The equator is not a circle either

This ladder’s whole subject is what the sphere costs. The Earth is a sphere, and when it is not opens it; geodetic against geocentric latitude prices the eleven and a half arcminutes that follow from replacing one with the other; the flattening is not a free parameter shows the shape is determined by the rotation rather than chosen; and eight more rungs work out the consequences.

Every one of them replaces the sphere with a surface of revolution. An ellipsoid of revolution has one equatorial radius, one polar radius, an equator that is a circle, and a meridian section that is the same ellipse whichever meridian is taken. That last property is what makes an auxiliary latitude possible, what makes the transverse Mercator series a function of one variable, and what makes almost every formula on this site short enough to write down.

The Earth’s equator is not a circle.

The three axes, stated #

The shape model is an input, as it is for every other body this collection maps. Vesta’s three semi-axes are stated because the IAU publishes them; the Earth’s are stated here for the same reason and with the same status:

The seventy metres between the two equatorial radii is the shape expression of the degree-two, order-two term of the gravity field, and satellite geodesy has measured it since the early 1960s. The equatorial flattening is about one part in ninety-one thousand, against the polar flattening’s one part in 298 — a factor of three hundred smaller, which is why nobody uses it and why it is worth asking what “nobody uses it” costs.

Everything below is computed from those three numbers.

Thirty-five metres, on the equator, four times per turn #

Thirty-five metres is not small by the standards of this ladder. It is more than half the vertical part of the datum shift between OSGB36 and WGS84; it is nearly ten times the residual a seven-parameter fit leaves over a country; and it is very much larger than any projection error this site has measured, all of which are millimetres to centimetres over the regions they are used on.

It is also, and this is the whole of the answer, smaller than the geoid.

What it would do to a latitude #

A separation in height is not the quantity that matters most. A geodetic latitude is defined by the direction of the reference surface’s normal, so changing the surface tilts the normal, and a tilted normal is a moved latitude.

That is the same kind of quantity as the deflection of the vertical — the angle between the plumb line and the ellipsoid’s normal — and it is about a fifth of its typical size. A geodesist who adopted a triaxial reference surface would be trading one two-arcsecond correction for another, and the two are not independent.

The auxiliary latitudes stop existing #

The consequence that matters for a projection is not the size of anything. It is that a whole class of formula loses its meaning.

Every one of the six auxiliary latitudes — conformal, authalic, rectifying, parametric, geocentric, isometric — is an integral along the meridian. Walk from the equator to a parallel, accumulate something, invert. On a body with three unequal axes every meridian is a different ellipse, so there is no the, and the quantity each latitude is defined by becomes a function of longitude as well.

A map of a body with three axes reaches the same conclusion for Vesta and Phobos and puts it starkly: there is no the, and so there is no auxiliary latitude to write a formula in. That essay measures the wobble in an exactly equal-area map’s top edge at 1.58 per cent for Vesta and 7.3 for Phobos, against 3 × 10⁻¹³ per cent for the Earth.

The three-in-ten-to-the-thirteenth is the rotational Earth. On the triaxial one the corresponding wobble is not zero, and this rung is the measurement of it: ten metres out of ten million, which is one part in a million and is a wobble rather than a straight line. So the Earth is on the same list as Vesta and Phobos and is at the far end of it.

Why ignoring it is right, and not for the reason it looks #

The temptation is to say seventy metres is small and stop. It is not small — this ladder has spent eleven rungs on effects a tenth of the size — and the real reason is better.

Any reference ellipsoid is wrong by the geoid. WGS84’s surface is not sea level and does not claim to be: the geoid departs from it by up to about a hundred metres, and every serious computation that needs a height carries a geoid model to correct for exactly that. Adopting a triaxial reference would remove seventy metres of that departure and leave the other hundred and twenty, at the cost of making every formula on this site a function of two variables instead of one.

