Bending and stretching are one failure

A projection can bend a route and it can change its scale along the route, and the first rung of this ladder computed both. The obvious next question is whether they are two failures or one, and the obvious answer is that they are two: bending is a turn and stretching is a scale, they have different units in every ordinary sense, and a map ought to be free to do either without the other.

That answer is wrong for a quarter of the library, and the reason it is wrong is a two-line proof that the measurement found first.

The measurement that arrived first #

The rose is a polar plot of a signed quantity against the bearing of the geodesic being followed. Reading the top three: the solid curve’s maximum falls exactly where the light curve crosses zero, and the two curves reach the same value.

Written as numbers, across the library at 20° east, 40° north:

Four decimal places of agreement, on projections whose forward formulae have nothing in common — one exponential, one a rational function of a half-angle, one a power of a colatitude — is not a coincidence and is not the differencing being kind. It is a theorem, and it was not looked for.

The Lambert conformal conic’s pair is small because the test point sits near one of its standard parallels, where the scale is momentarily stationary and its gradient is nearly zero. Small and equal is still equal: the coupling is a statement about the ratio, and the ratio is one wherever the projection is conformal, including where there is almost nothing to couple.

The proof, which is two lines once the pattern has been seen #

On a conformal projection the local scale factor $\sigma$ is a function of position alone. That is exactly what conformal means: there is no direction in it, because a projection that scaled differently in different directions would change the angle between two curves crossing at the point.

So the only second-order object available at that point is the gradient $\nabla\ln\sigma$ — one vector, with a length and a direction. Now walk a geodesic in direction $\hat t$:

• the scale changes at the rate $\nabla\ln\sigma\cdot\hat t$, and that is the skewness;
• the image turns at the rate $\nabla\ln\sigma\cdot\hat n$, up to a sign, and that is the flexion.

Two components of one vector. Their extremes are equal, because both equal the vector’s length; they occur ninety degrees apart, because a component along and a component across are extreme at right angles; and where one is largest the other is zero.

The sharper way to say it is the one worth carrying: conformality removes the second-order freedom as well as the first-order one. A general projection has two independent second-order numbers at each point. A conformal projection has one.

The direction in which a ruled line is honest #

The coupling has a consequence a reader can use, and it is not the one intuition offers.

On a conformal map there is exactly one direction at each point along which a geodesic is drawn without bending at all, and it is the direction in which the scale is changing fastest. Where the map is stretching hardest along the route, it is not bending the route.

On Mercator that direction is north–south, everywhere. The gradient of $-\ln\cos\varphi$ points north, so the zero of the flexion curve is at bearing zero, and the prediction is that a meridian — which is a great circle — is drawn perfectly straight.

That is a familiar fact arriving from an unfamiliar direction. Anybody who has used a Mercator chart knows the meridians are straight; what the second-order machinery adds is that this is the only honest direction, that it is a zero of a curve which is nowhere else zero, and that the same rule identifies the honest direction on any conformal projection, including ones nobody has a picture of.

The same rule applied to the other two conformal members of the library returns two facts that are already known, which is the strongest kind of check a new instrument can pass.

On a stereographic projection the scale depends only on the distance from the centre, so the gradient of its logarithm points radially, and the flexion’s zero is along the radius. The prediction is that great circles through the projection’s centre are drawn straight — which is exactly what a stereographic projection does and is one of its defining properties.

On a conformal conic the scale depends only on latitude, so the gradient points north and the honest direction is the meridian. Conic meridians are straight rays from the apex, which is again a first-order fact about the family.

Three conformal projections, three predictions, three properties anybody could have looked up. That is the point of running the test on them: a second-order instrument that rediscovered a fact its user already had is an instrument that can be trusted on a projection where nobody has the fact — which is what the solved conformal maps of the condition ladder are, and what a numerically constructed map of an irregular body is.

Half the coupling is cheap, and the half that is cheap is the one that looks impressive #

The right angle is the striking part of the table, and it is the part that proves least.

