The worst point is not on the grid

Two hundred and eighty-six essays here measure maps. The founding rule is that no property is printed until it has been computed from the projection’s own derivatives, and the machinery that does the computing has been extended, audited and argued with many times since. What has never been audited is the last step, which is the same in every one of those measurements and is not a derivative at all: the numbers are read off a finite set of points.

A maximum angular deformation is a maximum over a sample. An Airy number is a quadrature. A ranking is an ordering of quadratures. Those are three different estimators with three different failure modes, and this ladder is about all three. It starts with the one that is easiest to state and hardest to notice, because its error is invisible in every direction a reader might look for it.

The estimator, stated plainly #

To measure the worst distortion a projection does over a region, sample the region and take the largest value. That is what distortion over a region does, it is what every published comparison of projections does, and there is no alternative that is not itself a sample of some kind.

The estimator has one property that decides everything else about it. A maximum over a subset cannot exceed the maximum over the set. So the number it returns is a lower bound on the truth, always, with no exceptions and no cases where it happens to come out high. It is not noisy. It is biased, in one direction, by an amount nobody reports.

That is the difference between this and an ordinary measurement error, and it is why the failure survives every check a careful person would run. Refining the grid moves the answer up. Refining it again moves it up again. Nothing ever moves it down, so there is no oscillation to notice, no scatter to plot, and no point at which the sequence looks unstable.

The case with an answer in closed form #

Auditing an estimator requires an answer the estimator had no part in producing. Mercator supplies one.

Its areal factor is $\sec^2\varphi$ exactly — the meridian scale and the parallel scale are both $\sec\varphi$, which is what conformality means for a cylindrical projection, and the areal factor is their product. Over a band from the equator to 60° north the largest value is therefore $\sec^2 60° = 4$, attained along the whole top edge and nowhere else.

Four. Not approximately four.

Nineteen per cent at the density a quick comparison uses. Two per cent at a density that costs nine thousand evaluations of a Jacobian. And every one of those numbers is a plausible-looking areal factor that somebody could print with three decimal places.

Why it never arrives #

The reason is in the hero figure and it is entirely geometric. A grid of cell centres over a box puts its samples at $(i + \tfrac12)/n$ of the way across, so the outermost row sits half a cell inside the boundary. Mercator’s areal factor increases with latitude everywhere, so its largest value over any northward band is on the band’s top edge, and no cell centre is ever on an edge.

The samples approach the edge as $n$ grows, so the estimate approaches the truth, but the approach is linear in the spacing rather than quadratic: halving the grid spacing halves the remaining distance to the edge, and halves the shortfall with it. The table above is exactly that — each row roughly half the one above.

The rate says where the extreme is #

The interesting quantity is not the shortfall at one grid size but how it falls, because that number carries information the estimate itself does not.

An extreme in the middle of a region behaves differently. The grid can straddle it: some sample lands within half a spacing of the peak on each side, and because the quantity is smooth and stationary there, being half a spacing away costs only the second-order term. So the shortfall goes as the square of the spacing, and refinement is four times as effective per halving.

An extreme on the boundary cannot be straddled, only approached, and the quantity is not stationary there — it is still climbing when the region stops. So the shortfall goes as the first power.

That is a diagnostic rather than a curiosity. Fitting the exponent of the shortfall says whether the extreme being estimated is inside the region or on its edge, without knowing where it is, and the answer changes what a refinement is worth. Around one, every doubling of the grid buys a factor of two, and the estimate is expensive to improve. Around two, every doubling buys four, and three refinements settle it.

The measured exponents are 0.88 and 2.48 rather than 1 and 2, and the departures are worth naming. The boundary case is a shade under one because the quantity’s curvature contributes a small second-order term as well. The interior case is over two because the Lambert conformal conic’s departure between its standard parallels is not a single smooth peak but a broad flat one, and a flat maximum is even easier to find than a round one.

What was computed, and how #

The truth in every comparison here comes from a different algorithm rather than a finer grid, which is the only arrangement that says anything. A pattern search starts from the best point of a coarse grid and then moves continuously, clamped to the region rather than to the sample, halving its step whenever no neighbour improves. It is therefore allowed to sit exactly on an edge, which is the whole difference. Against Mercator’s closed form it returns 4.000000000011 and locates the maximum at 60.000000° north.

Where no closed form exists, the reference is a one-dimensional golden-section solve on the zonal quantity, run to twelve digits. That is again a different estimator rather than more of the same one.

Both agree with each other and with the closed form wherever all three exist, which is what licenses using either where only one does.

The control that costs nothing #

Five projections in that figure return a shortfall of exactly zero, and they are exactly the equal-area ones.

The reason is not that the grid gets lucky. An equal-area projection has an areal factor of one everywhere, so the field has no maximum to miss: every sample returns the same number, and so does every point between the samples. The estimator is exact because the quantity is constant, and it would be exact on a grid of one point.

That is the cleanest possible statement of what is being measured. The error belongs to the field, not to the map and not to the grid. A sampled maximum of a constant is perfect; a sampled maximum of a quantity that climbs to the region’s edge is short by an amount proportional to the spacing; and no property of the projection other than the shape of the field it produces enters anywhere.

The same figure drawn for angular deformation instead makes the point from the other side, because there the equal-area projections are the ones with something to miss.

The corner, which is worse than the edge #

Eight of those ten projections have their worst angular deformation at 60° north and 30° from the central meridian: a corner of the box, as far from the region’s centre as it is possible to get.