And the seventy metres it would remove is already in the geoid model. The equatorial ellipticity is the degree-two, order-two harmonic of the gravity field; a spherical-harmonic geoid contains that term along with thousands of others, and a reader who has a geoid has already been handed the correction. Building it into the reference surface instead would move one term from one model to another and change no answer at all.

That is the argument, and it is a different shape from the argument this ladder usually makes. Everywhere else the finding is this assumption costs more than it looks. Here it is: this assumption costs about what it looks, and the reason it is safe is that a better model already contains it, not that it is negligible.

What a two-variable formula would cost #

The reason the triaxial term is not adopted is stated above as an inequality against the geoid. There is a second reason and it is arithmetic rather than physics.

Every closed form on this site that involves the ellipsoid is a function of latitude alone. The meridian arc is a series in latitude; the six auxiliary latitudes are functions of latitude; the transverse Mercator’s coefficients are functions of the flattening and its argument is a latitude; the point scale factor of a conformal projection is a function of latitude. That is not a convenience, it is a consequence of the symmetry: a surface of revolution has one meridian, so anything computed along a meridian is computed once.

On a triaxial surface each of those becomes a function of two variables, and the cost is not doubling — it is that the series stop being series. A meridian arc in latitude has a five-term expansion in the third flattening that is good to a millimetre; the corresponding object on a triaxial body has no such expansion, because the integrand’s dependence on longitude is not small in the same way. Where the series stops being the map is about the limit of one such expansion; the triaxial version has no expansion to have a limit.

So the practical position is not that the seventy metres is too small to bother with. It is that correcting for it costs the closed forms, and the closed forms are what makes a geodetic library a library rather than a numerical integrator.

The one place it is not ignorable #

There is a case where seventy metres of equatorial shape matters and it is not on the ground.

A satellite in a low orbit feels the degree-two, order-two term directly, and the resulting perturbation is what made the measurement possible in the first place. It also has a consequence anybody who has read about geostationary orbits will recognise: a satellite parked over the equator drifts towards one of the two minima of the equatorial potential, and has to be nudged back. The stationkeeping budget for that drift is a direct expenditure of fuel against the seventy metres in this essay.

So the same term is negligible for a map and decisive for an orbit, and the difference is what each of them is a function of. A map wants the shape of the surface a coordinate refers to, where the term is one part in ninety-one thousand of the radius. An orbit wants the gradient of the potential, where the same term produces a tangential acceleration with nothing to compete against — because the round part of the field pulls straight down and contributes nothing along track.

That is the pattern purpose before property states as a rule, arriving in the one place on this site where the object being described is not a map at all.

The same question, on other bodies #

The Earth is the least triaxial body this collection maps, and the sequence is worth seeing in one place.

Ordered by how far from a surface of revolution each body is, the Earth sits at the bottom of a list that runs to Phobos, where the departure is 7.3 per cent of the map’s own height. What changes along that list is not whether the question is worth asking but what the answer costs: on Phobos the closed forms are gone and the maps are solved on a mesh; on Mars they survive with a stated error; on Earth they survive with an error nobody has to state.

And the Earth’s own place on that list moved once. Before satellite geodesy the question was open, several eighteenth- and nineteenth-century figures of the Earth were published as triaxial, and the arc measurements available could not resolve the answer. It was not settled by a better argument; it was settled by an orbit, and the orbit measured the gravity field rather than the shape.

Where the model stops #

The three axes are an input and the essay is only as good as they are. The seventy metres and the 14.9° west are the values reported for the Earth’s degree-two, order-two term expressed as a shape; different reductions give figures a few metres apart and a degree or two of longitude apart. Nothing here would change qualitatively for any of them, and no number here should be quoted to more figures than the input carries.

The comparison with the geoid uses a stated range rather than a model. About 190 metres from the lowest to the highest point, which is the same kind of stated parameter this site uses when it needs a geoid separation and declines to import one. What is computed is the triaxial term; what is quoted is the thing it is compared against, and the conclusion depends on the comparison only through an inequality that a factor of two either way would not disturb.

