A ring can be drawn whole, and only one way

Seven essays about what a sphere does to a flat page all begin from the same fact: the sphere has no edge and no hole, and a page has an edge. No map of the whole sphere is one to one makes that the first impossibility, and what a cut buys prices the cut every world map pays for it in kilometres.

Most maps are not maps of the whole sphere. They are maps of a region, and a region can have a different shape in the sense that matters here — the topological sense. A country is a disc: it has one edge and no hole, and nothing topological stands between it and a page. A band of latitude is not a disc. The roaring forties, the Southern Ocean, the Arctic between its circle and 80° north, and the tropics are all rings: each has two edges, one on either side, and a hole in the middle that contains a pole.

A ring can be drawn on a page without a cut. A flat ring exists. The question is what that costs, and it turns out to be decided by one number the ring carries and no map can change.

The number a ring carries #

Two regions can be mapped onto each other conformally — keeping every angle — only if they have the same topology. For discs that is the whole story: any two discs, however shaped, are conformally the same, which is why a map maker choosing an angle-keeping projection for a country has a free choice among infinitely many.

Rings are different. Two rings can be mapped onto each other conformally only if one further number agrees, their conformal modulus. For a flat ring between two circles, the modulus is fixed by the ratio of the outer radius to the inner one; stretch one radius without the other and the ring can no longer be reached from the first by any angle-keeping map at all.

For a band of latitude the number is easy to read off. Along a meridian, the coordinate in which a conformal map of the sphere is simplest is the isometric latitude, ψ = ln tan(45° + φ/2), the same coordinate Mercator uses for its northing. A band from latitude a to latitude b has a modulus fixed by the difference ψ(b) − ψ(a), and any flat ring it is drawn onto, without a cut and keeping angles, has outer radius over inner radius equal to e raised to that difference.

Only the stereographic page hits the number, and it hits it to six decimal places, because only the stereographic of these four keeps angles. The other three are free to draw the ring at any ratio they like, and each draws a different one. A page that does not keep angles is not bound by the modulus, and a page that does has no choice.

The same number on Mercator, with the ring cut open #

The modulus is not an exotic quantity that only a pole-centred page shows. Every reader has seen it, on the most familiar map there is, without being told.

Why Mercator exists is the essay about the projection whose northing is the isometric latitude itself. On Mercator a band from latitude a to latitude b is drawn as a rectangle: its width is the full circle of longitude, 2π in the page’s own units, and its height is ψ(b) − ψ(a). So the rectangle’s height divided by its width is exactly the modulus of the band, to the last digit, at any scale the chart is printed at.

That is the same invariant the stereographic carries as a ratio of radii, read on a picture that has cut the ring open along the 180th meridian. Mercator is conformal, so it is bound by the modulus too; it simply spends the number on a rectangle’s proportions rather than on a ring’s, because a cut ring is a rectangle and a whole one is an annulus. The two pictures are the two faces of one number, and every conic between them — cut, conformal, with a gap — spends it on a sector.

Any ring, not only a band of latitude #

Nothing above needs the ring to run round a pole. A ring round any point of the sphere — the band of ocean between two distances from an island, the zone between two radii of a radar — is a band of latitude in a sphere whose pole has been moved to that point, and the aspect is a free choice is the essay about moving it.

So the result is a statement about every ring-shaped region anywhere. Drawn whole and keeping angles, it must be the stereographic centred on the ring’s own centre; cut once, it can be the oblique conformal conic with the ring’s two edges as its standard circles. Two charts are enough, and one is not counts the fewest sheets a whole sphere needs; for a ring the count is one, and this is the price of insisting on it.

On a body with a hole, north can be up everywhere meets the same invariant on a torus, where a conformal world map with no cut exists and its rectangle’s aspect ratio is fixed. It notes in passing that a ring-shaped region on an ordinary body is the same kind of object. This is that region, measured.

Without a cut, the map is forced #

Keeping the ring whole and keeping every angle leaves room for many maps. It does not leave room for a good one other than the stereographic.

