Where the worst point is

Ask where a map of a country is worst and the answer sounds like it must depend on the country. It does not, provided the map is conformal.

The claim is exact, it holds for every region and every conformal projection, and it is the reason a criterion that only inspects the frontier can settle a question about the interior.

The statement #

Let a conformal projection carry some region of the sphere to the plane, and let $k$ be its scale factor — one number at each point, because conformal means the two principal scale factors are equal and there is nothing left for a direction to distinguish.

Then $\log k$ attains its maximum on the boundary of the region and nowhere strictly inside it.

Not usually. Not for well-behaved regions. Always, for any region whatever, and for every conformal projection at once — Mercator, the stereographic, any Lambert conformal conic, the transverse Mercator series, an oblique aspect of any of them. The worst point of a conformal map of France is on the French frontier.

Where it comes from #

The proof is two lines of differential geometry and the site has both of them already.

Write the sphere’s own metric in the map’s coordinates. For a conformal projection the map is a similarity at every point, so the sphere’s line element in terms of the page’s $x$ and $y$ is

$$ds^2 = \frac{1}{k^2}\left(dx^2 + dy^2\right)$$

with no cross term — that absence is what conformality is. Coordinates in which the metric takes this shape are called isothermal, and for a metric written this way the Gaussian curvature of the surface has a closed form in terms of the conformal factor alone. Applied here it gives

$$K = k^2\,\Delta \log k$$

where $\Delta$ is the ordinary flat Laplacian on the page. The sphere has $K = 1$ in units of its own radius, so

$$\Delta \log k = \frac{1}{k^2} > 0$$

everywhere. A function whose Laplacian is positive is subharmonic, and a subharmonic function has no interior maximum. That is the whole argument, and the curvature of the sphere is doing all the work in it: the same calculation on a plane gives $\Delta \log k = 0$, the scale factor is harmonic, and the theorem becomes the ordinary maximum principle of complex analysis.

The identity, measured #

An identity quoted is an identity nobody checked, so it is computed here by two routes that share no algebra.

The first route is the derivation above. The second walks the page: pick a point, step a small distance in each of the four directions in map coordinates, ask the projection’s inverse where each of those four points came from, evaluate the scale factor there, and assemble the five-point Laplacian. Nothing in that procedure knows about isothermal coordinates or about Gauss.

The Mercator row at 45° is the one worth pausing on. There the scale factor is $\sec 45° = \sqrt{2}$, so $1/k^2$ is exactly one half, and a numerical Laplacian taken by differencing on the page returns 0.50000. Nothing was arranged: the half is $\cos^2 45°$ arriving from a completely different direction.

The check must also reject, and it does. Put a projection that is not conformal through the same machinery — using its larger principal scale in place of $k$, since there is no single scale factor to use — and the identity fails by 175% on Mollweide, 462% on Gall–Peters, 486% on the plate carrée and 1,165% on Robinson. Mollweide is the interesting refusal: at one of the four sample points it agrees to within 5%, purely by coincidence, and a check written against a single point would have passed a broken implementation.

What the picture shows, and what it took to get right #

The figure at the head of this essay plots the range of the scale factor over each ring of a region — the set of points a given fraction of the way from the middle to the edge — so that the horizontal axis is position within the region rather than latitude or longitude. A claim about where the extreme is needs an axis that measures where.

The first version of this measurement asserted something stronger and false: that both extremes lie on the boundary. That is the statement for a harmonic function, and $\log k$ is not harmonic — the curvature term makes it subharmonic, which constrains the maximum and says nothing at all about the minimum.

The Lambert conformal conic refused the assertion immediately. Over Europe its scale factor is smallest 65% of the way out from the middle, which is not an anomaly and not a numerical artefact: it is the standard parallel, the latitude where a conic is built to be exactly true to scale. A projection with a line of true scale running through the interior of a region has its minimum on that line, by construction, and there is no theorem forbidding it.

So the result is one-sided, and the corrected version is the stronger one. It is also the version that matters, because nobody minimising distortion cares where the scale is smallest.

