The places where a map is exactly right

Every rung of this ladder is a negative. No map is faithful; the trade-off is two lines of algebra; what can be unrolled is exactly what has no Gaussian curvature; an isometry is impossible in two derivatives and possible in one. Nine essays, all of the form this cannot be done.

There is a positive version of the same statement and the ladder has never made it. A map is not right everywhere; it is right somewhere, and the somewhere is a set with a shape. Finding that set for each member of the library turns the impossibility from a theorem into a measurement, and the measurement has a number in it that no projection of a sphere can produce.

What “exactly right” has to mean #

A projection has two principal scale factors at every point, a and b, and this collection computes them from the four partial derivatives rather than from anything a projection is called. The map is exactly right at a point when both are one: a ruler laid on the page at the map’s stated scale then measures the true ground distance in every direction at once, not merely in two directions or on average.

That is a stronger condition than any this site usually applies. Conformal is a = b, and says nothing about their size. Equal-area is ab = 1, and permits a = 4 with b = ¼. True scale is a = b = 1, which is both conditions at once — and both at once is an isometry, which the curvature forbids.

So the theorem already says what the answer must be. What it does not say is what the set looks like for any particular map, and that is measurable.

Why the minima and not the crossings #

A detail of method, because it changed one of the numbers above and is the kind of thing that produces a wrong table quietly.

The first version of this measurement looked for sign changes of a − 1 along a meridian, which is the natural way to find where a function is zero. It reported seventy true-scale parallels for the plate carrée within four degrees of the equator, and none at all for Mercator.

Both answers are artefacts and they are opposite ones. The plate carrée’s a − 1 is already zero to the last bit of the arithmetic across a band round the equator, so its sign flips on floating-point noise and every flip reads as a crossing. Mercator’s a − 1 has a minimum of exactly zero at the equator and never changes sign at all, so a crossing search finds nothing on the one projection whose true-scale set is the most famous line in cartography.

Searching for minima of (a − 1)² + (b − 1)², and then requiring the minimum to be below a tolerance, fixes both: a tangency is a minimum, a crossing is a minimum, and noise round a zero produces one minimum rather than seventy. That distinction is the same one the exponents then measure — a tangency shrinks as √ε and a crossing as ε — so the method and the finding are two views of one fact about how the scale approaches one.

Fourteen projections, and what each is right about #

The set is found by locating the zeros of (a − 1)² + (b − 1)² — the minima, not the sign changes, because a scale can touch one without crossing it — and then asking whether the set runs along a parallel or stops at a longitude.

Three things in that table are worth stopping on.

The conics’ standard parallels are recovered rather than read. The library is told to build a conic with standard parallels at 20° and 60°; the measurement of where a = b = 1 returns 20.000° and 60.000° from the derivatives, with no access to the parameters. That is the check that the instrument works, and it is the same discipline that recovers a projection’s family from its symmetry group rather than from its name.

Four common projections are right at exactly two places. Mollweide, Eckert IV and the Winkel tripel are printed in atlases at wall size, and the true-scale set of each is two points on the central meridian: a place, not a line. Everywhere else on the sheet, in every direction, the scale bar is wrong.

Robinson is right nowhere at all. Its own definition names ±38° as standard parallels, and along those parallels the parallel scale is one while the meridian scale is not. There is no point on a Robinson map where both are one, so a ruler laid on it is never measuring the ground, in any direction, anywhere.

The instrument, and why it is a dimension #

A set of measure zero cannot be measured by its area, because its area is zero. What can be measured is the neighbourhood: the ground on which both scales are within ε of one, which is a band round the set and does have an area.

How that band shrinks says what the set is. Near a curve the scale passes through one at some rate, so a tolerance of ε admits a band of width proportional to ε and an area proportional to ε. Near a curve the scale merely touches, the departure is quadratic, the band has width √ε. Near an isolated point the band is a disc of radius proportional to ε, so its area goes as ε².

Measured over tolerances from 0.08 down to 0.01:

And the exponent this ladder is about is 0. A set with area — a patch of ground the map is exactly right on — would give a fraction that stops falling as the tolerance tightens, and a slope of zero on that chart. Nothing in the library produces it, nothing ever can, and this is the impossibility appearing as a measured number instead of as a citation.

The refusal, which has to come from off the sphere #

An exponent that is never zero is only evidence if zero is a value the instrument can return.

It is, and the witness has to be something that is not a map of a sphere: the identity map of a flat patch onto itself. Its (a − 1)² + (b − 1)² is identically zero, so the fraction within any tolerance is one at every tolerance, and the fitted exponent is 0.000000000000. That is what the measurement looks like when the answer is everywhere, and it is available only to a map whose source has no curvature.

The plate carrée, measured by the same code on the sphere, returns 0.485.

What a tolerance can and cannot buy #

The obvious response to a set of measure zero is that nobody needs exactness, and a tolerance is what real work uses. That response is correct and it is worth pricing, because the numbers are smaller than it suggests.

At a part in five hundred — five hundred metres in two hundred and fifty kilometres, which is loose by any surveying standard — the fraction of the world within tolerance in both directions is under one per cent for every member of the library. The band is narrow because the set at its centre has no width to start from, and a band round nothing is thin however generous the tolerance.

This is what a scale bar is right in one place states as a fact about legends, given its geometry: the place is not merely small, it is a set with no area, and the bar is right on it and nowhere else. The tolerance decides how large a neighbourhood of that place a reader is willing to call right, and this ladder’s contribution is that the neighbourhood shrinks to nothing rather than to a patch.

Two directions, and why the conjunction is the hard part #

It is worth separating the two halves of the condition, because each alone is easy and only the conjunction is hard.

