Chebyshev's criterion

Almost every comparison of projections on this site ends in a table and a judgement. The criteria disagree, the weightings are unstated, and the honest conclusion is that a ranking is a statement about somebody’s region and somebody’s preferences.

There is one exception. It is from 1856, it has a proof, and it names a single projection as the answer to a question that is genuinely well posed.

The statement #

Fix a region $\Omega$ on the sphere. Among all conformal projections of it, consider the ratio between the largest and smallest scale factor over the region, and ask which projection makes that ratio smallest.

Chebyshev’s criterion: the minimising projection is the one whose scale factor is constant on the boundary of the region.

That is the whole theorem. It does not say what the projection is; it gives a condition that picks it out, and the condition is checkable.

Why the question is well posed and the others are not #

Worth being precise, because the difference between this and every other “best projection” claim is structural rather than a matter of rigour.

Three things are fixed before the question is asked.

The candidate family is the conformal projections of $\Omega$. That is a genuine, infinite, well-defined set — conformal maps of a region correspond to analytic functions, so there are as many as there are analytic functions, and the question is a real optimisation over a real space.

The objective is the ratio $k_{\max}/k_{\min}$ over the region. That is a worst case, not an average, so it needs no measure on the region and no weighting. The weighting is the step that makes an averaged criterion a preference, and a minimax objective does not take that step.

The region is named. Every distortion statement is about a region and this one says which.

Compare that with “which projection is best for a world map”. The candidate family is every smooth map of the sphere, the objective is appearance, and there is no theorem. The difference is not that Chebyshev was more careful; it is that he asked a question that has an answer.

The closed form for a cap #

For a general region the criterion names a projection and computing it is a boundary-value problem with no elementary solution. For one shape it is immediate.

Take a spherical cap of angular radius $\rho$. By symmetry the optimal projection must be symmetric about the cap’s centre, so its scale factor depends only on distance from that centre — and a scale factor depending only on distance is automatically constant on the boundary. That is the criterion, satisfied by construction.

The conformal projection with that symmetry is the stereographic projection centred on the cap, whose scale factor at angular distance $c$ is $\sec^2(c/2)$. So the least achievable scale ratio over the cap is

$$\frac{k_{\max}}{k_{\min}} = \sec^2\frac{\rho}{2}$$

The site measures the optimum’s achieved ratio against the closed form and requires agreement to a part in a million: at 30° it comes out at 1.0717968 against a bound of 1.0717968.

The rivals, and why they matter #

An optimality claim is worth nothing unless something loses, and the choice of rivals is where a check of this kind earns its keep.

The sharpest rival is the same projection, pointed somewhere else. A stereographic projection centred 20° from the cap’s centre is conformal, is the same construction, and differs only in where it was aimed. Over a 30° cap it manages 1.164 against the optimum’s 1.072 — worse by a factor of 2.3 in the excess.

That comparison isolates the content of the theorem cleanly. The winner is not a special projection; it is an ordinary one, correctly aimed, and the criterion is what says where to aim it.

The site also requires every rival to be conformal and to actually cover the region. Transverse Mercator was the first rival tried and it appeared to win, because its declared three-degree longitude window means the sampler saw a sliver of the cap and a sliver has almost no scale spread. A projection covering four per cent of a region is not a candidate to be the best map of it, and the check now requires ninety per cent coverage before a rival is admitted.

Why the criterion is a boundary condition #

The form of the answer is the interesting part, and it is worth unpacking because it explains why the theorem exists for conformal maps and for no other family.

A conformal map of a region is an analytic function, and analytic functions have a rigidity ordinary smooth functions do not: their values in the interior are completely determined by their values on the boundary. That is the maximum principle, and it is what makes the criterion possible.

The scale factor of a conformal projection is $|f'|$ for the analytic function $f$, and $\log|f'|$ is harmonic. A harmonic function attains its maximum and its minimum on the boundary of its domain and nowhere inside. So the largest and smallest scale over the region are both boundary values, and minimising their ratio is a question entirely about the boundary.

If the boundary values are not all equal, the ratio is the spread of a non-constant function and can be reduced by flattening it. When they are all equal the spread is zero on the boundary, the interior values are pinned between them, and there is nothing left to improve.

That is why the criterion reads as it does, and why nothing comparable exists for equal-area projections: an equal-area map has no such rigidity, its interior is not determined by its edge, and there is no maximum principle to argue from.