Why the one-sidedness is the useful half #

Chebyshev’s criterion says that the best conformal map of a region — best in the sense of the smallest ratio between the largest and smallest scale factor — is the one whose scale factor is constant on the boundary.

Read cold, that is a strange thing for a criterion to be. Distortion happens everywhere in the region; why should a condition on the frontier settle anything about the middle?

Because of the theorem above. The worst point is on the boundary whatever the projection, so minimising the worst point means minimising something that lives on the boundary, and levelling the scale round the boundary is the only way to stop one part of it being worse than another. A projection with a bulge in its boundary scale can always be improved by pushing that bulge down; a projection with no bulge cannot.

The boundary constancy of the winner is measured, not assumed — the ratio of the largest to the smallest scale factor around the rim of the cap comes out at 1.00000003.

Where the maximum goes when the map is not conformal #

Nothing above survives the loss of conformality, and the failure is easy to exhibit.

Gall–Peters has its standard parallels at 45°, so within a band from 23.5° south to 23.5° north the exaggeration of shape is worst at the equator and improves towards the edges: 1.41× at the middle against 1.30× at the tropic lines. Behrmann, standard at 30°, does the same less dramatically. The Albers conic over the conterminous United States puts its maximum 85% of the way out — inside, but only just, and by an amount that would be easy to mistake for a sampling artefact if the conformal cases were not pinned to exactly 1.00 for comparison.

That contrast is what makes the measurement a measurement. Nine pairings of conformal projection and region produce nine maxima at $t = 1.00$ with no exceptions; three non-conformal pairings produce maxima at 0.05, 0.00 and 0.85. A run in which everything landed on the boundary would prove that the sampler could not see the interior.

What it means for a national grid #

A country designing a projection for itself is solving exactly the problem the theorem constrains, and the theorem tells it where to look.

The practical consequence is that the design question reduces from a two-dimensional one to a one-dimensional one. Instead of asking how the scale behaves over the whole territory, the designer asks how it behaves around the border, and adjusts the free parameters — the standard parallels of a conic, the scale factor of a transverse Mercator, the centre of an oblique aspect — until the border readings are as level as they can be made.

That is precisely what UTM’s constant of 0.9996 does in miniature. A zone is a rectangle and its worst scale error is on the two long edges, so the scale factor is chosen to trade the error on the edges against the error on the central meridian until the largest of them is as small as possible. The zone edges are the boundary; 0.9996 is the levelling; the 981 parts per million that results is the minimised maximum.

The boundary of the region, not the edge of the sheet #

Two boundaries are in play and confusing them turns the theorem into nonsense.

The one the result is about is the boundary of the region being mapped — a frontier, a coastline, the rim of a cap, the four sides of a UTM zone. It is a curve on the sphere chosen by whoever commissioned the map.

The other is the edge of the sheet: the place where a cylindrical projection is cut open, or where an interrupted projection tears. That edge is a property of the projection and can be moved anywhere by changing the central meridian, and the scale factor is perfectly well behaved across it — the two sides of a Mercator seam are neighbouring ground drawn at identical scale.

The distinction matters because a region can contain a seam. A map of the Pacific on a normal Mercator has the sheet’s edge running through the middle of the region, and the theorem is untouched: the maximum is still on the frontier of the region. What the seam does is make the picture discontinuous while leaving every scale factor continuous, which is the same separation the aspect essay draws between what a rotation moves and what it does not.

That Albers reading is the useful cautionary case. A single measurement at $t = 0.85$ could be a sampling artefact, and the only reason to believe it is not is that every conformal pairing in the same run lands on exactly 1.00. A measurement is discriminating when its two populations do not overlap, and here they do not.

Two things the theorem does not say #

It does not say the maximum is on the boundary for non-conformal projections, which is the whole content of the previous section and is not a corner case: most projections in use over large regions are not conformal.