The reason is short. Write any conformal map of the ring as the stereographic followed by an angle-keeping rearrangement of the flat ring it draws. The logarithm of the rearrangement’s local magnification is a harmonic function on that ring — the kind of function that equals its own average round every circle — and for a map that keeps the ring whole, is one to one and does not turn the ring inside out, its average round every circle is a single constant. So the logarithm of the composite map’s scale, averaged round any parallel of the band, is exactly the stereographic’s, shifted by a constant.

A function cannot vary over a region by less than its own averages vary. So every uncut, one-to-one, angle-keeping map of the ring varies in scale at least as much as the stereographic does, and the stereographic — or its mirror from the other pole — is the least-varying picture of a whole ring there is.

The figure is the argument tested rather than trusted. Each family is a different way of disturbing the stereographic without cutting the ring, each at six sizes, and not one member varies less than the map it disturbs. The families whose power adds more to the derivative — those with larger k + 1 — disturb more at the same ε, but the direction is the same for all of them.

What the stereographic’s scale does across the band #

The north-pole stereographic’s scale at latitude φ is proportional to 1/(1 + sin φ). It is constant round every parallel, which is what makes the ring close up, and it changes across the band from one edge to the other.

On the forties, the southern stereographic’s scale is 1.075 times larger at 40° S than at 50° S — larger on the edge further from the pole it is centred on, as a stereographic’s scale always is. That factor is not a choice anyone made. Keeping the ring whole and keeping angles fixed it, and the modulus of the band fixed the modulus of the flat ring it lands on.

One cut frees the cone constant #

Cut the ring along a single meridian and the constraint that forced the stereographic goes.

The general angle-keeping map whose scale depends only on latitude draws a parallel at radius proportional to e^(−nψ) and a meridian at angle nλ, for some number n — the cone constant. When n is 1 the meridians fan out through a full 360 degrees and the ring closes. For any other n they fan through 360n degrees, which is less than a full turn, and the ring cannot close: it opens along one meridian into a sector with a gap. That map is the conformal conic, and a cut is exactly what it needs.

The conic is the whole family is the essay that shows n running continuously from the cylinder at 0 to the plane at 1. Read against the ring, the family has a topological reading it did not have there: the cylinder and every conic between it and the plane need a cut, and only the endpoint at n = 1 does not.

With the cut, n can be chosen to make the scale equal on the band’s two edges:

$$n^{*} = \frac{\ln(\cos a / \cos b)}{\psi(b) - \psi(a)},$$

which is the conformal conic with the band’s edges as its standard parallels. What a standard parallel buys describes that choice as deciding where the distortion is zero; here it is the choice the ring could not make while it was whole.

Scanning every cone constant from −1 to 1 finds the least-varying one within two thousandths of that closed form on every band tried. On the forties n* is 0.708, the conic spans 255 degrees, and its scale varies by 1.0038 against the stereographic’s 1.0750.

The criterion a whole ring cannot meet #

Chebyshev’s criterion says that the conformal map of a region with the least scale variation is the one whose scale is constant along the region’s whole boundary. On a cap round a pole that is the stereographic, whose scale is constant round the cap’s one edge.

A ring’s boundary has two pieces. The stereographic keeps its scale constant along each of them, and the constants differ — 1/(1 + sin a) on one edge and 1/(1 + sin b) on the other — and nothing about the stereographic can make them agree, since the modulus has already fixed their ratio. Making them equal takes the exponent n*, which takes a cut. So a whole ring is a region on which the criterion’s one condition cannot be met, and the cut is what restores it.

The criterion is also a caution on the numbers that follow. The conformal conic is the best angle-keeping map whose scale depends only on latitude, and a cut band might be mapped better still by one whose scale varies along the parallels too. Nothing here rules that out, so every price below is a floor: not cutting costs at least that much.

The price of not cutting #

Both ends of the curve are forced, and the middle is not obvious.

