The same projection on a different body

This site opened with a measurement: Web Mercator is not conformal, because it applies spherical Mercator formulae to ellipsoidal latitudes, and the maximum angular deformation that produces is 0.3848° against a noise floor of 1.6 × 10⁻⁶° for the genuinely conformal projections.

That number has been quoted in a dozen essays since as though it were a property of Web Mercator. It is not. It is a property of a flattening.

Distortion cannot depend on the size, and the check says so #

A projection’s distortion quantities are ratios. The principal scale factors are the singular values of a Jacobian that maps ground displacements to page displacements, the areal factor is their product, and the angular deformation is a function of their ratio. Multiplying the body by a constant multiplies every length in the problem, and a ratio of lengths does not notice.

That is dimensional analysis and it is worth ten seconds of thought and one line of code, because a claim that is obviously true is exactly the sort that is worth testing on the machinery that is going to be trusted with the non-obvious ones. Given a body ten times the size of Mars with Mars’s flattening, the measured worst angular deformation agrees with Mars’s to twelve decimal places, which is not a check on geometry but on the code having no radius in it anywhere.

So a map of Mars and a map of the Earth are the same problem with one number changed, and the number is the flattening.

The law, and the size of its failure #

Measured across the bodies whose flattenings span two orders of magnitude:

The Moon and Venus have flattening zero by convention and produce 1.2 × 10⁻⁶° — the noise floor of the differencing, which is what a projection that really is conformal measures.

The error is 2f radians, to a third of a per cent for the three nearly-spherical bodies, and the departures for the two giants are as necessary as the agreements: a first-order law that held at f = 1/10 would be a law with no first order in it. Both halves are asserted, so the check would fail if the site’s arithmetic ever started agreeing with the prediction where it should not.

That reframes the site’s opening number. 0.3848° is not a fact about a web mapping convention; it is what 2f looks like when f is the Earth’s, and any body’s version can be predicted before it is computed.

What the mistake looks like on Mars #

The pattern is identical to the terrestrial one and is 1.76 times as large everywhere, which is the ratio of the two flattenings to within half a per cent.

Whether that matters is the usual question and has the usual answer: it depends on the tolerance. Mars is mapped at resolutions where 0.68° of angular deformation is a real shear — an orbital image resampled onto a spherical-formula Mercator grid has features rotated relative to their neighbours by up to that much — and the correct fix is the same as the terrestrial one, which is to use the ellipsoidal formulae and the conformal latitude.

Why the same law governs every failure of this kind #

The error being 2f is not a coincidence of Mercator’s, and the reason is worth stating because it predicts the size of a whole family of mistakes.

Applying a spherical formula to an ellipsoidal latitude is, in effect, substituting the geodetic latitude where the conformal latitude belongs. Those two differ by f·sin 2φ to first order — the same quantity the previous rung measures between the geodetic and geocentric latitudes, with a different coefficient — and a projection differentiates its input, so an error of order f in the latitude becomes an error of order f in the local scale ratio, which is an angular deformation of order f radians.

The factor of two comes from the difference being in one direction only: the meridian scale is wrong and the parallel scale is not, so the ratio is wrong by twice the departure. Nothing in that argument mentions Mercator, and the same 2f should govern any spherical formula fed geodetic latitudes.

That is checkable and it was checked expecting a different constant. It is the same constant.

So the rule of thumb has no projection in it at all: substituting a geodetic latitude into a spherical formula costs about 2f, whichever property that formula was supposed to preserve. Equal-area on the wrong body measures the areal version on Earth and finds 0.67 per cent; that is 2f for f = 1/298, and Mars’s is 1.19 per cent, which is 2f for f = 1/170.

The two ways the flattening enters a map #

There is a distinction here that is easy to lose and that the numbers make sharp.

The flattening changes what the coordinates mean. That is the previous rung: two latitude conventions, differing by f radians, and a coordinate that does not say which is ambiguous by twenty kilometres on Mars.

The flattening changes what a projection does. That is this rung: a formula derived for a sphere, applied to an ellipsoid, is wrong by 2f radians of angular deformation regardless of which convention the latitudes are in.