It does not say the boundary maximum is small. Mercator over Europe reaches 2.92× at the frontier, and it is still perfectly true that its worst point is on the frontier. Knowing where the worst point is says nothing about how bad it is, and the two questions are answered by different machinery — this one by the curvature of the sphere, the other by the shape and size of the region.

Confusing them produces a specific error: the belief that a conformal projection is safe in the interior of the mapped area. What is true is that the interior is better than the worst of the boundary, which over a large region is a very weak guarantee.

What it does to a ranking #

A ranking of projections over a region is an average, and an average over a region is somebody’s weighting. The boundary result says something the averages cannot: that for the conformal candidates the worst case is decided entirely by the frontier, so two criteria that weight the interior differently will still agree about which conformal projection has the smallest maximum.

This is the practical division of labour between the two kinds of question the choosing field keeps separating. How much distortion does this map impose on average is a question about the interior and about a weighting nobody states. How bad does it ever get is a question about the boundary, and for a conformal projection it is a question about a curve.

The same argument in a different subject #

The maximum principle is one of the standard tools of potential theory, and cartography’s use of it is unusually concrete: the subharmonic function is a scale factor, the region is a country, and the boundary is a frontier drawn on the ground.

The connection is worth stating in the other direction too. Because $\log k$ is subharmonic with a known Laplacian, a conformal projection of a region is completely determined by its scale factor on the boundary, up to a rotation and a translation — the same statement as the solvability of the Dirichlet problem. Choosing a conformal projection for a region and choosing a boundary condition are the same act, and the enormous freedom the conformal family appears to offer is exactly the freedom to pick a function on a curve.

The conformal family is the one the boundary result belongs to, and membership of it is a measurement rather than a name.

What was computed here #

The scale factor of each projection was evaluated at 21 rings of 72 points across each region, using the principal scale factors derived from the projection’s four partial derivatives by Richardson-extrapolated central differences, and the ring at which the largest and smallest values occur was recorded.

The Laplacian identity was checked by walking the page: five points in map coordinates, inverted back to the sphere by Newton iteration on the forward map, with the scale factor evaluated independently at each. Agreement with $1/k^2$ is within 1.1% across three conformal projections at four points each, with the residual set by the finite step in the five-point stencil.

The refusals are four non-conformal projections at four points each, each of which must miss the identity by more than 50% somewhere, and three non-conformal pairings that must put a maximum in the interior.

What the pictures cannot show #

The figures show a range over each ring, which compresses the two dimensions of a region onto one. A ring is not a level set of the scale factor and was never meant to be; two points at the same $t$ can have very different scale factors, and the band’s thickness is exactly that spread.

Nor can the figures show that the result is a theorem rather than a survey. Nine pairings agreeing is evidence; the argument from $\Delta \log k = 1/k^2$ is the reason, and no number of green ticks would establish it.

Who found it, and when #

The maximum principle for subharmonic functions is nineteenth-century potential theory, and the relation $K = k^2 \Delta \log k$ between curvature and a conformal factor is in Gauss’s Disquisitiones generales circa superficies curvas of 1827 in a different notation — it is the same paper the Theorema Egregium comes from, which is not a coincidence, since both are statements about what the metric alone determines.

Chebyshev stated his criterion in 1856 and Grave proved it in 1896. The proof rests on the boundary result, and reading the criterion without it makes it look like an arbitrary condition rather than the direct consequence it is.

It is worth noting what the boundary result does to how the criterion is taught. Stated on its own, the best conformal map of a region has constant scale on its boundary looks like a condition somebody imposed for tractability. Stated after the maximum principle, it is the only place the condition could be: the extremes of the scale factor are on the boundary whatever the map does, so making them equal there is making them equal everywhere they can be found. The criterion is a consequence rather than a stipulation, and a reader given it in the wrong order has no way to see that.

The same is true of most conditions that look arbitrary: somewhere behind them is a theorem that makes them inevitable.

Where this goes next #

The scale factor has a size and a location, and it has one more property nobody reports: a direction. Where a map stretches the ground most is a question with an answer at every point, and on most projections that answer is not along the meridian — which is the next rung.