A thin band costs little because a thin band hardly varies in scale under any reasonable map. The stereographic’s 1/(1 + sin φ) changes little over two degrees, and the conic’s advantage over it is a small fraction of a small number.

A band that reaches the pole costs nothing at all, and for a reason that is topological rather than numerical. At 90° the band’s inner edge shrinks to a point and the hole closes; the region is no longer a ring but a cap, which is a disc. The cone constant that makes the edges’ scales equal tends to 1 as the upper edge approaches the pole, the conic’s gap closes, and the best cut map and the best uncut map become the same map. The modulus, meanwhile, runs to infinity: a disc has no modulus to preserve.

In between, the price peaks at a band from 40° to about 70° north, at 1.14 — fourteen per cent more scale variation for keeping the ring whole. That band is broad enough for the stereographic’s scale to change substantially across it and still far enough from the pole for the best conic to be well away from n = 1.

Where the ring sits matters more than how wide it is #

The equatorial band is the ring that suits the stereographic least, and the reason is visible in the formula. The stereographic’s scale 1/(1 + sin φ) changes fastest where sin φ changes fastest, which is at the equator, and a band centred there has its best cut map at n = 0 — the cylinder, as far along the family from the stereographic as a conic can be.

That is the ordinary cartographic rule — cylinders for the tropics, conics for middle latitudes, azimuthal maps at the poles — with a topological edge to it. The rule is usually justified by where each family’s distortion is smallest. The ring adds that a map of a whole band of latitude, drawn without a cut, has no rule to follow at all: it must be azimuthal from a pole, wherever the band is, and pays for it in proportion to how far the band is from that pole.

Five rings on the Earth #

The tropics pay most, a factor of 2.13: an uncut conformal map of the band between them is a stereographic from a pole sixty-seven degrees away from its nearest edge, and its scale varies more than twofold across the band. The cylinder cut along one meridian varies by nine per cent.

The circumpolar oceans pay three to seven per cent, which is small and is not nothing. A chart of the Southern Ocean drawn whole — the natural picture of a body of water that circles a continent — accepts about four per cent more scale variation than the same ocean opened along one meridian, and the choice between them is the choice between showing the ocean as the ring it is and showing it at its most uniform scale.

The cut each band needs is a single meridian from edge to edge: 1,112 kilometres on the forties and on the Southern Ocean band, 1,494 on the Arctic band, 5,213 across the tropics. What a cut buys finds that a world map’s first cut along a full meridian buys shape at 8.72 degrees of mean angular deformation per ten thousand kilometres. On a ring the currency is scale variation rather than shape, and the rate is set by where the ring lies.

Where the model stops #

The sphere is a sphere. The isometric latitude and the modulus both have ellipsoidal forms, and the prices change by parts per thousand; nothing about the topology changes.

The cut maps compared are conics. The best cut map is taken as the best angle-keeping map whose scale depends only on latitude, which is the conformal conic. A cut map whose scale also varies with longitude might do better, so the prices are lower bounds on what staying whole costs.

The uncut result is a theorem, and the perturbations are a test of it. The families disturbing the stereographic are a check that the averaging argument has been applied correctly, not a proof that no other uncut map does better; the proof is the argument.

And the score is the scale’s largest value over its smallest. A score that weighted the band’s area, or measured angular deformation instead of scale, would be a different question — though for angle-keeping maps angular deformation is zero everywhere, and scale is all there is to compare.

Still open: a region with more than one hole #

A ring has one hole and one number. A region with two holes — an ocean with two islands in it, a country with two enclaves — has three numbers that no angle-keeping map can change, and from two holes upwards a region with k holes has 3k − 3 of them.

That count is classical. What has not been measured is what those numbers force. For one hole the answer was a single map and a single price. For two, an uncut conformal map has to satisfy three constraints at once with the freedom a flat domain with two holes leaves, and whether the least-varying such map is still something with a name — or whether the price of keeping a many-holed region whole grows with each hole — is a question the ring cannot ask.