A map with no formula

Every projection in this site’s library is a formula. Mercator is ln tan(π/4 + φ/2); Mollweide is a Newton solve for an auxiliary angle and then two lines; the polyconic is longer and still a formula. A formula can be quoted in a standard, implemented independently twice, and checked against itself — which is the entire basis on which a coordinate reference system is a system rather than a piece of software.

The map the previous rung solves for is not a formula. It is a list of complex coefficients produced by a least-squares fit to a particular boundary, and there is no expression for it in the ordinary functions.

That is a different kind of object, and this essay is about what changes.

The first question is how many numbers #

A map that has to be shipped as data is worth having only if the data is small, and the coefficient spectrum answers that directly.

For the elongated region, the boundary residual against the number of terms kept:

Two things in that table matter more than the residual column.

The spread stops moving at six terms, four rows before the residual does. The quantity a cartographer cares about — how much the scale varies across the map — is settled long before the boundary condition is met to machine precision, because the criterion is a statement about an optimum and an optimum is flat near its minimum. Nine numbers is a projection.

And the residual falls geometrically: each pair of terms buys two decimal orders. That is what makes the whole approach practical, and it is a property of the region’s boundary rather than of the method.

The second question is whether it inverts #

A projection that cannot be inverted is not much use: every reverse geocode, every click on a map, every reprojection of a raster is an inverse.

The solved map has no closed-form inverse and this is not a special defect. Neither does Mollweide’s; neither does the Winkel tripel’s; neither does the polyconic’s. The site’s library already carries a general two-dimensional Newton iteration on the forward map for exactly this reason, and it uses the same Jacobian the distortion calculation needs.

Pointed at the solved map, starting from the centre of the region rather than from the answer, it converges everywhere in the region and the worst round-trip error over 1,250 interior points is 8.6 × 10⁻¹⁴ radians — 0.55 micrometres on the ground.

That is a stronger result than it looks, and the reason is the analyticity. A conformal map’s Jacobian is a rotation times a scale, so its determinant is the square of the scale factor and can vanish only where the scale does. Inside the region the scale is bounded away from zero by construction — the fit holds it at one on the boundary and the maximum principle bounds the interior — so Newton has no singular point to fall into. A solved conformal map is better behaved under inversion than most named projections, which have poles, seams and points where the determinant is zero.

The third question is what it means to publish one #

This is where the object genuinely differs from every other projection on the site.

An EPSG entry for a projected coordinate reference system names a projection method and lists its parameters: a central meridian, a scale factor, a false easting. Two implementations reading that entry produce the same coordinates because the method is a formula and the parameters are numbers.

A solved map cannot be published that way. What would be published is either the coefficients themselves — which is a table, and a table has a precision, an ordering convention and a chart convention that all have to be stated — or the recipe: this boundary, this many terms, this fitting method. And the recipe is not reproducible unless every choice in it is pinned down, because a different sampling of the boundary or a different scaling of the normal equations gives coefficients that differ in the last digits and a map that differs by micrometres.

Micrometres do not matter. What matters is that two implementations no longer agree exactly, and exact agreement is the property a coordinate reference system is for. A published coordinate is a result makes this argument about survey coordinates and it applies unchanged here: the moment a definition becomes a computation, its output becomes a realisation of the definition rather than the definition itself.

The data is not larger than the thing it replaces #

Nine numbers is a projection is stated above as a conclusion about the spectrum, and it invites a comparison the essay does not make: how large is the definition of a named projection?

An EPSG entry for a projected coordinate reference system carries a method name and its parameters — a latitude of origin, a central meridian, a scale factor, a false easting, a false northing, and for a conic two standard parallels — which is five to seven numbers, plus the datum and unit declarations that a solved map would need as well. The well-known-text form of such a definition runs to a few hundred characters.

