A map that cannot be read backwards

Every projection so far on this site is a function with an inverse. That is not stated as a requirement anywhere, because it has never needed stating: a map is something a reader locates a place on, and locating is the inverse operation.

There is a condition whose solution has no inverse, and it has been in print since 1909.

The condition, and why it is not the one people assume #

Azimuthal on this site has meant a projection with a family — the one built on a plane touching the sphere — and an azimuthal projection preserves the direction from its centre to everywhere else. Read a bearing off the centre of an azimuthal equidistant map and it is right.

That is the wrong way round for the question this projection answers. Somebody in Jakarta wanting the direction to Mecca needs the bearing from Jakarta, measured at Jakarta, and an azimuthal map centred on Mecca gives the bearing at Mecca — which differs, sometimes by tens of degrees, because the two ends of a great circle have different bearings. The site’s own great-circle vertex essay is about exactly that difference.

So the condition here is retroazimuthal: at every point, the direction to one named centre is readable at that point. It is a different sentence from the azimuthal one and it has a different answer.

James Ireland Craig, working in the Egyptian survey, published a solution in 1909. It keeps longitude as the horizontal coordinate and solves for the vertical one, so the map is a distortion of a plate carrée in the vertical direction alone.

The condition holds exactly #

The check is worth doing rather than trusting, because it is easy to write a formula that nearly works.

The bearing drawn on the map is the direction of the straight line from the plotted point to the plotted centre, computed from the two plane positions. The bearing on the sphere is the initial azimuth of the great circle, computed by a spherical formula that shares no code with the projection. Over 676 points spanning 200° of longitude and 110° of latitude, the worst disagreement between the two is 5.7 × 10⁻¹⁴ degrees, which is the arithmetic.

That is the whole of what the map is for, and it is exact.

And the map has no inverse #

Craig’s vertical coordinate along a fixed meridian offset is $C\sin(\varphi - \delta)$ for a $\delta$ that depends on the offset. A sine turns over. If the turning point falls inside the mapped band of latitudes, every height on the inner side of it is reached twice, by two different latitudes.

Sampled at 121 longitudes across the map, every single one carries a fold. The projection is not injective anywhere along its southern half, so there is no function from the map back to the sphere: given a point on the paper, there is no way to say which of two places it is.

The test is the site’s usual shape, which is to make it reject. The same fold-detector run over Mercator, Mollweide, the equirectangular and the sinusoidal reports zero folded meridians on all four — those maps are monotone in latitude along every meridian, which is what having an inverse looks like when it is measured rather than assumed.

How far apart the two places are #

The pair of latitudes sharing an image is not a fixed pair: it depends on the level chosen and on the meridian. Taking, at each longitude, the level halfway between the turning value and the value at the edge of the band gives a representative pair, and the separations are not small:

The same construction gives a pair at every longitude once the level is chosen on the branch both sides reach: 32.6° apart at 95° west, 59.6° at 150° east. Nine thousand kilometres is not an ambiguity anybody could resolve from context, and neither is three. The pairs are closest together near the centre’s own meridian and open out either side of it, which is the geometry of the fold rather than a property of the arithmetic.

Where the fold line runs, and how much ground is under it #

The table of colliding pairs carries more than it says. Craig’s vertical coordinate is a sinusoid in latitude along every meridian, so a pair drawn at the same height is symmetric about the sinusoid’s own turning point — which means the midpoint of each pair is the turning latitude on that meridian, and the table is a sampling of the fold line without having been presented as one.

Reading the five rows that way, with each meridian’s offset from the centre at 39.826° east:

Two things fall out immediately. The fold is deepest on the centre’s own meridian, at 68.6° south, and rises towards both edges — and the two rows sixty degrees either side of the centre give 52.1° and 51.8°, agreeing to a third of a degree. That near-equality is the symmetry the construction has to have and is a check on the five numbers rather than a finding, but it is the sort of check that would have caught a sign error in the offset.

It also explains the pattern the essay reports about the pairs. Where the fold is deep the band south of it is narrow, so the two latitudes sharing an image are close together — 3,303 kilometres on the Greenwich meridian. Where the fold is shallow the band is wide and the pair straddles the equator, at 9,174 kilometres. The pairs are not variably confused; they are uniformly confused about a band whose depth varies, and the depth is a smooth function of the offset from the centre.