A grid misses a corner in both coordinates simultaneously. Its nearest sample is half a spacing away in latitude and half a spacing away in longitude, so the shortfall picks up both first-order terms, and the constant in front of the rate is correspondingly larger. That is visible in the numbers: the areal shortfalls at $n = 12$ run from 4 to 13 per cent, and the angular ones over the same grid run from 10 to 17.

It is also a fact about how projections are built. A projection is usually designed to behave well near the centre of what it is drawn for, and the corner of a rectangular region is where the design has the least to say — so the worst point is at the corner more often than not, and the corner is where the estimator is weakest. The two facts point the same way, which is the shape of a trap rather than the shape of a coincidence.

What it does to a comparison #

A shortfall that were the same for every projection would be harmless: it would shift every number down together and leave every comparison intact. It is not the same for every projection, and the spread is the part that bites.

Over the band measured here the angular shortfalls at a twelve-by-twelve grid run from 10.4 per cent for the sinusoidal to 16.9 for the Winkel tripel — a spread of six and a half points of percentage, on quantities that differ from one another by less than that. Where the worst point is ranks projections by exactly this number, and two projections separated by five per cent in their true worst deformation can change places when both are estimated on the same grid, because the one whose extreme sits in a corner loses more of it than the one whose extreme sits on an edge.

So the estimator does not merely lower the numbers. It lowers them by an amount that depends on where each projection’s worst point happens to be, which is a property nobody is comparing and which correlates with nothing anybody cares about. That is worse than a bias and it is the reason this rung is not a footnote: which projection a weighting can make best shows how much freedom there already is in scoring maps, and this is a further degree of freedom that arrives without anybody choosing it.

The fifth rung of this ladder measures how much of it actually reaches this collection’s own published rankings. The answer is less than this section implies, for a reason worth waiting for.

Where the model stops #

The pattern search is not a proof. It finds a local maximum, from a start decided by a coarse grid, and a field with several separated peaks could hide one from it. Every case in this essay was additionally checked against a much finer grid and against a one-dimensional solve where the quantity is zonal, and where a closed form exists all of them agree. That is evidence rather than a theorem, and a genuinely multi-peaked distortion field — an interrupted projection, say — would need the search restarting from every grid point rather than from the best one.

A latitude–longitude box is the easy case. Its boundary is two parallels and two meridians, all of which a grid is aligned with, and that alignment is why the shortfall has such a clean rate. The regions in distortion over a region include caps and ellipses whose boundaries cross the grid at every angle, and the effective spacing along such a boundary depends on where it is being crossed.

A quantity has to be continuous for any of this to hold. The rates assume the field being sampled is smooth enough to have the Taylor expansion the argument is built on. A projection defined by a table has an interpolation in it, and a field with a kink in it obeys neither rate — which is the fourth rung of this ladder, where a refinement sequence converges beautifully to a number that is wrong by a factor of twenty-seven.

Nothing here says the published maxima are wrong. They are lower bounds, computed correctly, and stating them as maxima is the only defect. The next rung takes a case where the number reported is not a bound on anything, because the quantity being averaged has no average.

The generalisation #

The rule is not about maps and it is not about grids. It is about which side of an answer an estimator sits on, and whether anybody has said so.

The collection’s founding move was measuring instead of naming — refusing to call a projection conformal until the angular deformation had been computed and found to be zero. That refusal is only as good as the computation, and this rung is the first place the computation itself has been asked to show its working. It survives, in the sense that nothing it has printed is false; it does not survive in the sense that some of what it printed was a bound rather than a value.

This collection is careful about the direction of an error elsewhere: the tolerance that decides the verdict is explicit about which way a threshold can be wrong, and a coordinate is a number with a width treats a position as an interval rather than a point. A maximum over a sample is exactly the same kind of object — a number with a width, all of it on one side — and it has been printed as a point value in every essay that computed one.

The repair is one line of arithmetic and it is now in the machinery: report the sampled maximum with the refined one beside it, or report the shortfall’s fitted rate so a reader can see how much room is left. A statement like “at most 26.5°, sampled; 30.9° by a search that can reach the boundary” costs nothing and stops the reader from believing a digit that was never there.

Who found it, and when #

Nobody found this, because it is not a discovery. That an extremum over a finite set understates an extremum over a continuum is arithmetic, and that a smooth interior maximum is second-order accurate while a boundary one is first-order is standard in every numerical analysis text.

What is missing is the join, and the join is where every one of this collection’s failures of the same shape has been. The cartographic literature quotes maximum distortions constantly — a projection is “good to within 8 per cent” over some region — and does not state the sampling scheme. The numerical analysis literature has the rates and has nothing to say about maps. So a number computed by one field using a method audited by the other passes through unaudited, and it is a lower bound presented as a value.

The one place cartography does think about this is the design of an aspect search, where somebody optimising a projection cares a great deal whether the objective is being evaluated accurately. That is the same estimator with the roles swapped: an optimiser’s objective, rather than a reader’s number, and it gets attention because a bad objective visibly wrecks the optimisation while a bad number just looks like a number.

A cheap habit follows from the rates, and it is the one thing a reader of such a number can do without recomputing anything: ask what the sampling was, and if it is not stated, treat the quoted maximum as a lower bound of unknown tightness. A figure from a coarse graticule and one from a dense adaptive search are different kinds of claim wearing the same words, and the difference is larger at a boundary than in an interior — which is where a region’s worst distortion usually is.

Where the ladder goes next #

A sampled maximum is a lower bound, which is at least a statement about something. A sampled mean is not always even that, and the next rung takes the case where the quantity being averaged has no mean at all — where the estimator returns a number, a refinement returns a larger one, and the sequence has no limit to converge to. The projection this happens on is Mercator, and the quantity is the one whose average is quoted more often than any other in the subject.