A circle of a distance is not a circle

Every essay in this field so far has drawn a curve. The shortest route between two places is a curve; the rhumb that Mercator straightens is a curve; the leg an aircraft actually flies is a curve. The questions have all been about length, bearing, time and uniqueness, and every one of them is a property of a one-dimensional object.

A map is asked a second kind of question just as often, and this collection has never put it. What is within. Everywhere a helicopter can reach before it runs out of fuel. Everywhere closer to this hospital than to that one. Everywhere within two hundred kilometres of a pipeline. Those are sets — two-dimensional regions of the sphere — and nothing about them follows from the route essays, because a map can put a route in exactly the right place and still be badly wrong about the region round it.

The simplest such set is the disc: everywhere within a stated distance of one place. Its edge is the range ring, and it is drawn on more maps than almost any other constructed curve.

The claim, and what is wrong with it #

The claim a range ring makes is that a reader can take a ruler to it. It is drawn as a closed curve round a place, and it says: everything inside this is within so many kilometres. A legend often adds a scale bar, and a scale bar is right in one place — but the ring’s own promise is stronger than the scale bar’s, because it is a promise about every direction at once.

On the ground the promise is exact. The set of places at a fixed geodesic distance from a point on a sphere is a circle in the only sense that matters: it is the locus of a constant distance, and it has the full rotational symmetry of the sphere about that point. Nothing distinguishes one bearing from another.

On the page the promise is a claim about the projection, and it is almost never checked. What is drawn is the image of that locus, and the image is a circle only if the map treats every bearing from the centre alike — which is a much stronger property than any of the ones this site normally measures. It is not conformality: a conformal map preserves angles at a point, which says nothing about what happens ten degrees away.

Sixteen projections, one ring #

The spread is large and it is not the spread of any distortion measure this collection already carries. The Winkel tripel, a compromise projection nobody would call faithful, returns 1.12. Mercator returns 2.04 — and Mercator is conformal, which is supposed to be the property that keeps shapes right. The Lambert cylindrical, which is exactly equal-area, returns 4.17: it is the worst of the sixteen on this test and one of the best on any test of area.

The three sentences that follow from that are worth separating.

Conformality does not buy a round range ring. Conformal means the angular deformation is zero at a point, which fixes the ratio of two scales in two directions there. A ring 3,000 kilometres out is not there. Mercator’s scale grows as sec φ, so the top of the ring is drawn at a larger scale than the bottom, and the ring comes out as a lopsided oval that is nonetheless made of locally undistorted pieces.

Equal area does not buy one either, and buys the opposite. The Lambert cylindrical holds area exactly by compressing north–south precisely as much as it stretches east–west, so the two directions are pulled apart as hard as they can be while the product stays fixed. That is why it is the worst on this test: preserving the product of the two principal scales is a way of guaranteeing they differ.

The projections that do return one are the azimuthal family, and only about their own centre.

What the azimuthal family actually gives #

The azimuthal family is one function — a single radial function ƒ(ρ) turning a ground angle into a page radius, with the bearing carried through unchanged. That construction has rotational symmetry about the centre built into it, so every member draws a ring about that centre as an exact circle. The stereographic does. The orthographic does. The gnomonic does. So does the Lambert azimuthal, which is equal-area, and the equidistant one, which is not.

This is where the plan for this essay was wrong, and the machinery said so on the first run. The sentence in circulation — the one a search returns, the one this rung was written to test — is that the azimuthal equidistant is the projection that draws range rings correctly. What it draws correctly is not the shape. The shape is a property of the whole family, and it has nothing to do with distance.

The number the family does not share #

Draw eight concentric rings instead of one and the family separates immediately. On the azimuthal equidistant the page radius is proportional to the ground distance, exactly, by construction: ƒ(ρ) = ρ. Twice as far away is twice as far out, and a ruler on the page is a measurement.

