Solving for the map instead of choosing it

Chebyshev’s criterion states the answer to the question this whole field is about. Among conformal projections of a region, the one whose scale varies least is the one whose scale is constant on the region’s boundary. Chebyshev stated it in 1856; Grave proved it in 1896; and the site has been able to exercise it on exactly one shape.

That shape is the spherical cap, where symmetry does the work: the stereographic projection centred on the cap has a scale of sec²(ρ/2), which depends only on distance from the centre and is therefore constant on every circle about it. The criterion is satisfied by inspection and the optimum has a name.

For any other region the site has quoted the criterion and stopped, and the expansion phase recorded why: the general case is a boundary-value problem this site does not solve. That was true when it was written and it is wrong now, because the problem is linear.

Why it is linear #

Three facts, and the third is the one that does the work.

A conformal map of the sphere is an analytic function of a conformal chart. Take any coordinate in which the sphere’s metric is a scalar multiple of the flat one — the isometric coordinate, or the stereographic one used here — and conformality is exactly analyticity.

The log scale of such a map splits into two pieces. With f the analytic function and μ the chart’s own metric factor, ln k = Re ln f′ + ln μ, and μ is known.

So Chebyshev’s condition is linear in the coefficients. Write ln f′(ζ) = Σ cₖ ζᵏ. Then Re ln f′ is a linear function of the coefficients, ln μ is a known function of position, and scale constant on the boundary reads

$$\operatorname{Re} h(\zeta_j) - C = -\ln \mu(\zeta_j)$$

for every boundary sample ζⱼ, with the constant C an unknown of the same fit. That is an over-determined linear system in the coefficients and C, and least squares solves it.

The map itself is then f = ∫ exp(h), obtained as a series by the standard recurrence rather than by quadrature, so the solved projection evaluates by Horner at one multiplication per term — as fast as any named projection.

What licenses it: the case with an answer #

A solver that returns a plausible map is worth nothing. The check is the cap, where the answer is known and nothing in the code knows it.

The agreement is not approximate in the way a numerical result usually is. For a cap the exact solution has h constant, so the fit finds a constant, and the residual is at the level of the arithmetic rather than of the method. The boundary residual on a 20° cap is 4 × 10⁻¹⁵.

What the criterion looks like when it is being met #

The condition is about the boundary, so the boundary is where to look.

Everything else about the solved map — its spread over the interior, its shape, the fact that its meridians are curves nobody has a name for — is a consequence of that flat line. The criterion says nothing directly about the interior at all, which is what makes it surprising that it settles the interior completely.

The mechanism behind that is worth stating because it explains why the criterion is a theorem rather than a heuristic. The logarithm of a conformal map’s scale factor is a harmonic function plus a known term, and a harmonic function attains its extremes on the boundary of its domain — the maximum principle. So the interior spread is bounded by what happens on the boundary, and flattening the boundary is the only lever there is.

What it is worth against a projection with a name #

A cartographer choosing a projection for a region is not choosing among all conformal maps; they are choosing among about a dozen with names, most of which have a free parameter or two that can be fitted to the region. The honest question is how much the optimum beats the best of those.

One per cent. That deserves to be stated plainly rather than buried, because the essay could have been written as though solving a boundary-value problem transforms cartography and it does not.

What it does do is settle a question that was previously unanswerable. Before this, a claim that some named projection was the best conformal map of a region could not be checked; there was nothing to compare it against except other named projections. Now there is a floor, and the interesting result is that the floor is close — that a century and a half of cartographers choosing projections by judgement got within a per cent of the theoretical optimum for a region shaped like this one.

It is not always close. Turning the same region 40° away from the graticule moves every named projection off its best aspect while the solved map is unaffected, because the solved map has no preferred direction to be off.

What was computed, and how #

The chart is the stereographic one centred on the region, with the sphere’s metric 2|dζ|/(1 + |ζ|²), checked against arc length by differencing rather than assumed.