Scale one along a curve is available to almost every projection and is what a standard parallel is. The parallel scale of a cylindrical projection is one at the equator; of a conic, at its two standard parallels; of the sinusoidal, everywhere. Nothing forbids that.

Scale one in every direction at a point is available too, and it is what the table’s first column measures.

Both, on an open set, is an isometry, and that is the whole of the obstruction. So the ladder’s negative statement and this rung’s positive one are the same sentence read in two directions: the true-scale set can be as large as a curve — one dimension — and adding the second dimension is exactly what the curvature costs.

That also explains the exponent 2 rows. Mollweide is exactly right at two points and not along the parallel through them, because it is an equal-area projection: ab = 1 everywhere, so a = b = 1 requires only a = b, which is conformality, and a projection that is equal-area and conformal at a point is right there and nowhere near it. Its true-scale set is the set where its angular deformation vanishes, which for a pseudocylindrical is a pair of points rather than a curve.

What happens on a body that is not a sphere #

The obstruction is curvature rather than sphericity, so the same measurement on a different body is a check that the argument travels.

The Earth’s curvature is not one number — it varies by 1.35 per cent from equator to pole on the ellipsoid, and by a factor of 2.7 on Vesta. None of that changes the conclusion here, and the reason is worth stating: the Theorema Egregium forbids an isometry from any surface with curvature to any surface without it, whatever the curvature’s value, and it forbids it on any open set however small. A body whose curvature is a thousand times smaller has a true-scale set of exactly the same dimension.

What does change is the band. The neighbourhood within a tolerance is set by how fast the scale leaves one, which is set by the curvature, so a nearly-flat body has a wide band and a strongly curved one a narrow band. How small is flat enough is that statement for a survey, and its answer is a size rather than a shape.

So the two halves separate cleanly. The dimension of the true-scale set is topological and identical on every curved body. The width of the band round it is metric and is a number the body decides. This rung measures the first and prices the second, and the pairing is the same one the impossibility field’s other anchor keeps meeting: what cannot be made small at all, against what can be made small at a price.

What this means for a reader with a ruler #

Three practical readings, in the order a reader meets them.

A stated scale is a statement about a set, and the set is never printed. “1:40,000,000” is true on the equator of a Mercator map, at two points of a Mollweide, and nowhere on a Robinson. No legend distinguishes those three cases and all three legends look identical.

“True along the standard parallels” is a half-truth with a measurable other half. The phrase is used of conics and cylindricals and it usually means the parallel scale is one there. For a conic it is the whole truth — both scales are one at 20.000° and 60.000°, which the measurement confirms. For the sinusoidal it is not: every parallel is true east–west, and only one meridian’s worth of places is true in both directions.

And a map with no true-scale set at all is not a defective map. Robinson is a compromise, scored honestly on this site against the alternatives, and its designer gave up exactness deliberately in favour of a shape that reads well. What is worth saying is that the trade was made and is not visible: a Robinson map carries a scale bar like any other, and there is no place on the sheet at which that bar is right.

What a scale bar is claiming #

The finding about Robinson makes a general question unavoidable: if the set of places where a map is exactly right can be a curve, two points, or empty, what exactly is a printed scale bar asserting?

On most sheets, something that is true almost nowhere. A bar is a single length labelled with a single ground distance, so it is a claim of constant scale — which no projection has, and which the true-scale set says holds at best along a curve. A reader measuring anywhere off that curve is reading a number that is wrong by the local scale factor, with nothing on the page to say by how much.

Nautical charts solved this and nobody else adopted it. A Mercator chart carries no distance scale bar at all. Distance is read off the latitude scale in the margin, at the latitude of the line being measured — one minute of latitude to a nautical mile — which is exactly right because the Mercator’s meridional scale factor and its parallel scale factor are equal at each latitude, so the latitude graduation is a scale bar calibrated locally. The navigator applies the correction by reading it in the right place.

That is a general solution wearing a particular disguise. The chart replaced a constant scale bar with a graduated one whose calibration varies down the sheet, and it did so because the people using it were measuring for a living and could not afford the error. Every atlas plate has the same problem and prints the constant bar.

The three honest alternatives are all available. A graduated bar, as on the chart. A bar drawn with its true-scale curve marked, so the reader can see where it applies. Or a stated range — this bar is correct at 45° and reads 12 per cent short at 65° — which is one sentence and covers the whole sheet.

It is worth saying why the chart’s solution did not spread, since it is not obscurity: an atlas plate is read rather than measured, and the constant bar is a gesture at scale rather than an instrument. The trouble is that nothing distinguishes the gesture from the instrument on the page, so a reader who does decide to measure is given no warning that they have crossed from one to the other.

And for a map whose true-scale set is empty, the only honest option is the third, because there is no place to mark and no graduation that would be right anywhere. A Robinson sheet with a scale bar is making a claim that has no locus at all, and this rung’s measurement is what says so.

What this rung establishes #

Every projection’s true-scale set is a curve, a pair of curves, a pair of points, or empty, found from the derivatives rather than from the parameters, and the conics’ declared standard parallels come back at 20.000° and 60.000° as a check on the instrument.

The set’s neighbourhood shrinks with a measured exponent — 0.48 for a curve of tangency, about 1 for a curve crossed, about 2 for isolated points — and the exponent that would mean a patch is 0, which nothing on a sphere returns and a flat map returns exactly.

One projection in common use is exactly right nowhere. Robinson has no point at which both scale factors are one, so no ruler on it measures the ground at any place in any direction.

The rest of this ladder says a map cannot be an isometry. This rung says how nearly it can be one and in what shape: along a line, at a point, or not at all — and the missing dimension is the whole of what the curvature takes.