The doubled region can then be estimated rather than left as owed. Everything south of the fold line is drawn on top of something north of it, so the area at issue is the spherical area under that curve. Integrating the five sampled turning latitudes across the 160 degrees of longitude they span, and using the symmetry to close the interval, gives about 0.52 steradians — roughly four per cent of the sphere, and a larger fraction of the band the map actually draws.

That is an estimate from five meridians and it is quoted as one. What it settles is the order of magnitude, which is the part the earlier claim needed: the overlap is not a sliver along a line and it is not most of the map. It is a cap of a few per cent, centred under the antipode of the map’s centre and pulled towards it.

Why a century of users did not mind #

The four per cent is also the answer to a question the essay’s account of the map’s uses leaves open: how a projection with no inverse stayed in print and in use since 1909.

The fold on the Mecca-centred map sits between 50° and 69° south across the whole span of longitudes its users occupy. What is under it is the Southern Ocean and Antarctica. Not one place with a qibla to establish falls inside the doubled region, and the same is true of the radio use, since directional antennas are pointed from transmitter sites rather than from the Weddell Sea.

So the defect is real, exact, and located where the map has no readers. That is worth saying plainly because it is a different thing from the defect being small: it is not small, it is 3,300 to 9,200 kilometres of ambiguity over a few per cent of the world, and it is irrelevant because of where the centre happens to be.

The London-centred and New York-centred versions in the figures above are the test of that reading, and they fail it. Moving the centre moves the fold — the London map’s colliding pairs span 86 degrees of latitude — and there is no reason for the doubled cap of an arbitrary centre to land on empty ocean. A retroazimuthal chart drawn for a transmitter in the southern hemisphere would put its fold across inhabited ground, and its users would meet the failure the qibla map’s users never have.

A projection’s tolerability is therefore a fact about the centre as much as about the construction, which is one more instance of the thing this field keeps finding: a map’s adequacy is a joint property of the map and of what is being asked of it, and neither half can be assessed alone. Craig’s map is unreadable backwards over four per cent of the world, and the four per cent is chosen by a parameter that was set for an entirely different reason.

Why the fold is not a defect of Craig’s solution #

The natural next thought is that a cleverer construction would satisfy the condition without folding. It would not, and the reason is a counting argument of the kind the previous rung turned on.

The retroazimuthal condition fixes, at every point, the direction from that point to the centre — one equation per point, on a map with two coordinates per point. That leaves one function’s worth of freedom, which Craig spends by keeping longitude as the horizontal coordinate. Any other choice spends it differently and none of them can change the fact that the condition is about directions to a single point, which is a family of curves that converge.

The convergence is the fold. Two places on opposite sides of the centre’s antipode have nearly opposite bearings to the centre, so the condition drags their images through each other, and the same happens for any solution of the same condition. Hammer’s retroazimuthal projection, published in 1910, folds too, in a different place.

There is a shorter way to say it: a map that shows every direction to one point is a map of a family of curves through that point, and a family of curves through a point cannot be flattened without something running over something else. That is the same statement of impossibility the rest of this site makes about area and angle, arriving in the least expected of the three.

What the first-order machinery says #

Since the map exists and is differentiable, Tissot’s apparatus applies, and it says the map is very bad — which is neither surprising nor the point.

That last sentence is worth keeping. The whole first-order description of a map — the indicatrix at every point — is compatible with the map being one-to-one and with it not being. Injectivity is not a derivative of anything.

The same construction round a different centre #

There is nothing Mecca-specific in the mathematics; the centre is a parameter.

The invariance is the argument. A property that survives moving the centre through 40° of latitude and 40° of longitude is a property of the condition and not of a particular solution’s accidents, which is the reason to compute a second one rather than to reason about the first.

What the second derivative says #

The flexion machinery applies to Craig’s map wherever it is differentiable, and it returns the largest numbers anywhere in this collection.

