Four radii of the Earth

Ten rungs of this ladder handle the ellipsoid by an auxiliary latitude — a change of the angle that makes one property come out right when the formula is spherical. The conformal latitude for a conformal map, the authalic for an equal-area one, the rectifying for a distance along a meridian.

Every one of those constructions also needs a radius, and the radius is where the arithmetic actually happens. A latitude decides which point of the sphere a point of the ellipsoid corresponds to; a radius decides how large the sphere is. Substituting one and not the other gets the shape of the answer right and its size wrong.

There is no single right radius, and the four in common use are all called the mean radius of the Earth.

The four, and what each is for #

The authalic radius is √(A/4π), the sphere of the same total surface area. It is the radius an equal-area projection of the ellipsoid must be drawn on, because the authalic latitude only fixes where the areas go and the radius fixes how much of them there is. For WGS84 it is 6,371.0072 km.

The rectifying radius is the quarter-meridian times 2/π, the sphere on which a meridian is the same length. It is what a distance along a meridian needs, and it is 6,367.4491 km.

The volumetric radius is (a²b)^⅓, the sphere of the same volume. Nothing on a map wants it; a gravity model does. 6,371.0008 km.

The arithmetic mean is (2a + b)/3, which is what the IUGG publishes and where the familiar 6,371 comes from. 6,371.0088 km.

Three of the four are within eight metres of each other. The rectifying radius is three and a half kilometres away from all of them, and the reason is not arbitrary: an ellipsoid’s meridian is a shorter curve, relative to its area and its volume, than a sphere’s is.

Why they differ at all #

The three-way agreement and the one-way disagreement both have reasons, and they are worth a paragraph because they say which radius to distrust in an unfamiliar case.

An oblate spheroid is a sphere squashed along one axis. To leading order in the flattening $f$, its area falls short of the sphere of radius $a$ by about $2f/3$ of the equivalent, its volume by $f$, and its meridian is shorter by about $3f/4$ — and taking the corresponding root of each brings area and volume to a shortfall of $f/3$ in radius while the meridian’s stays at $3f/4$. Two of the three land in the same place and one does not.

The arithmetic mean $(2a + b)/3$ is $a(1 - f/3)$ exactly, which is why it agrees with the authalic and volumetric radii to a part per million rather than by coincidence: it is the same leading term, and the difference is second order in $f$, which is 11 parts per million squared. The published mean radius is an area radius wearing a neutral name.

That is a useful thing to carry, because it says the agreement is structural rather than lucky. Any body whose flattening is small will have its area, volume and arithmetic-mean radii within $O(f^2)$ of one another and its rectifying radius $f/6$ below them, and the four-way table above will look the same on Mars as it does here.

What the wrong one costs #

The numbers matter because 559 parts per million is not a rounding error in any surveying context. It is 56 centimetres in every kilometre, 5.6 metres over ten, and 56 times a first-order survey’s proportional tolerance of one part in a hundred thousand.

The pattern is the useful thing. The published mean radius is safe for the two uses nobody computes by hand and unsafe for the one everybody does. An area or a volume of the whole Earth is a number looked up rather than derived; a distance along a meridian is what a person with a coordinate pair and a spherical formula computes every day, and it is the one where 6371 is 559 parts per million wrong.

Using the rectifying radius the other way round is worse still: an area computed on a sphere of 6,367.4 km is 1,117 parts per million short, because the error enters squared.

The gap has a closed form #

The three-and-a-half-kilometre gap looks like an accident of which ellipsoid was adopted. It is not.

$$\frac{R_{\text{authalic}} - R_{\text{rectifying}}}{R} \;\simeq\; \frac{f}{6}.$$

Clarke 1866, with $f = 1/295.0$, gives 565.0 parts per million; WGS84, with $f = 1/298.257$, gives 558.8. The predicted values are 564.9 and 558.8. The line in that figure is $f/6$ rather than a fit, and every ellipsoid in the library lies on it to within a hundredth.