Nothing else in the family does that. Over rings from 1,000 to 8,000 kilometres:

• the Lambert azimuthal falls short by 6.35 per cent at the outside — the 8,000 km ring is drawn where 7.49 belongs;
• the stereographic overshoots by 15.37 per cent, drawing the 8,000 km ring where 9.23 belongs;
• the orthographic falls short by 23.97 per cent, because it is a picture of a globe seen from infinitely far away and the far rings are foreshortened;
• the gnomonic overshoots by 142.30 per cent, drawing the 8,000 km ring at 19.38 units where 8 belongs.

All five pass the roundness test. Four of the five fail the test a reader is actually applying, which is that the rings are evenly spaced. That is the equidistant member’s own property, it is what its name says, and it is the half of the claim that is true.

The qualifier every published version of the claim drops #

An azimuthal projection has a centre, and it is a parameter of the map rather than a property of the projection. Move it away from the place the ring is about and the roundness goes:

So the azimuthal equidistant draws range rings as circles is not a statement about a projection at all. It is a statement about a projection and a place, and it holds for exactly one place per map. A chart with two range-ring diagrams on it, centred on two different airfields, has at most one of them right.

That is a general pattern this collection keeps meeting. The aspect is a free choice and a free choice has to be made; what a standard parallel buys is bought at one latitude and paid for everywhere else. A property attached to a parameter is not a property of the map.

What the ratio costs a reader with a ruler #

An axis ratio is an abstraction until it is turned back into ground. Do that the way a reader does: take the map’s own scale at the centre — which is what a scale bar states, and what a legend means by “1:40,000,000” — and use it to read the two extreme radii of the London ring.

On the Lambert cylindrical the ring’s shortest page radius stands for 3,000 kilometres of ground and its longest stands for 3,000 kilometres of ground, and the two page lengths differ by a factor of 4.17. A ruler calibrated on the short one reports the far edge of the ring as 12,517 kilometres away. A ruler calibrated on the long one reports the near edge as 719. Neither reading is a small error and neither is a random one: the direction of the ruler decides the answer, and the map gives no indication that it does.

Mercator’s factor of 2.04 is the one that will actually be met, because Mercator carries almost every map on the internet. A 3,000 kilometre ring drawn on it and read north–south returns 6,105 kilometres. The mistake is not exotic — it is what happens when anyone measures anything on a web map with the browser’s own coordinates, which is the ordinary case rather than the careless one.

Where the reader is, changes the answer #

The ratios above are all for London. Move the centre and every number moves with it, because the projections that fail here fail as a function of latitude:

At the equator every cylindrical projection is nearly innocent — the worst of the four is 4.4 per cent out. At 70° north the Lambert cylindrical stretches one ground distance into thirteen times another. The projections that look interchangeable on an equatorial test bench are not interchangeable, and a range-ring product tested at the equator and shipped to the Arctic has a defect that no test at the equator could find.

Mollweide is the interesting row. It is the only one of the four whose failure does not grow by an order of magnitude, and it is also the only one that is not cylindrical — its meridians curve toward the poles, which shortens the parallels there and partly undoes the stretch. It is not good; it is 2.1 at Tromsø. It is less bad in a different way, which is what a compromise projection is for.

Why this is a different ladder from the routes #

It is worth being exact about what separates this from the eleven essays that precede it in this field, because they share a subject and share none of their machinery.

A route question is a question about a curve, and the test is this: replace the route with a different route of the same length and the answer changes. The shortest path from London to Tokyo is not the shortest path from London to Cape Town, even though the two might be the same length; the bearing is different, the vertex is different, the time is different.

A reach question is a question about a set, and the same test comes out the other way. Everywhere within 3,000 kilometres of London is a fixed region, and no route enters its definition at all. The quantities it has — area, boundary length, whether a place is in it, which of several sets a place belongs to — are quantities a curve does not have, and the failures they meet on a page are failures a curve does not meet.