Nine complex coefficients is eighteen real numbers. How many digits each needs follows from the spectrum: the coefficients fall by 1.4 × 10¹³ from the first to the fourteenth, so reproducing the map to a relative 10⁻¹⁰ asks about ten significant figures of the leading coefficient and correspondingly fewer of the rest. Call it ten digits apiece, and the whole projection is under two hundred characters.

So a solved map is the same size as the definition it would replace. That is worth saying because the objection to shipping a projection as data is usually framed as though it were an objection about size, and it is not. It is entirely an objection about authority: a parameter list points at a formula that exists independently of any implementation, and a coefficient list points at nothing but itself.

What a corner costs, and it is not a factor #

The essay reports that a region with corners costs an order of magnitude more coefficients, which is true at one tolerance and is the wrong shape of statement. A geometric rate and an algebraic one do not differ by a factor; they diverge.

The smooth boundary buys two decimal orders per two terms. The square’s residual falls as N^−2.17. Setting each to reach a stated residual from the same starting point of 1.9 × 10⁻³ at two terms:

At drawing accuracy the corner costs a factor of two and is a nuisance. At the accuracy a coordinate reference system is held to it costs a factor of a hundred and sixty, and the series has stopped being a practical object.

The penalty for a corner is therefore not a number; it is the tolerance’s own exponent. Halving the target residual costs a smooth region one term and costs a square region a fixed multiplicative factor on however many it already had. That is the difference between a series that converges and a series that merely decreases, and it is why the polyhedral face map of the other ladder is measured by its exponent rather than by its term count.

It also explains a decision this collection has taken twice without connecting the two cases. A conformal map onto a polyhedral face has a branch point at each corner and is therefore in the algebraic regime; Lee’s closed form exists precisely because it puts the singularity into an elliptic function rather than approaching it with a polynomial. A closed form is not merely more elegant than a series here — it is the only object that reaches the corner at all, and the series can approach it forever without arriving.

What the object is good for #

Three uses, and they are not the ones a named projection has.

A one-off map of an awkward region. The construction costs a fit and produces the best conformal map of that region there is. Nothing has to be published, because nothing else needs to read the coordinates back.

A floor for comparison. A claim that some named projection is a good choice for a region can now be checked against the best possible, which was not previously available. Over a square region 20° across the solved map reaches a spread of 1.019 and the best named projection — the stereographic, centred on it — reaches 1.031, so the named choice gives away 1.2 per cent of scale spread.

A design tool. The solved coefficients say what the optimal map does, and a cartographer can read the shape of the answer off it and then choose the named projection that behaves most like it. That is how the criterion has actually been used in the literature for a century: not to publish the optimum, but to know how far from it a practical choice sits.

What a reader is supposed to do with it #

The awkward question, and it deserves an answer rather than a shrug.

A reader looking at a map wants to know what projection it is in, and every figure on this site says so — every figure here is a projection is one of the site’s own threads, and naming the instrument is the point of it. A solved map has no name to give.

What it has instead is a description: the minimum-distortion conformal projection of this region, fitted to this boundary, with a scale spread of 1.0138 and a boundary residual of 2 × 10⁻⁷. That is more informative than a name, not less — a name tells a reader which formula was used and this tells them what the map does — and it is the form every caption in this essay takes.

It is also the form a caption should probably take anyway. Naming Mercator tells a reader nothing about how badly the map they are looking at distorts the region it shows, which is the whole burden of measuring instead of naming. A projection that cannot be named forces the honest caption, and the site’s own habit was already to write one.

One more thing the coefficients decide #

A projection that is shipped as data has one property no formula has: it can be truncated by whoever receives it. Dropping the last two coefficients of a fourteen-term map is a legitimate operation that produces a slightly different, still exactly conformal map — and it is not available to somebody handed Mercator’s formula, who can only implement it or not.

That is a real convenience and it is also a hazard, because the truncated map is still conformal and still looks right. Nothing in it declares which terms were kept. The boundary residual is the only quantity that tells a reader how long the series was, which is the argument for publishing it beside every fitted map.