The two are independent failures with the same parameter, and they compound. A Martian image reprojected with the wrong latitude convention and the spherical formulae carries a twenty-kilometre positional shift and a two-thirds of a degree shear, and neither is visible in the output unless something else is available to compare against.

What this says about the site’s own habit #

The site’s founding rule is that no projection is called conformal until the angular deformation has been measured. This rung adds a second-order version of the same discipline: no measured failure should be quoted without knowing what it is a function of.

0.3848° was correct, was measured rather than recalled, and was still being used as though it were a constant. It is a value of 2f. The distinction matters because a constant cannot be predicted for a new case and a function can, and because a number quoted without its parameter invites exactly the reasoning this site exists to refuse — that the error is a property of the software rather than of the geometry.

There are three other numbers on this site with the same character, and they are worth naming as candidates for the same treatment: the meridian-arc series’ truncation error, which the series that computes it gives for WGS84; the UTM scale factor’s balance point, which the scale factor was chosen derives as 1/√k_max; and the Molodensky shortcut’s residual in Molodensky’s shortcut. Each is quoted for WGS84 and each is a function of the flattening.

The general form of the complaint is one this site makes about other people’s numbers all the time — a measurement without its parameter is a measurement that cannot travel — and this rung is where it caught the site making it.

The pattern this makes available #

Once the failure is known to be a function of f, three questions become answerable in advance rather than by experiment.

Is a spherical formula good enough for this body? Compute 2f, compare against the tolerance. For the Moon, whose adopted figure has f = 0, it is exactly good enough by construction. For Mercury at 0.107°, a mapping tolerance of a tenth of a degree is marginal. For the giants it is hopeless.

How much better is the ellipsoidal formula? By the same factor everywhere: the ellipsoidal treatment removes the whole 2f, leaving the series truncation, which is smaller by orders of magnitude on any body a series is used for.

Which bodies need which treatment? A body with f under about 1/500 can be mapped spherically at a tenth of a degree of angular tolerance; between 1/500 and 1/50 needs ellipsoidal formulae; above that needs them and needs care about which latitude is which, because the auxiliary latitudes are then degrees apart.

The one place the analogy to Earth breaks #

Everything above transfers with one constant, and there is a single respect in which planetary cartography is not terrestrial cartography rescaled.

The Earth’s ellipsoid is fitted to the Earth to a few metres, because two centuries of geodesy went into it. A planetary reference ellipsoid is fitted to far sparser data and the body departs from it by a great deal more: Mars’s topography spans about 30 kilometres from the summit of Olympus Mons to the floor of Hellas, against a mean radius of 3,390 — nearly one per cent, where the Earth’s equivalent range is under a third of that relative to its own radius.

So the reference figure is a smaller part of the answer there. On Earth, choosing the wrong ellipsoid is a real error against a well-known surface; on Mars, the ellipsoid is a convention and the interesting variation is in the topography measured against it. That reverses which part of the chain deserves attention, and it is the reason planetary products carry a shape model or a digital elevation model as a first-class object rather than as an afterthought.

None of that changes the 2f law, which is about the reference figure. It changes how much of a real error the reference figure accounts for — the same distinction datum shifts dwarf projection errors draws between two sources of error that are usually discussed as though they were one.

Where the model stops #

Only the flattening is varied. Two bodies with the same flattening and different mass, rotation or composition give identical numbers here, which is correct for the geometry and says nothing about anything else. A body’s actual figure departs from its reference ellipsoid by topography of tens of kilometres — Olympus Mons is 21 kilometres above the Martian datum — and none of that is in these measurements.

2f is the first term. The measured departures at 1/15 and 1/10 are 3.3 and 5.1 per cent, and no attempt is made here to fit the next term. A body with f above about 0.05 needs the full ellipsoidal treatment rather than a corrected first-order rule, which is what the flattening is not a free parameter argues about the terrestrial case from the opposite direction.

The bodies’ radii are cited, not computed. They are the IAU working group’s fitted values; how they were obtained belongs to a different subject and is linked out for rather than restated.