That is why a projection can be excellent at one and hopeless at the other. The gnomonic draws every great circle as a straight line, which is the single most useful property any projection has for route work and is why the gnomonic is Mercator’s companion. It also draws a set of concentric range rings at radii 142 per cent out at the edge of this test, which makes it useless for reach. Nothing in the route ladder would have found that, because nothing in the route ladder measures a set.

The ring is only half of it — the disc is the other half #

Everything so far is about the shape of the boundary. The set the boundary encloses has two numbers of its own, its area and its edge length, and both are quoted in planar form on maps that have no business doing so.

On a sphere of radius R, a disc of geodesic radius r has

$$\text{area} = 2\pi R^2\,(1 - \cos(r/R)) \qquad \text{boundary} = 2\pi R\,\sin(r/R)$$

and both tend to the planar πr² and 2πr as r goes to nothing. The machinery here checks the closed forms against a direct sample of the set rather than quoting them: at 4,000 km the sampled area agrees with the formula to 9.3 × 10⁻⁵ and the sampled edge to 2.7 × 10⁻⁷, which is the discretisation of the sampler and not a disagreement.

The two errors are different sizes at every radius, which matters because a report that quotes a catchment’s area and its perimeter takes both from the same planar assumption and is wrong by different amounts:

The boundary error is almost exactly twice the area error over this whole range, and it is the boundary that is usually the more surprising of the two, because a perimeter feels like a length and lengths feel safe. Both are second order in r/R, which is the same statement how small is flat enough makes about a survey: the error of treating a patch as flat grows as the square of its size, and the size at any stated tolerance is a number.

The quantity that is not monotone #

Push the radius out and the two quantities stop resembling each other entirely.

The area is monotone. It rises from nothing to the whole sphere — 510.07 million square kilometres — and it never falls, because adding to the radius never removes a place from the set. At half the circumference, 10,007 kilometres, the disc is exactly half the sphere.

The boundary is not monotone. 2πR sin(r/R) peaks at exactly that same 10,007 kilometres, at 40,030 kilometres of edge, and then falls back: at 15,000 km the edge is 28,356 km and at 19,000 km it is 6,351. Everything within 15,000 kilometres of London has a shorter boundary than everything within 10,000, while covering 85 per cent of the world against 50. In the plane both quantities rise for ever and the derivative of a perimeter is never negative.

The picture makes the reason obvious once it is drawn, and it is genuinely hard to hold in the head otherwise: the far edge of a very large reach is a small ring round the antipode. This is the same fact that a query disc is not a cell has to work around at small scales and that a polygon on a sphere has no outside states in its general form: a closed curve on a sphere bounds two regions, and once the disc is more than half the sphere the ring is better understood as the boundary of the complement.

What this rung establishes, and what it does not #

The set of places within a distance is the simplest reach question there is, and four things about it survive the page badly:

1. The shape of its boundary is preserved only by the azimuthal family, and only about the map’s own centre.
2. The spacing of several boundaries is preserved only by one member of that family, and this is the property the family is usually credited with as a whole.
3. The area it encloses is smaller than πr² at every radius, by an amount that is second order in the reach.
4. The length of its boundary is shorter than 2πr, by about twice the relative amount, and it is not even a monotone function of the reach.

None of that is a statement about routes. The shortest route from London to a place 3,000 kilometres away is drawn perfectly well on the Lambert cylindrical — it is a curve, and it lands where it should. It is the set that the same map gets wrong by a factor of four, and the reason is that a set has an extent in every direction at once while a curve has an extent in one.

What this rung does not establish is what happens when there is more than one centre. A reach question with several sources is not several discs — it is a partition, every place belonging to whichever source is nearest, and the boundaries are not range rings at all. Whether a query returns the nearest site depends on the metric is the closest this collection has come, and it asks about one query point at a time. The next rung asks about the whole surface at once, where the failure is not a shape but a misassignment with an area.