What was computed, and how #

The coefficient sizes are moduli of the complex coefficients of f, the map itself, after the exponentiation and integration — not of h, the series that was fitted. That distinction matters because the spectrum of h is the answer to the fitting problem and the spectrum of f is what would have to be shipped.

The round trip uses the site’s general inverse, unchanged, with the initial guess fixed at the frame’s centre for every point so that the test cannot be flattered by starting near the answer. Starting at the answer, the reported error is exactly zero, which is a measurement of nothing.

The truncation table is generated by re-fitting, not by dropping terms from a long fit. Those are different objects: a least-squares fit with six terms is not the first six terms of a fit with twelve, and treating them as though they were is the standard way to make a convergence table look better than the method is.

The comparison this ladder was built to make #

The condition ladder began with maps defined by conditions rather than constructions — the two-point equidistant map, the trimetric residual, the retroazimuthal map that satisfies its condition exactly and cannot be read backwards. All three of those are still formulae; their conditions happen to have closed-form solutions.

Chebyshev’s condition does not, and that is the difference this rung isolates. Four maps, four conditions, and only the last of them forces its map out of the class of things that can be written down.

The contrast is sharpest against the trimetric residual, which is what a condition costs when it cannot be met at all: three exact distances asked for where only two are available, and a shortfall growing as the cube of the region’s size. Chebyshev’s condition is different in kind. It can be met, exactly, by a map that exists, and what it costs is not accuracy but expressibility — which is a price nothing in the earlier vocabulary knows how to quote.

One consequence of that divergence is worth stating for the practice rather than for the theory. A specification that fixes a term count is safe for a smooth region and meaningless for one with corners, because the accuracy the count delivers depends on the tolerance being asked for and the two are no longer related by a fixed factor. A fitted map of a region with corners has to be specified by its residual rather than by its length, and the residual is the one number this essay recommends publishing anyway.

Where the model stops #

A truncated series is not the optimum, it is a map near it. Everything published here is the truncation’s own measured spread, so no claim depends on the difference; but the phrase the optimal projection should be read throughout as the best map the fit found, with its residual stated.

Nothing is said about how the coefficients behave under a change of region. A cartographer wanting a family of maps for neighbouring regions would want the coefficients to vary smoothly with the boundary, and they should, since the fit is linear and the boundary enters continuously — but that is an expectation and not a measurement.

Reproducibility is discussed and not demonstrated. A genuine test would be an independent implementation from a written specification, which is a thing one person cannot do to their own code.

The chart is a choice and the coefficients depend on it. The same map fitted in the isometric coordinate rather than the stereographic one has entirely different coefficients, which is another way of saying that a table of coefficients is meaningless without its chart — a species of the units problem that the units are part of the coordinate is about.

Who found it, and when #

Numerically defined projections are older than they look. The Robinson projection of 1963 is a table of twenty values with interpolation between them and nothing else — it has no formula at all, by design, because Robinson chose the values by eye until the map looked right. It has been published, standardised and implemented widely, and the fact that implementations disagree about how to interpolate between its knots is a live and well-documented nuisance.

The Winkel tripel, adopted by the National Geographic Society in 1998, is a formula forwards and a Newton solve backwards, so half of it is a computation. The National Geodetic Survey’s NADCON and the various grid-shift files are tables of offsets, and are the accepted definition of a datum transformation for exactly the reason a solved map cannot be: the table is the definition, published once, with an interpolation rule stated alongside it.

So the practice of shipping a projection as data has precedent, and the precedent says what makes it work: the table has to be the definition rather than an approximation to one, and the interpolation rule has to be published with it.

Where the ladder goes next #

The condition ladder is finished as an argument: a projection can be specified by what must be true of it, that specification may or may not have a formula behind it, and the cost of the ones that do not is expressibility rather than accuracy.

What has been quietly assumed throughout is that a projection is fitted to a region in a given frame — that the region sits where it sits and the map is built around it. Rotating the sphere first is free and changes everything, and the site’s machinery for searching that freedom turns out to have been searching one line of it.