There is a consistency in that worth naming. The fold, which is a global failure of injectivity, shows up locally as an enormous second derivative in the neighbourhood where it happens. The first-order description does report something at the fold — the Jacobian is singular exactly on the turning line, so the two ways a map is wrong has a degenerate indicatrix to show — but a singular Jacobian is a local statement about one line, and the doubling it produces is a claim about two regions on opposite sides of the sphere. Nothing local distinguishes a map that folds from a map that merely has a fold line and stops there.

Where it is used, and by whom #

Retroazimuthal maps are working documents rather than curiosities, and the reason is that a compass course to a fixed place is a genuine daily requirement for two groups.

The larger is anybody establishing the qibla — the direction of prayer toward Mecca — which is the use Craig built for, and which is why his map is often called the Mecca projection. The other is radio: pointing a directional antenna at a fixed transmitter is the same question with a different fixed place, and retroazimuthal charts have been drawn centred on broadcast sites.

Both uses share the property that makes the fold tolerable. Neither of them ever reads a position off the map. The user finds their own place — which they already know — lays a straight edge to the centre, and reads an angle. The information flows the opposite way from an ordinary map, and the projection is built for that direction only.

That is the same argument the operation decides the coordinate system makes about data in a machine, arriving here in an atlas: what the map is going to be asked determines what it must preserve, and there is no way to know whether a projection is adequate without knowing the operation.

What this ladder has established #

Three rungs, three conditions, and the pattern the count predicts each time:

The last row is the one that does not follow from the arithmetic alone, and it is the finding of this essay: satisfiability and readability are independent. A condition can be satisfiable exactly and still leave an object that is not a map in the sense the word usually carries.

The general lesson for the choosing field is the one it was built to make. Every projection minimises something argues that naming the objective is more honest than naming the construction; a conditioned projection is the case where the objective is the whole definition, and where naming it is unavoidable. Nobody has ever called Craig’s projection a good map, because there is no way to describe it without saying what it is for.

What a one-way map is still good for #

Non-invertibility sounds disqualifying and is not, because the operation this map exists for runs in one direction only. A reader puts a finger on their own place and reads a bearing off the graticule; nothing in that asks the map which place a given point of paper stands for. The inverse is what a machine needs — to digitise a feature off the sheet, to georeference a scan, to answer a click — and none of those was available when the construction was published or wanted by the people who used it.

So the fold is a defect relative to a use the map was never put to, which is worth separating from a defect in the construction. What it does mean is that the map cannot be treated as a coordinate system: two places share a point, so a point is not a location, and every rule this collection states about a coordinate needing its system applies here in the strongest possible form.

Where the model stops #

The condition is about initial bearings on a sphere. On the ellipsoid the initial azimuth of a geodesic differs from the spherical one by up to a few hundredths of a degree at continental distances — small, and larger than the 6 × 10⁻¹⁴ this construction achieves against its own model. A qibla computed to the arcsecond needs the ellipsoidal azimuth and this map does not provide it, which is a limit of the model rather than of the drawing.

The fold is located here only in latitude. The full description of where the map overlaps itself is a region in two dimensions, and what is measured above is the turning latitude along each meridian, which is its boundary. Mapping the overlapping region as a region is work this collection has not done.

The projection is not in the site’s main library, and that is deliberate rather than an omission. PROJECTIONS holds maps that are asked for their distortion and their properties by every gate the site runs, and several of those questions have no answer for a map that folds — where the worst point is would report the fold line and nothing else. Conditioned projections are constructed on demand instead, which is the same treatment the aspect is a free choice gives to a rotated projection and for the same reason: they are parameterised objects rather than named ones.

Nothing here says the fold is the only failure. Injectivity was tested by looking for a turning point along each meridian, which finds folds of the kind this projection has. A map could fail to be one-to-one in a way that no meridian scan reveals — two distant regions overlapping without either folding — and detecting that needs a different test.

Where the ladder goes next #

The anchor now holds a construction with a unique solution, one with no solution, and one whose solution is not invertible. What it does not hold is a conditioned projection whose condition is about area or angle rather than distance or direction — the areal factor must be exactly one along these three parallels, say — where the counting argument is the same and the analysis is a boundary-value problem rather than a circle intersection.

That is a solver this site does not have, and the honest place to say so is here rather than in a plan.