The Chebyshev connection #

The name on the criterion and the name on the polynomials are the same person and the same idea, which is worth noticing because it makes the result feel less like a coincidence.

Chebyshev polynomials are the solution to a minimax approximation problem: among all polynomials of a given degree with a fixed leading coefficient, the one whose maximum deviation from zero on an interval is smallest. The answer has the equal-ripple property — the extreme deviation is attained the same number of times, alternating in sign, spread evenly across the interval rather than concentrated at the ends.

The map criterion is that property in two dimensions. The optimal conformal projection is the one whose scale error is spread evenly around the boundary rather than concentrated somewhere on it, and “spread evenly” for a quantity attaining its extremes only on the boundary means constant there.

The same reasoning produces UTM’s scale factor: a tangent projection is exact in the middle and worst at the edge, and multiplying by a constant below one equalises the two extremes and reduces the worst case. Chebyshev’s criterion is that trade taken to its limit, over a region rather than along a line.

Where it has been used #

The criterion is not only a theorem. Several national mapping systems descend from it directly.

The Laborde projection of Madagascar (1928) is the best-known case. Madagascar runs north-east to south-west, so no normal-aspect projection fits it; Laborde constructed an oblique Mercator satisfying Chebyshev’s condition for the island’s outline, and it is still Madagascar’s official projection.

The Rosenmund oblique Mercator used for Switzerland, and several other national systems for awkwardly-shaped countries, follow the same reasoning: pick the aspect and parameters so that the scale is as nearly constant on the country’s boundary as the family allows.

The pattern is that Chebyshev’s criterion is used to choose parameters within a family rather than to construct an arbitrary conformal map. The exact optimum for a real coastline would be a conformal map with no name, computed numerically; a national grid needs a formula that can be published, so the practice is to take a parameterised family and optimise inside it.

That is a compromise the theorem does not sanction and everybody makes, and it is a fair one: the gain from the exact optimum over a well-fitted oblique Mercator is small, and the cost of a projection with no closed form is a legal system that cannot state its own coordinates.

What the optimum looks like on the map #

Worth describing, because the winning projection is not exotic and the reason it wins is visually obvious once stated.

The optimal conformal map of a cap is the stereographic projection centred on it, and the stereographic projection has a property no other conformal projection has: it maps every circle on the sphere to a circle on the plane. So a cap — which is bounded by a circle — becomes a disc, and the scale factor depends only on the distance from the centre.

That radial symmetry is the whole of it. A projection whose scale depends only on distance from the centre has, by definition, the same scale everywhere on any circle about the centre, and in particular on the boundary. The criterion is satisfied not by fine tuning but by the projection’s symmetry matching the region’s.

Which suggests the practical form of the criterion for regions that are not caps: make the projection’s symmetry match the region’s. A region with an axis of elongation wants a projection with a line of true scale along it, which is an oblique cylindrical; a compact region wants a radially symmetric one. That is not the theorem, and it is the theorem’s shadow, and it is the reasoning behind every well-fitted national projection.

Its indicatrices are all circles — the projection is conformal, measured — and they grow with distance from the centre. That growth is the quantity the criterion minimises, and conformality says nothing whatever about it.

What it does not settle #

Four limits, and naming them is what keeps this from being an answer to a question nobody asked.

Conformal only. The theorem is about the conformal family. Give up conformality and the scale ratio can go lower still, at the cost of angular deformation, and there is no comparable theorem for the general case.

Scale ratio only. The objective is the worst-case scale spread. Two projections with the same ratio can distribute the variation completely differently — one uniformly, one concentrated in a corner — and the criterion does not distinguish them. It is a minimax, with the usual minimax property of being insensitive to everything except the extreme.

No construction for a general region. For a cap the answer is stereographic. For an arbitrary boundary the criterion is a condition rather than a formula, and finding the map satisfying it means solving a boundary-value problem numerically. Which is why it is used to tune families rather than to build maps.

Nothing about world maps. A region with no boundary has no boundary condition, and the criterion has nothing to say about the sphere as a whole. The world-map question stays a judgement, and that is not a gap in the theorem — it is the reason the theorem needs a region.

The shape of a defensible “best” #

The transferable part, since this is the only place in the subject where the word is earned.