The reason is symmetry rather than conformality. An azimuthal projection is rotationally symmetric about its centre, so at any point the radial and tangential directions are principal directions of everything the map does, and any extreme of any directional quantity has to occur along one of them. Two perpendicular directions, and hence a right angle, for free.

So the test the assertion actually applies has two clauses and requires both: the ratio must be one and the angle must be ninety degrees. Applied to the library, the conformal members pass both and every other member fails at least one:

The orthographic row is worth a second look. Its ratio is below one, so it bends less than it stretches, which no conformal projection can do — for a conformal projection the ratio is one by the theorem, so a ratio away from one in either direction is a refutation of conformality. This is the sort of statement the site is built to make: a property that has been checked by four partial derivatives can also be checked, independently, by two second derivatives, and the two checks agree.

The azimuthal equidistant is the other instructive row. Its skewness is fifteen times smaller than its flexion because both of its principal scales are constant along their own directions — the radial scale is one everywhere by construction, and the tangential scale varies radially rather than tangentially. It is the least-skewed projection in the library over a cap, at 0.044, and it pays in bending. That is the projection the route with no shortest path and a scale bar is right in one place both lean on for its exact radial scale, and the exactness that makes its scale bar honest along one family of lines is precisely what drives its skewness to nothing.

What the same numbers say about a whole map, which is a different question #

Measured across a whole map instead of at one point, flexion and skewness look strongly related on almost every projection:

Correlations from 0.89 to 1.00 look like a decisive answer to the essay’s opening question, and they are an answer to a different question. Both quantities grow away from wherever a projection is well behaved, so a scatter of points across a map mostly measures distance from the good part — and any two badness measures on the same map will correlate for that reason alone.

The relation this essay is about is at a point, over directions, and it is the one that has a proof. The distinction between a relation at a point and a relation across a region is the same one distortion over a region draws for the first-order quantities, and it is worth keeping sharp: a map’s worst point, its average and its structure at a point are three different objects, and only the last of them can carry a theorem. The correlation across a map is a description of where each projection’s errors live, which the field pictures already show:

Mercator’s exact 1.000 in that table is the coupling again, showing up as a correlation: on a conformal map the two fields are the same field, so measuring both over the sphere produces two copies of one number. That is a consistency check on the correlation code rather than a finding, and it is the reason the row is there.

Why the essay this replaced was wrong #

This ladder was laid out with a different second rung in mind: bending and lopsidedness are different failures, with the correlation across a map as the measurement that would show it. Both halves of that plan turned out to be errors, in opposite directions.

The correlation across a map is high, not low, so the planned measurement would have argued the opposite of the planned claim. And the relation at a point is not independence but an exact coupling, which is a stronger and more useful result than either. The site’s rule for this situation is to publish what was measured and to say what was expected, because a plan that survives contact with the arithmetic unchanged is usually a plan that was not tested by it.

The finding that survives is more specific than the one that was sought: the two second-order failures are independent on a general projection and are one failure on a conformal one.

The departure from the coupling is not a measure of anything first order #

There is a tempting further step — that a projection’s distance from the coupling grades how far from conformal it is — and the essay’s own tables refuse it.

Set the two columns side by side:

The projection with the smallest angular deformation in the list has by far the largest departure from the coupling, and the projection with the largest ω has a ratio smaller than two of the maps that beat it on ω. There is no monotone relation, and the two extreme rows are inverted.

Nor could there be one, which is the reason this is a correction rather than a surprise. Angular deformation is a first-order quantity — a statement about the Jacobian — and the ratio of the two second-order extremes is a statement about the second derivatives. The whole thesis of this ladder is that the second derivative is not predicted by the first: the first rung exists because Tissot’s construction stops one derivative short, and a graded measure of conformality read off second-order data would have contradicted it.

So what the coupling supplies is a test rather than a measure, and it is one-sided.

Failing it refutes conformality, exactly. A ratio away from one, or an angle away from ninety degrees, proves the map is not conformal at that point, by the contrapositive of the two-line proof. That is a genuine and unusual instrument: a property normally checked with four partial derivatives, checked instead with two second derivatives and no angles at all — and the orthographic’s ratio of 0.720 is a refutation just as much as the azimuthal equidistant’s 15.634 is.

