A projection defined by a table has an interpolation in it

Eight rungs of this anchor treat a projection family as a construction with derivatives — cylinders, cones and planes to begin with, and a symmetry rather than a shape by the end. Cylinders, cones and planes; conditions; symmetries; the aspect as a free parameter. Every one of them assumes there is a formula to differentiate.

One member of this library has none. Arthur Robinson designed his projection by eye in 1963, adjusting nineteen pairs of numbers at five-degree steps until the map looked right to him, and the table is the definition. There is no expression it approximates and no closed form it is a truncation of.

Everything between the rows therefore belongs to whoever ran a curve through them — and so does every derivative anywhere, which is to say every distortion figure ever published for the projection.

The definition, in full #

Two columns. The first scales the length of each parallel relative to the equator; the second gives the distance of each parallel from the equator. Multiply by 0.8487 and 1.3523 respectively and the projection is complete.

Nothing else is specified. Not a rule for intermediate latitudes, not a smoothness class, not a preferred interpolation — nineteen rows and two constants.

Three published schemes #

The three interpolations here are the ones in actual use.

Linear is what several implementations ship, on the reasoning that the difference from anything smoother is below the width of a drawn line. That reasoning is correct about the drawing.

Aitken’s quadratic through the three nearest rows is what the published implementations derived from the National Geographic and USGS descriptions use, and it is what most software inherits.

Catmull–Rom is a cubic through four rows, which is the usual smooth default when somebody decides the projection ought to be differentiable.

All three agree at every tabulated row, exactly, because all three interpolate rather than approximate.

The drawings agree #

The graticules the three produce differ by at most 4.25 parts per thousand of the map’s own span.

On a page 200 millimetres wide that is 0.85 millimetres, which is about two line widths — visible if the two are overlaid precisely, invisible otherwise, and far below the accuracy of any use anybody puts a Robinson map to.

So the reasonable-sounding defence of linear interpolation is right. Nobody looking at a Robinson map can tell which scheme drew it.

And the distortions do not #

Thirty-seven of eighty-seven sampled latitudes disagree by more than half a degree of angular deformation, and the worst disagreement is 8.42 degrees. The areal scale factor differs by 0.567 between schemes at its worst point.

Those are not small numbers by the standards of this collection. This site’s own conformality tolerance is $10^{-4}$ degrees and the smallest real failure it detects is 0.385 degrees — so the disagreement between two implementations of one projection is twenty times the site’s headline finding.

A scale factor that is a staircase #

This is the sharpest form of the result and it is not a matter of degree.

A piecewise-linear interpolant has a derivative that is constant inside each interval and undefined at the knots. So the meridional scale factor of a linearly-interpolated Robinson is a step function: 27.3 per cent jumps at every five-degree row, and flat between them.

A projection whose scale factor is a step function is not a projection anybody designed. Robinson set his numbers to produce a smooth-looking map, and a smooth-looking map with a piecewise-constant scale factor is a description of the interpolation rather than of the design.

Why the drawing agrees and the derivative does not #

The gap between those two facts is worth an explanation, because it is the general lesson of the essay rather than a fact about Robinson.

Interpolation controls the value and does not control the slope. Three curves through the same nineteen points can agree to a thousandth of the span everywhere and have wildly different tangents, because the error in a value is bounded by how far the curves can wander between knots and the error in a slope is bounded by nothing of the sort — a curve can return to a knot from any direction.

The knot spacing here is five degrees and the map spans ninety, so the values are constrained at every eighteenth of the range. That is tight enough to pin the picture and loose enough to leave the derivatives free, and the derivative is where every distortion measure lives.

That is why “the difference is below a line width” is a true statement that supports none of the conclusions it is used for. It is a statement about values, and every quantity in this collection is a statement about derivatives.

What this does to a distortion comparison #

Every published comparison that includes Robinson — every table of angular deformation, every Tissot audit, every ranking of compromise projections — has a number in it that depends on an interpolation the comparison does not name.