A claim that something is optimal requires three things stated, and Chebyshev’s case has all three: a candidate family, an objective, and a proof that one candidate minimises it. Robinson and Winkel tripel have none of the three — their family is “any smooth map of the sphere”, their objective is appearance, and there is no theorem.

That is not a criticism of them. The objective of a general-purpose world map is genuinely not expressible as a functional, and optimising a proxy for it would be rigour about the wrong thing.

What the comparison does show is where the boundary lies. A bounded region has a boundary for the criterion to live on; the sphere does not. The difference between a subject with optimality theorems and one with rankings and arguments is, in this case, exactly one boundary curve.

Measured along a meridian, a conformal rival has its angular deformation on the noise floor and its areal factor — the square of its scale factor — varying far more across a compact region than the optimum’s does. Being conformal is what the two have in common; how much the scale moves is what separates them.

The criterion has a closed form for the one case it is usually stated for, and reading that closed form as a function of the cap’s size answers a practical question.

And the optimum’s own spread against the radius of the cap is a straight line of slope exactly two on logarithmic axes, reaching five parts per million at 28.5 kilometres — the size at which a survey can stop thinking about projections altogether.

The slope of two in that figure is the closed form expanded rather than a fit. Since sec²(ρ/2) − 1 = tan²(ρ/2), the optimum’s excess over unity is (ρ/2)² for a small cap, which is (r/2R)² in ground units — so the least achievable scale spread over a region of radius r is the square of half its angular radius, and nothing whatever can beat it.

Inverting gives the size at which a survey may stop caring, in one line: r = 2R√τ. At five parts per million that is 2 × 6,371 × 0.002236 = 28.5 kilometres, which is the figure the ladder reports. At one part per million it is 12.7 kilometres and at a hundred, 127 — a square-root law, so tightening a tolerance a hundredfold costs a factor of ten in extent and not a factor of a hundred.

What was computed here #

The bound $\sec^2(\rho/2)$ is a closed form and is evaluated directly. Everything else is measured from the projections’ own derivatives.

Three assertions hold the result. The stereographic projection centred on the cap must achieve the bound, to a part in a million — it does, to 1.4×10⁻⁸ at 30°. Its scale must be constant on the cap’s boundary, which is the criterion, and comes out at a ratio of 1.00000003. And every conformal rival must be strictly worse by at least a part in a thousand, which they are by margins of 8% to 250%.

The measurement was wrong the first time in a way worth recording. The region sampler used equal-area rings, so its innermost ring sat half a step outside the cap’s centre and its outermost half a step inside the rim — and the reported scale ratio came out below the proved bound by three parts in a thousand. A measured quantity that beats a theorem is a sampling artefact every time, and the fix was to sample the extremes where they occur rather than near them.

The rivals are also chosen deliberately rather than for convenience. Two of them are different projections and the third is the winning projection pointed twenty degrees away, which is the sharpest available comparison: it differs from the optimum in one parameter and in nothing else, so the margin between them measures the value of the criterion rather than the value of the projection.

What the pictures cannot show #

The optimisation. The figure shows four curves and a line, and the theorem is a statement about an infinite family of which four members are drawn. Nothing in the picture indicates that the line cannot be crossed; that is carried by the proof and by the caption.

The figures also cannot show a region that is not a cap, which is every real region. The criterion applies to all of them and the closed form does not, and the difference between those two facts is where most of the practical difficulty lives.

Who found it, and when #

Pafnuty Chebyshev stated the criterion in 1856, in a paper on the drawing of geographical maps, and did not prove it. He is better known for the polynomials that bear his name, and the connection is direct: both come from minimax approximation, where the optimal approximation is the one whose error is equalised rather than concentrated.

Dmitri Grave supplied the proof in 1896, forty years later. That gap is not unusual for a result of this kind — the statement is easy to believe and the proof requires the theory of conformal mapping to have matured.

Laborde’s Madagascar projection of 1928 is the first major practical application, and the technique of tuning an oblique Mercator to a country’s outline by this criterion has been standard for awkwardly-shaped territories ever since. A nineteenth-century minimax theorem, applied to an island, and still in force.

Where this goes next #

The framing this theorem is the exception to is every projection minimises something. The bound it establishes for a cap is total curvature and the scale rule. And the ordinary case — comparison without an optimum — is distortion over a region.