700 at 10°, and a factor of two for a hemisphere. Those numbers are floors, and every conformal projection of that region is at or above them.

So the quantity that neither standard measure reports is the one with the sharpest available theory attached. That is a reasonable case for naming it separately even though it is not independent.

Where the confusion shows #

Three practical errors, all of them versions of treating a scale as a number.

Measuring with a ruler on a small-scale map. The scale bar is valid where it was drawn. On a world map the same physical centimetre is a different ground distance at different latitudes, by factors of several, and the bar gives no indication of where it applies.

Computing area from a projected polygon. The planar area of a shape on a map is its true area times the areal factor, which varies across the shape. For a small region the mean factor is a fair approximation; for a country on a world projection it is not, and the correct calculation integrates on the ellipsoid rather than measuring on the map. That is why this site computes cell areas from closed forms rather than from projected polygons.

Buffering in projected coordinates. A “one kilometre buffer” drawn in a projected coordinate system is one kilometre of map, which is one kilometre of ground only where the scale factor is one. In a UTM zone that is an error of up to a part in a thousand; on a web map at high latitude it is a factor of several.

The last is the most common and the least noticed, because the result is a plausible circle of plausible size.

What a scale-free measure is for #

Worth defending the standard measures rather than only pointing at what they omit.

$\omega$ and the areal factor are constructed to be independent of the map’s overall size, and that is what makes them comparable. Two implementations of the same projection scaled differently, two maps printed at different sizes, two projections with different natural extents — all of them agree on $\omega$ and on the relative areal inflation, and none of them agrees on $a$.

That is the same tiering as everywhere else on this site: $h$ and $k$ depend on the parameterisation, $a$ and $b$ and the areal factor depend on the map’s overall scale, and only $\omega$ and the ratio $a/b$ depend on neither.

Which locates scale distortion precisely. It sits in the middle tier: a real property of the map together with a chosen unit, meaningless in absolute terms and entirely meaningful as a variation across a region. That is why every statement about it here is a ratio — the scale here against the scale there — and never a number on its own.

Why the classification usually stops at two #

A closing note on the vocabulary, because the three-way split is standard and the third member always gets dropped.

Conformality and equal area both have a clean zero. A projection either has $\omega = 0$ everywhere or it does not, and either has $ab = 1$ everywhere or it does not. Each is a falsifiable universal statement, each can be asserted, and each can be caught failing.

Scale distortion has no clean zero, because zero scale distortion everywhere is an isometry and an isometry does not exist. So there is no projection to point at as the case where this failure is absent, and there is nothing to assert.

That asymmetry is why textbooks list three distortions and then build the theory on two. It is a fact about what is checkable rather than about what matters, and the consequence is that the quantity a map user most often needs — how big is the map here, really — is the one with the least developed vocabulary.

The remedy is not a new classification. It is to report the scale factor alongside the other two, at stated points or as a range over a stated region, which is what a national grid specification does and what a world map almost never does.

The third failure has a floor as well as a ceiling. No map of a given patch can hold its scale better than a number fixed by the size of the patch alone.

And the reason the classification usually stops at two is a size. The smallest scale spread any map of a circular patch can have reaches fifty parts per million at ninety kilometres, and below about ten kilometres the whole effect is under a part per million — which is why plane surveying works at all and why the third failure is invisible to most of the people it applies to.

What was computed here #

Every scale factor on this page comes from the projection’s own derivatives, and the projections are normalised before comparison so that the geometric mean areal factor over the region is one. Without that normalisation the comparison would mostly measure each projection’s arbitrary choice of overall size.

The regional scale spread is taken over the region’s interior samples together with its boundary and, for a cap, its centre. That last detail was a bug: the equal-area ring sampler put its innermost and outermost rings half a step inside the centre and the rim, and the resulting spread came out below the proved Chebyshev bound by three parts in a thousand. A measured quantity that beats a theorem is a sampling artefact every time.

The bound itself is asserted rather than quoted: the stereographic projection centred on a cap must achieve $\sec^2(\rho/2)$ exactly, which it does to 1.4×10⁻⁸ at 30°, and every other conformal projection tested over the same cap must do worse.

The equidistance claims are checked the same way the other properties are. The azimuthal equidistant projection’s radial scale factor must equal one along every radius from its centre, which is what “distances from the centre are true” means when it is stated as a measurement, and it holds on the same noise floor as every other exact property in the library.

What the pictures cannot show #

The nominal scale. Every figure here is drawn at whatever size fits the page, so the absolute scale of each is meaningless by construction — which is the essay’s own point turned back on it. What the figures show is variation, and variation is the only part that survives being redrawn.

The figures also cannot show the qualification on an equidistance claim. The azimuthal equidistant map looks like a map on which distances work; the fact that they work only from one point is carried by the caption, and there is no way to draw a guarantee that applies to some pairs of points and not others.

Who found it, and when #

The distinction between the scale a map claims and the scale it has is as old as the printed scale bar, and it has been standard in the technical literature since Tissot.

What is less standard is treating scale variation as the primary quantity. The three-way classification of distortions into angular, areal and linear appears in most nineteenth-century treatments, and the linear one is usually mentioned and then dropped — because the other two have clean zeros and it does not.

Chebyshev’s 1856 criterion is the exception, and it is stated entirely in terms of scale variation. That a theorem exists for the measure everyone drops, and not for the two everyone keeps, is a fair indication that the vocabulary followed the arithmetic rather than the geometry.

Why it cannot name a class of projections #

There is a structural reason the third failure never became a category, and it is worth separating from the historical one.

Conformal and equal-area are pointwise conditions. Each is a statement about the two semi-axes at a single point — $a = b$, or $ab = 1$ — and a projection belongs to the class when the condition holds at every point. Nothing outside the point is needed to test it, which is what makes the class well defined.

Scale variation is not like that. At a point, the scale factor is simply a number, and calling it a distortion requires something to compare it against — the map’s nominal scale, which is an arbitrary global normalisation. Zero variation would mean $a = b = 1$ everywhere, which is the thing no map can do, so the only usable statement is a spread taken over a region, and the region is a choice the reader makes rather than a property the projection has.

A class needs a condition that holds at each point; the third failure has none. That is why “equidistant” always arrives with a qualifier naming the curves along which it holds, and why the only theorem about the quantity is a bound over a stated region rather than a classification.

Where this goes next #

The two measures this one sits between are the two ways a map is wrong. The construction all three come from is Tissot’s indicatrix. And the exact bound on scale variation is total curvature and the scale rule.