That is a reproducibility problem of a specific and unusual kind. Two authors running the same analysis on the same projection with the same software will agree; two authors using different software will not, and neither will be able to say why, because neither of their tools reports its interpolation scheme.

This collection’s own conformality audit includes Robinson and inherits exactly this. Its number for that projection is a number about Catmull–Rom or about linear interpolation, depending on which build produced it, and the honest thing is to say so.

Is Robinson unusual? #

Partly. It is the only member of this library defined by a table, and that is deliberate on Robinson’s part — he was explicit that the projection was designed by iteration on appearance rather than derived from a condition.

But the shape of the problem is general. Several widely used projections have piecewise definitions with matching conditions at the seams: the Goode homolosine is sinusoidal below 40°44′ and Mollweide above it, joined so that the positions agree and the derivatives do not. The Winkel tripel is an average of two projections and is smooth, but the average of two projections shows how easily an averaging construction produces properties nobody chose.

Giving up continuity is the other family with joins in it, and its seams are chosen rather than inherited. So the general lesson is: a projection defined by pieces has properties at the joins that belong to the joining rather than to the design, and the joins are where the derivatives live.

What Robinson himself said #

The projection was commissioned by Rand McNally and Robinson described the process plainly: he worked from appearance backwards to the numbers, rather than from a condition forwards to a formula. He called the result a compromise and declined to state an optimality property for it.

That is an honest and unusual position, and it makes the projection a genuinely different object from every other member of the library — which is what a compromise projection is, taken to its conclusion.

What he did not anticipate is that his table would be differentiated — that measuring instead of naming would become the way a projection is judged. A designer who is choosing an appearance has no reason to care about the second derivative of the curve through his control points, and a hundred later authors computing distortion measures have every reason to.

The fix, and why it is a choice rather than a repair #

There is no correct interpolation of Robinson’s table, because the table does not determine one.

What a publisher of a Robinson implementation can do is state the scheme, so that a distortion figure computed from it is reproducible. That is a metadata line rather than a mathematical advance, and it is the same remedy this collection reaches for whenever a number turns out to depend on an unrecorded choice.

If a smooth scheme is wanted, a cubic through four rows is the natural default and is what this collection uses. It has no more claim to correctness than any other; it has the property that the derivatives exist, which is what everything downstream assumes.

An implementation detail that is not one #

There is a category error worth naming, because it is why this went unexamined for fifty years.

The choice of interpolation looks like an implementation detail: a decision internal to a program, invisible in its output, of no interest to the user. Programs are full of such decisions and it is right that they are not documented.

This one is not internal. It is part of the definition of the projection, because the projection has no other definition, and a program that chooses it is not implementing a specification — it is completing one.

The tell is that the choice affects a measurable, published quantity. An implementation detail cannot do that; if it can, it was a specification gap and the program filled it. Every projection library that ships Robinson has silently filled the same gap in one of three different ways.

The refusal #

The comparison has to be capable of reporting agreement and it does: at every latitude that is a multiple of five, all three schemes give identical positions, identical scale factors and identical deformations, to machine precision.

That is what makes the disagreements elsewhere a measurement of interpolation rather than of arithmetic noise. If the three schemes disagreed at the knots too, something would be wrong with the implementation rather than with the definition.

What the site did about it #

This collection has carried Robinson from the beginning, interpolated linearly, with a comment in the source saying that a cubic would be closer to Robinson’s intent and that the difference is below the width of a drawn line.

That comment was right about the line width and wrong about the consequence, and it is the reason this rung exists. Every distortion figure this site has published for Robinson — in the conformality audit, in the compromise-projection comparisons, in the rankings by region — was computed from a step function.

The repair is not to switch the default silently, which would move every published number for that projection with no record of why. It is what has been done instead: the three schemes are now available by name, the comparison above is a figure, and the site’s own choice is stated where a reader can find it.

That is the general remedy for a number that turns out to depend on an unrecorded choice, and it is one this collection has had to apply to itself several times. Record the choice, publish the comparison, and let the number be a number about a stated thing.