So the disagreement between the two radii is the flattening, in the same way that datum shifts dwarf projection errors is the flattening and the normal section is not the geodesic is the flattening. The whole ladder keeps arriving at the same small number in different disguises, and this is one more of them.

The one that is not a constant #

The fifth radius is the one a conformal map needs, and it is not on the axis at all.

The radius of curvature is two numbers establishes that an ellipsoid has a meridional radius $M$ and a normal radius $N$ and they differ. Their geometric mean $\sqrt{MN}$ is the radius of the sphere with the same Gaussian curvature at that point, and it is what a conformal double projection — ellipsoid to sphere to plane — has to be built on if the sphere is to have the ellipsoid’s own curvature where it matters.

It varies by 6,739 parts per million between the equator and the pole. So there is no conformal sphere; there is one per latitude, and a construction that names a single radius has named a standard latitude without saying so.

A double projection standardised at 45° is 3,347 parts per million out at the equator and 3,370 out at the pole. That is a third of a per cent, which on a national sheet is metres. It is also exactly conformal everywhere, which is the trap: the property the construction was built to guarantee is guaranteed, and the property nobody mentioned is not.

What was computed, and how #

Each radius comes from its own closed form, and each is checked against the quantity it is supposed to reproduce rather than against another formula. The authalic radius is required to give back the ellipsoid’s surface area — computed from the exact $2\pi a^2(1 + \frac{1-e^2}{e}\operatorname{artanh} e)$ — to twelve digits. The rectifying radius is required to give back the quarter-meridian, which this file computes both by series and by quadrature and holds against each other.

The Gaussian radius comes from the same $M$ and $N$ the rest of the ellipsoid machinery uses, so the profile is not a new derivation; it is a quantity the site already computes, plotted against latitude for the first time.

The control is the sphere. In the library’s own flattening-zero ellipsoid all five radii agree exactly, which is what says the distinction belongs to the ellipsoid rather than to the arithmetic — a bug that made two of the formulae the same expression would pass every other check here and fail that one.

The assertions require six things separately: that the authalic radius reproduce the area and the rectifying one the meridian; that the four constants differ by hundreds of parts per million; that each use be exact under its own radius to machine precision; that the arithmetic mean be wrong for a meridian by more than fifty first-order tolerances and right for an area within one, which is the finding rather than a general failure; that the Gaussian radius vary by more than the constants differ; and that the gap follow $f/6$ across every ellipsoid in the library.

What it does to this collection’s own numbers #

The audit is worth running here, and the answer is mostly reassuring for a reason worth stating.

Almost nothing in these essays uses a radius at all. The distortion machinery works in scale factors — ratios of lengths — and a ratio is unchanged by which sphere it was computed on, which is why what survives a change of coordinates has been the site’s organising principle. A projection that is 1.4 times too large along a meridian is 1.4 times too large whatever radius the meridian was measured with.

Where the radius does enter is wherever a number leaves in metres: a degree is not a unit of length, the survey reductions, the cut lengths, the reach sets. Those use 6371.0088 km — the arithmetic mean — and every one of them is a distance rather than an area, which is exactly the use it is worst for.

The size of the consequence is small and it is not zero: 559 parts per million on a length is 56 centimetres per kilometre, so a stated 20,015 km of cut is out by 11 km, and a stated 500 km reach ring is out by 280 m. Neither changes any argument in the essay it appears in, and both are larger than the precision the numbers are printed to. The honest correction is to state which radius a metre came from, and doing that across every essay that prints one is a pass rather than a paragraph — so this rung records it as a shortfall in the site’s own ledger rather than claiming to have fixed it here.

Where the model stops #

The Gaussian radius is one choice among several for a double projection. Gauss’s own construction fixes the sphere by matching curvature at a standard latitude, which is what is measured here; other conformal spheres match the meridian arc, or minimise scale error over a band, and each moves the zero-crossing without changing the shape of the curve. The 6,739 parts per million is the range the choice is made within, whatever rule makes it.