The fit variable is scaled so that |ζ| ≤ 1 on the boundary, and the normal equations are solved with the columns scaled to unit norm. Without that scaling the high-order coefficients are lost to rounding and the residual stops falling as terms are added — which reads exactly like convergence and is its opposite.

The interior is sampled on a grid inside the boundary, and the reported spread is taken over the interior and the boundary, because a region includes its edge and on a cap the extreme is exactly there.

The measurement is made on the drawn map, not on the construction. The scale factor is |f′(ζ)|(1 + |ζ|²)/2, evaluated from the solved coefficients — and the projection is also handed to the site’s ordinary distortion(), which differences it numerically and returns an angular deformation of 10⁻⁶ degrees, confirming conformality by the same route that measures every other projection here.

What a per cent buys, and where it would be more #

The one per cent above is a statement about one region, and the number moves with the shape.

A round region is the case where the named projections are already optimal, because for a cap the optimum is the stereographic projection and a cartographer who chose it has chosen exactly right — the gap is zero, exactly, and the solver confirms it rather than improving on it. A long thin region running along a parallel is nearly the Mercator case; one running along a meridian is nearly the transverse Mercator case. The named projections were chosen, over three centuries, to cover the shapes that occur, and they cover them well.

Where the gap opens is where no named projection has the right symmetry: a region with a bend in it, a region with a hole, or a region whose long axis runs diagonally across the graticule. The solved map has no symmetry to be wrong about — it is fitted to whatever boundary it is handed — and that is the property worth having rather than the per cent.

The other place it opens is at the edges of a large region, where a conformal map’s scale grows fastest. For a 30° cap the best possible spread is 1.072 and for a 50° cap it is 1.223; a region of that size chosen badly can easily be at three or four times the optimum, and the criterion is then worth quoting rather than a rounding error.

Two things the fit refuses to do #

It will not fit a boundary with corners as well as a smooth one. The residual falls geometrically on a smooth boundary and like a power on a polygonal one, because the map has a branch point at each corner and a polynomial has none. For an eight-term fit that is the difference between 10⁻⁷ and 10⁻³.

It will not beat a rival that satisfies the same condition. Run against Mercator over a cap centred at 45° north, the solved map wins on spread by a factor of two; run against the stereographic projection centred on that same cap, it draws, because that projection is the optimum and the fit finds it.

The second is the more important refusal. A solver that beat everything would be measuring its own normalisation.

The residual is the honest part of the answer #

Every fitted map here is reported with the residual it achieved, and that convention is worth defending because it is unusual: a named projection is quoted without one.

A named projection has no residual because it has no fit — Mercator is exactly Mercator, and whatever its scale does over a region is what it does. A solved map is only as good as its boundary condition is met, and the number that says how well is the difference between a map and a claim. Quoting the spread of a fitted map without its boundary residual would be quoting the answer to a problem that may not have been solved.

The convention has a practical edge too. A fit whose residual is 10⁻³ has a scale that varies by a part in a thousand on the boundary, which is comparable with the interior spread it is claiming to minimise; at that point the optimality claim is empty even though the map is perfectly usable. Reporting both numbers lets a reader see which regime a given figure is in, and every figure in this essay is in the first.

Where the model stops #

One region, one map, no aspect. The solved map is fitted to a stated boundary in a stated frame. Nothing here searches over where to centre the frame, which for a region far from round is a real freedom — and it is the freedom the next essay measures for named projections.

Conformal only. Chebyshev’s criterion is a statement about conformal maps. The analogous question for equal-area maps has no comparable theorem, and the general minimum-distortion problem over all maps — Airy’s criterion minimised without a conformality constraint — is a genuinely harder variational problem that this site has not attempted and does not claim to have.

A boundary, not a country. The regions here are caps, ellipses and polygons on the sphere, defined by formulae. Fitting to a real national boundary would need a polygon from a dataset, with the simplification level that implies, and the site’s standing decision is to compute from stated shapes instead.