The gap between the two facts has a rate as well as an explanation. An interpolant of order p through knots spaced h apart is wrong in its values by something of order h^(p+1) and wrong in its slopes by something of order h^p — one power worse, always, for every scheme. So the ratio between the derivative’s disagreement and the value’s is of order 1/h, and with h five degrees, or 0.0873 radians, that ratio is about eleven.

Eleven is the factor by which a scheme’s invisibility in the drawing overstates its invisibility in the distortion, before any of the particulars of Robinson’s table enter. It is a general property of interpolating anything, and it is the reason the phrase below the width of a drawn line can be true of a picture and worthless as a guarantee about anything computed from it.

Where the model stops #

Three schemes are compared here and there are more. Cubic splines with various end conditions, monotone interpolants, and the trigonometric fits several authors have published as closed-form approximations to the table each give another answer, and each is somebody’s Robinson.

What is also not treated is the other column. Everything above measures the effect of interpolating both columns with one scheme; interpolating them with different schemes, which no sane implementation does but which nothing forbids, gives yet another family.

The five-degree spacing is doing the work #

One more structural observation, because it says which tabulated objects are safe.

The disagreements are largest where the table is coarsest relative to how fast the function is changing. Robinson’s columns change slowly near the equator and quickly near the poles, and the five-degree spacing is uniform — so the same spacing is generous at one end of the table and tight at the other.

Measured, the worst angular disagreements are at low latitudes, where the two columns are nearly flat and the relative differences between interpolation schemes’ slopes are largest. A flat function interpolated three ways gives three nearly identical curves and three quite different derivatives, because the derivative is small and the differences between them are not.

So a table with more rows would not simply reduce the problem in proportion. It would reduce it where the function moves and leave it where the function is flat, which is where the trouble already is.

What a well-specified tabulated projection would carry #

Three lines, and none of them is hard.

The interpolation scheme, named, as part of the definition rather than of any implementation.

The claimed smoothness class. A designer who cares about the map’s appearance cares implicitly about continuity of the first derivative, and saying so rules out the linear scheme immediately.

A reference implementation’s derivatives at a few stated points, so that anybody can check whether their software agrees. This collection publishes the number that matters most — the meridional scale at 60° longitude and 45° latitude — and it differs between schemes by more than a per cent.

None of that is a criticism of Robinson, who was not writing a specification for software that did not exist. It is a description of what the object has needed ever since.

Who found it, and when #

Robinson published the table in 1974. The observation that implementations differ appears in the projection-software literature from the 1990s, usually as a compatibility note rather than as a measurement, and the usual framing is that the differences are negligible.

They are negligible in the drawing and they are not negligible in the derivatives, and the reason nobody separated the two is that a projection is normally its formula — so the distinction between the map and the way the map was interpolated does not arise for any other member of the library.

Two implementations that cannot be told apart #

There is a pleasing symmetry with an earlier rung of the identify ladder that is worth drawing out.

Two projections that cannot be told apart asks how large a region has to be before two different projections become distinguishable from their drawings. The answer is that over a small enough region they are not, because the difference between them is below the noise of the reading.

This rung is the same question with the roles exchanged. Two implementations of one projection cannot be told apart from their drawings, at any extent, because they agree at every knot and differ between them by less than a line width. What separates them is not a bigger region but a different instrument — a derivative rather than a position.

So the pair of results says something general about what a drawing carries. A map’s positions are a weak constraint: they cannot distinguish two projections over a small region, and they cannot distinguish two interpolations over any region. Everything that separates maps sharply is in the derivatives, and the derivatives are exactly what a picture does not print.

Where the ladder goes next #

Nine rungs have asked what a family is: a construction, a condition, a symmetry, and now a definition that is none of those. The unasked question is whether a projection needs a definition of any of these kinds at all — a map is a pair of functions, and a family is a set of them, and nothing requires the set to be describable.