Passing it does not establish conformality. The proof runs one way. A non-conformal map could have its two second-order extremes equal and perpendicular at one particular point by coincidence, and nothing in the second-order data would say so; establishing conformality needs the first-order test, at the point, as it always did.

That asymmetry is the ordinary shape of a necessary condition, and it is worth stating because the table invites the opposite reading. Three conformal projections passing both clauses looks like a characterisation. It is three instances of a theorem, plus a list of refutations, and the refutations are the part that does work — a projection whose conformality is being claimed can be disproved by this test in two evaluations, and a projection whose conformality is being sought cannot be found by it at all.

What the first-order picture would have said #

Every projection in the panel figure has been drawn on this site with indicatrices scattered across it, and none of those pictures contains any of this.

There is no reproach in that. An indicatrix map is a first-order picture and is honest about being one. What the ladder adds is that a second-order picture exists, is computable from the same forward map, and says things the first-order picture cannot — including a test for conformality that uses no angles at all.

Where the model stops #

The coupling is proved for a conformal projection of a surface with a metric, at a point where the projection is twice differentiable. Two of those conditions do real work.

A metric is needed because both quantities are rates per radian of arc on the sphere, so the sphere’s own distances are in the definition. Substituting the ellipsoid changes both by of order the flattening, which is a third of a per cent and is below the differencing’s own residual — the same conclusion datum shifts dwarf projection errors reaches about first-order quantities, arriving here for the same reason.

Twice differentiable excludes the Robinson projection, whose defining table is interpolated linearly, and whose second derivative therefore does not exist at a table entry. Its correlation of 0.997 in the table above is computed at points that mostly miss the entries; its mean flexion of 6.98 against Mollweide’s 1.38 is not a statement that Robinson bends five times harder, it is the interpolation’s kinks being differenced. The indicatrix is a limit met the same object one derivative earlier and reached the same conclusion, which is that a table is not a function until somebody says how it is interpolated.

Conformal excludes the projections that are conformal along a line rather than at a point — a conic with two standard parallels is conformal nowhere, and its ratio at the test point is 1.741 rather than one. The coupling is pointwise or it is nothing.

Who found it, and when #

The second-order pair is Goldberg and Gott’s, from 2007. The coupling on conformal maps follows immediately from the Cauchy–Riemann structure that has been the standard way of writing a conformal projection since Lambert and Lagrange in the 1770s — a conformal map of the sphere is a holomorphic function of the isometric coordinate, and holomorphy is exactly the statement that only one complex derivative exists where a general map has two real ones.

That is the same fact the average of two projections uses in a different guise: because conformal maps are holomorphic functions of one coordinate and holomorphic functions form a vector space, the average of two conformal projections is conformal. Here the same structure removes a degree of freedom one derivative up.

Neither of those is a new theorem. What is new here is that both are measured on this site’s own library rather than cited — the vector-space property to two parts in a million, the coupling to four figures — and that the measurement is what raised the question.

One consequence of the coupling is worth stating as an instruction rather than as a result. On a conformal projection the two second-order quantities carry no information about each other, so measuring both is measuring one thing twice — and a score that adds them is therefore weighting conformal maps differently from every other map for a reason nobody chose. Any ranking built on the pair has to say whether it treats the coupled case as one number or as two, and none of the published ones does.

It also decides what a fair comparison across projections has to hold fixed. Two maps compared on a pair of quantities are being compared on two numbers on one of them and on one number twice on the other, and the difference is not a matter of degree — it is the difference between a plane and a line in the space the scores live in. A ranking that does not say which it is measuring has averaged two different measurements and reported the result as one.

Where the ladder goes next #

Two quantities are now available for any projection, over any region, computed from the projection’s own forward map. The obvious use of a pair of numbers is a ranking, and this site’s habit with a ranking is to distrust it: which projection is best refuses the question until an objective is named, and distortion over a region shows the answer changing when the region changes.

The third rung asks whether it also changes when the derivative changes, and it does. Ranking eight standard world maps by their second-order failure rather than their first-order one moves the Mercator projection up three places.