One more consequence: the noise floor is a body property too #

The site’s conformality tolerance is set 60 times above a measured noise floor and 3,850 times below the smallest real failure, and both margins are asserted. That floor — 1.6 × 10⁻⁶ degrees — is the residual of differencing a genuinely conformal projection, and it does not depend on the body at all: it is arithmetic, not geometry.

What does depend on the body is the margin. On Mercury the real failure is 0.107°, which is 67,000 times the floor; on Saturn it is 11.8°, which is seven million times. So the same tolerance discriminates on every body in the list, with room to spare, and would not on a body flattened by less than about one part in a hundred thousand.

That is worth recording because it is the one place where a planetary extension could have broken something quietly. A tolerance calibrated against one body’s failure sizes is a tolerance that might be meaningless on another, and the check that it is not is a comparison of two numbers that had never been put beside each other.

A note on what “the same projection” means #

The phrase in this essay’s title deserves one paragraph of care, because there are two things it could mean and only one of them is what is measured.

The same formula is what is applied here: Mercator’s log-tangent expression, evaluated at the same latitudes and longitudes, on every body. That is the sense in which the comparison is clean — nothing changes between the rows of the table except the metric the result is measured against.

The same map would be something else. A Mercator map of Mars and a Mercator map of the Earth show different things at different scales for different purposes, and comparing them as artefacts is a question about cartography rather than about geometry. Which projection is best is the essay that refuses that comparison without an objective, and nothing here supplies one.

What transfers is the arithmetic, and the arithmetic is what the 2f law is about.

Who found it, and when #

The ellipsoidal Mercator was derived in the eighteenth century and the substitution error this essay measures is as old as the practice of using spherical formulae for convenience, which is older than computers.

What is recent is the scale at which it is committed. Web Mercator was adopted as a rendering convention in 2005 because a spherical formula is cheaper, and it became the coordinate system of nearly every online map — so a deliberate approximation made for speed became the default frame for a generation of data. The planetary versions have the same shape: a mission’s processing pipeline picks a convention for reasons of convenience and the choice outlives its reasons by decades.

The IAU’s reports document that pattern explicitly, which is more than most terrestrial standards do.

A convenience that outlives its reason #

The pattern the essay closes on — a choice made for convenience becoming a frame nobody revisits — is worth stating as a general observation, because the two instances given are separated by four hundred years and are the same thing.

The convenience is always real at the time. A spherical formula is genuinely cheaper to evaluate than an ellipsoidal one, by enough to matter when the evaluation is by hand or on a machine rendering millions of tiles. Nobody who made either choice was being careless; they were trading accuracy for a cost that was binding.

What does not survive is the binding cost. Tables and slide rules gave way to machines, and the machines got fast enough that the ellipsoidal formula’s expense stopped being measurable. The approximation remains, not because it is still worth its price, but because by then it had become the thing everything else was written against.

And the transition is invisible from inside. There is no moment at which a convenience announces that its justification has expired. The formula keeps working, the data keeps being produced in that frame, and the reason it was chosen stops being part of what anybody knows about it.

Which is what makes the IAU’s practice worth singling out. Documenting that a body’s convention was adopted for a stated reason, in a stated report, leaves a record that a later reader can evaluate — and the evaluation is then possible without having to reconstruct the reasoning from the practice. Terrestrial standards mostly do not do this, and the result is a body of conventions whose rationales are folklore.

The practical form of the observation is a question to ask of any inherited choice: what was expensive when this was decided, and is it still expensive. Both instances here fail that question, and neither will be changed for it, which is the honest end of the argument.

Where the ladder goes next #

Everything in these two rungs assumes the body is an ellipsoid of revolution — that there is an axis, that latitude is a function of position, and that the whole terrestrial apparatus transfers with two constants.

For Vesta and Phobos it is false. Their three axes are all different, the surface normal at a fixed planetocentric latitude swings by up to six degrees as one walks round in longitude, and the word latitude has no definition that does not also depend on longitude. That is the last rung.