The optimum is local in the same sense every least-squares answer is. The linear system has one solution and it is the best fit within the truncation. Longer series get closer to the true optimum and, on a smooth region, converge geometrically to it; the residual reported with every fit is the honest statement of where it stands.

Who found it, and when #

Pafnuty Chebyshev stated the criterion in 1856, in a lecture to the Imperial Academy of Sciences, and the statement is one sentence: the best conformal projection of a region is the one whose scale is constant on its boundary. He did not prove it. Dmitri Grave, his student, proved it in 1896.

The constructive side has been slower and is mostly twentieth-century numerical analysis rather than cartography. The problem of finding a conformal map onto a prescribed region is the parameter problem of the Schwarz–Christoffel literature; the least-squares-on-the-boundary method used here is a cousin of the collocation methods that literature calls Symm’s and Bergman’s, adapted by the fact that on a sphere the metric factor is known in closed form.

That the two literatures answer the same question is not widely noticed, and it is the reason this essay is short of citations: the cartographic side states the criterion and the numerical side solves the problem, and they were not written for each other.

Verifying it is much cheaper than solving it #

A projection that exists only as a list of coefficients raises an obvious worry about trust. A named projection can be checked by anybody: write the formula out, implement it independently, compare. A fitted one appears to offer nothing of the kind — the only way to confirm the coefficients seems to be to re-run the fit, and re-running the fit reproduces every bug in it.

That reading is wrong, and the reason is the shape of the criterion.

The criterion is a property of the answer, not of the search. Chebyshev’s statement is that the best conformal map of a region has constant scale on the region’s boundary. Constancy on the boundary is something a second party can measure directly from the shipped coefficients: evaluate the scale factor at a few hundred boundary points and take the spread. One evaluation pass, no optimiser, no starting guess, no convergence question — and if the spread is small, the map satisfies the criterion whatever route produced it.

So the object ships with its own certificate. The expensive half is finding the map and the cheap half is confirming it, which is the same asymmetry that makes a factorisation checkable by multiplying and a root checkable by substituting. A projection defined by a search is not less verifiable than one defined by a formula; it is verifiable by a different operation, and the operation is cheaper than the one a formula needs.

What has to be shipped alongside the coefficients is therefore small, and it is not the code. It is the specification: the region’s boundary, the criterion, the basis and its degree, and the sample the residual was measured on. Given those, a second party can do three separate things, in increasing order of cost.

• Check the certificate — evaluate the boundary spread of the published coefficients. One pass.
• Check the map — sample the projected coordinates on a stated lattice and compare against a published checksum, which is what reporting the map rather than the parameters asks of every fitted object on this site. One pass.
• Re-solve — run their own optimiser from their own start and compare the resulting maps, not the resulting coefficients. Expensive, and the only one of the three that a reader would need a reason to do.

The first two are what makes the third optional, and the third is what makes a disagreement resolvable when it happens.

One caution about the certificate, since a one-sided test is the only kind on offer here. A small boundary spread confirms that the criterion is met; it does not confirm that the region, the basis or the sample were the right ones to have chosen. A map fitted to the wrong boundary will satisfy Chebyshev’s condition on that boundary beautifully and be the wrong map, and no amount of checking the certificate will say so. That question is answered by the specification being published rather than by any measurement of it.

And the residual is where the honesty lives, which the essay has already said in its own terms and which this sharpens. A published residual is a claim about how nearly constant the boundary scale is, and it is the one number in the whole object that a reader can refute in a minute. A named projection offers nothing comparable — it is exactly what its formula says, and whether that formula was a good choice for the region is a question the formula cannot be interrogated about.

Where the ladder goes next #

There is now a map. It has no name, no formula anybody could quote, no inverse in closed form, and it exists only as a list of coefficients.

That is a different kind of object from every other projection on this site, and it raises questions the named ones never do: how many numbers does it take to ship, what happens when it is inverted, and what a reader is supposed to do with a projection that cannot be written down.