Nothing here is about the ellipsoid-to-sphere step being worth it. Where the series stops being the map prices the alternative — computing on the ellipsoid directly with a truncated series — and that comparison is the one an implementer actually faces. This rung is about the constants, and it applies to whichever of the two routes is taken.

And a distance along a meridian is the easy case. A general geodesic is not a meridian arc and does not have a single radius that makes it exact; geodesics on the ellipsoid, and why they are hard is what that costs. The 559 parts per million measured here is the meridian’s own error, which is the largest of the family and the only one with a closed form.

The case that is worse than any of these #

One use of a radius is missing from the table because it does not fit in it, and it is the one where the choice does the most damage.

A projection’s own scale is a ratio and is radius-free, as above. A projection’s output in metres is not: a grid coordinate is a scale factor times a radius times an angle, so a national grid built on a spherical formula inherits the radius three times over — in its eastings, in its northings and in its scale factor. The three do not inherit the same error, because eastings run along a parallel and northings along a meridian, and the radius that makes one exact is not the one that makes the other exact.

That is the situation the rectifying radius exists for and it can only fix the northings. Choosing it makes a meridian arc exact and leaves the parallel out by the same 559 parts per million with the sign reversed; choosing the authalic radius does the reverse. No single sphere makes both exact, which is a small and exact instance of the impossibility the whole site is about: the ellipsoid’s meridian and its parallel have different curvatures, and a sphere has one.

The escape is not a better radius. It is to stop substituting a sphere, which is what the transverse Mercator series does — and the series’ own truncation error at the order in use is under a part per million, which is three orders below the error the best possible single radius leaves behind.

The generalisation #

The rule is that an approximation has as many parameters as the thing it is approximating has properties, and fixing one of them is usually presented as fixing the approximation.

An auxiliary latitude is a real and elegant device: it makes one property of the ellipsoid exact on a sphere. What it does not do — and what its name does not warn anybody about — is make the sphere the right size. The two halves are independent, they are usually presented together in a way that suggests one implies the other, and the second half is a single constant that is easy to take from a different source than the first.

The same shape recurs wherever a model is fitted to one property and used for another. Equal-area on the wrong body is the version of it this collection opens with: a projection that is exactly equal-area on a sphere is not equal-area on the ellipsoid, and the failure is invisible because the property is still being called exact.

The rule also says where to look for the next instance of it. Any construction described as “using the auxiliary X” has a second constant beside the X, and that constant is the one to check — because the construction is named after the part that was thought about.

The habit is a question with two halves. When a spherical formula is used on an ellipsoidal body, ask which latitude was substituted and which radius. If the answer to the second is 6371, ask what the number is being used for — and if it is a distance, the answer is out by more than half a metre in every kilometre.

Who found it, and when #

None of this is obscure. Every geodesy textbook lists the auxiliary radii, and the IUGG’s definition of the mean radius as (2a + b)/3 is a published standard with a stated purpose. The closed forms are old.

What travels badly is the pairing. The radii live in a table of constants and the auxiliary latitudes live in a chapter on conformal and equal-area mapping, and a person implementing one of those constructions reaches for a radius from wherever is nearest — which is nearly always the 6371 that is in every reference, on every calculator, and in the first line of most spherical-formula code.

The place the distinction is taken seriously is in the specification of national grids, where the sphere’s radius is stated explicitly along with the latitude it was taken at. The Dutch RD grid, the Swiss system and several others are conformal double projections and every one of them names its own sphere. That is exactly the practice this rung argues for, and it is a convention of the projection specifications rather than of the textbooks the specifications are built from.

The cheap habit: a spherical formula applied to the Earth has two constants in it and one of them is usually invisible. Write both down, name what each makes exact, and the 559 parts per million becomes a decision rather than an accident.

Where the ladder goes next #

Eleven rungs have handled the ellipsoid by substitution — a latitude, a radius, a series — and every one of those substitutions has been exact for one property and approximate for the rest. What none of them has faced is the case where the property being approximated is not a scalar at all: a direction, carried across a region, where the thing that has to be preserved is a relationship between two points rather than a quantity at one.