The triaxial surface here is an exact ellipsoid with three unequal axes. The real Earth is not that either — it is a geoid, which is not any smooth algebraic surface — so this is one more model in a sequence of models, each closer than the last and none of them the thing. The sequence is the point: sphere, ellipsoid of revolution, triaxial ellipsoid, geoid, and each step is worth taking only if something downstream uses it.

And nothing here is about gravity. The equatorial ellipticity is inferred from the gravity field and is expressed here as a shape, which is a substitution that only works because the two are related by a model. A reader who wants the mass distribution rather than the surface should read the plumb line’s own ladder, which does the arithmetic the other way round.

Who found it, and when #

The suspicion is old and predates any means of testing it. Newton’s Principia argues for polar flattening from rotation, and the possibility of a triaxial figure was debated through the eighteenth and nineteenth centuries on the basis of arc measurements that could not possibly have resolved seventy metres. Several nineteenth-century reference ellipsoids were published as triaxial — Clarke considered one, and the debate over whether the equator was measurably elliptical ran for decades on data that could not settle it.

Satellite geodesy settled it in about 1961, from the way an artificial satellite’s orbit precesses under a non-axisymmetric field, and the answer was that the effect is real and about seventy metres. What happened next is the interesting part: nothing. No reference ellipsoid adopted since has been triaxial, and the reason given was always that the gain does not justify the complication — which is the argument above, made without the comparison being written down.

Why a measured refinement is not adopted #

Nothing happened is the finding worth generalising, because the reasoning that produced it is not the reasoning it appears to be.

The obvious explanation is that the effect is too small, and it is wrong. Seventy metres is not small. It is far larger than the accuracy of the coordinates every national datum publishes, larger than the geoid corrections everybody does apply, and vastly larger than the sub-millimetre precision of a modern baseline. By the usual test — is the correction bigger than the error of the things it corrects — the triaxial term passes easily.

The real objection is analytic rather than numerical. An axisymmetric ellipsoid lets every quantity in geodesy be a function of latitude alone: the meridian arc, the radii of curvature, the conformal, authalic and rectifying latitudes, the series that compute the transverse Mercator. Break the symmetry and each of those becomes a function of two variables, and the auxiliary latitudes stop existing as objects — there is no single conformal latitude, because the conformality condition now depends on the longitude as well.

So the cost is not arithmetic, it is the loss of a whole apparatus. Every closed form in the subject is built on the symmetry, and adopting a triaxial reference surface does not perturb them; it invalidates them and replaces them with two-dimensional numerical solutions.

Which is why the decision was right and why it may not stay right. The apparatus was load-bearing when a projection had to be evaluated by hand from a table; the argument for keeping it is much weaker when every coordinate transformation in the world is performed by a library on a machine that would not notice a two-variable quadrature. What blocks adoption now is not the mathematics but the standards, the file formats, the EPSG registry and the sixty years of data written against them — which is a real obstacle and a different one.

It also explains why the debate died rather than concluding. Nobody published an argument that the equator is round enough; the measurement arrived, the apparatus stayed, and the question stopped being asked because no decision had to be taken.

The general rule is worth carrying out of this essay. A refinement is adopted when it is larger than the errors it corrects and cheap in the framework it lands in. The second condition is the one that decides most cases, it is rarely stated, and it changes over time in a way the first does not.

Where the ladder goes next #

The ladder has now examined the sphere assumption, the flattening, the series, the inverse, and the last unexamined symmetry. What is left is the assumption underneath all of them, which is that the reference surface is a smooth algebraic surface at all.

It is not. Every practical computation uses an ellipsoid for the geometry and a geoid for the physics, and the two are different kinds of object joined by a model. Where the join is, what it costs, and which questions have to be asked of which surface is the thing this ladder has been circling for eleven rungs and has never asked directly.