The curvature of the Earth is not one number

The impossibility this site is built on is stated for a sphere: a sphere has constant curvature $1/R^2$, a plane has zero, and no isometry can exist between surfaces whose curvatures differ. Applied to the actual Earth, the same argument says something more, and it is not usually said.

Two radii, not one #

At any point of a surface of revolution there are two principal radii of curvature, and on an ellipsoid they have names because geodesy uses both constantly.

$M$, the meridional radius, is the radius of curvature of the meridian ellipse at that latitude:

$$M(\varphi) = \frac{a(1-e^2)}{(1 - e^2\sin^2\varphi)^{3/2}}$$

$N$, the prime-vertical radius, is the distance along the normal from the point to the axis of rotation:

$$N(\varphi) = \frac{a}{(1 - e^2\sin^2\varphi)^{1/2}}$$

On WGS84 these run

Both increase towards the pole and they meet there, where the surface is locally spherical. They differ by 43 km at the equator, which is 0.67% — the flattening, arriving in the form it takes in practice.

The Gaussian curvature, and why it falls towards the pole #

The Gaussian curvature of a surface is the product of the two principal curvatures, so on an ellipsoid

$$K(\varphi) = \frac{1}{M(\varphi)\,N(\varphi)}$$

Both radii grow towards the pole, so $K$ falls: from $2.4747\times10^{-14}\,\mathrm{m^{-2}}$ at the equator to $2.4417\times10^{-14}$ at the pole.

That direction catches people out, including the first version of the assertion that guards this calculation, which claimed the opposite and was refused in the fourth significant figure. The intuition that misleads is the outline: seen from outside, an oblate ellipsoid looks blunter at the poles and pointier round the equator, which suggests the poles are more curved. The outline is the meridian ellipse, and the meridian ellipse turns most sharply at the ends of its major axis — which are on the equator. The polar region is the flattened part.

Writing it as a radius makes it memorable. The Gaussian radius $\sqrt{MN}$ runs from exactly $b$, the semi-minor axis, at the equator, to exactly $a^2/b$ at the pole. The Earth’s least curved point is the pole, and the sphere it matches there is 43 km larger than the sphere it matches at the equator.

The spread, and what it forbids #

$$\frac{K_{\text{equator}}}{K_{\text{pole}}} = \frac{1}{(1-e^2)^2} = 1.013524$$

so the curvature varies by 1.35% across the meridian. That number is $2e^2$ to first order, which is four times the flattening, and it is asserted against the closed form to a part in $10^8$ rather than against a tolerance somebody chose.

The consequence follows in one line from the same theorem the site opens with. A surface whose curvature varies cannot be mapped isometrically onto a surface whose curvature is constant, because curvature is intrinsic and an isometry preserves it. So the sphere is not developable onto the ellipsoid, and the spherical approximation of the Earth is a map projection in exactly the sense the rest of this site means, with its own unavoidable distortion.

That is the part that is rarely stated. Every geodesy text explains that the ellipsoid replaced the sphere because the sphere is not accurate enough; few put a floor under how inaccurate it must be.

The floor, computed #

For a conformal map with scale factor $\lambda$ between two surfaces, the curvatures are related by $K' = K/\lambda^2$ at every point. Rearranged, the ratio of the scale factors at two points is the square root of the ratio of the curvatures there:

$$\frac{\lambda_1}{\lambda_2} = \sqrt{\frac{K_2}{K_1}}$$

That is not a property of a clever map or a stupid one. It holds for every conformal map between the two surfaces, so the scale factor of any conformal map from a sphere to WGS84 must vary by at least

$$\sqrt{1.013524} = 1.006739$$

between the equator and the pole: 6,739 parts per million, and nothing gets under it.

For comparison, the entire design effort that produced UTM’s six-degree zone and its scale factor of 0.9996 holds the worst-case error to 981 ppm. The spherical approximation over the whole Earth costs seven times as much as the thing an international standard was engineered to control.

Why the spherical approximation survives anyway #

Because 6,739 ppm is the figure for a map covering the whole range of latitudes, and almost nothing does.

The bound scales with the spread of curvature over the region mapped, and curvature varies slowly: from the equator to 30° the Gaussian radius moves by 11 km in 6,357, so a sphere fitted to a region 30° tall need only accommodate about 1,700 ppm, and over the six-degree strip a UTM zone occupies the spread is a few tens of parts per million. A sphere chosen for a region rather than for the planet is a good approximation of that region, and choosing it is exactly what the conformal latitude does.

That is the standard device of practical geodesy and it is worth naming precisely. A conformal map from the ellipsoid to a sphere exists — the Gauss conformal double projection — and it is conformal exactly, everywhere. What it cannot be is isometric, so it carries a scale factor that varies with latitude, and that varying scale factor is the 6,739 parts per million above, made explicit and carried through the computation rather than ignored. Several national grids are built this way: ellipsoid to sphere conformally, sphere to plane conformally, with the two scale factors multiplied.

Every surface with varying curvature has the same problem #

The ellipsoid is not a special case, and putting it beside the surfaces the impossibility argument was first made on shows what it has in common with them.

That placement is the useful one. A torus and an ellipsoid are both non-developable onto a sphere for exactly the same reason, and the entire difference between “geodesy uses a sphere all the time” and “nobody maps a torus with a sphere” is the size of the variation. The theorem does not grade; the number does.

The other ellipsoids agree #

The spread is a function of the flattening alone, so every reference ellipsoid in daily use gives nearly the same figure:

The variation between them, 1.5% of the spread, is a measure of how much the nineteenth century’s estimates of the flattening disagreed. It is also a reminder that the ellipsoid is a datum’s definition rather than a measurement of the Earth: these are five different agreements about the shape of the planet, and the difference between the agreements is far larger than the precision each was quoted to.

What it does to the two claims the site already makes #

Two of this site’s results are stated for a sphere and both are changed in a way worth recording.

The trade-off. Conformal and equal-area cannot both hold because both together force an isometry, and an isometry between surfaces of different curvature does not exist. On the ellipsoid the argument runs identically and forbids slightly more: a map that is conformal and equal-area between the ellipsoid and the sphere is also impossible, so even the intermediate step geodesy takes for convenience must give something up.

The unrollable surfaces. What can be unrolled is a question about zero curvature, and it is unaffected — a cylinder and a cone have $K = 0$ whatever body they were fitted to. What changes is the other side of the comparison: a cone tangent to an ellipsoid along a parallel is tangent to a surface whose curvature at that parallel is one of a range of values, so the fit deteriorates away from the line of tangency at a rate that depends on where the tangency is. A conic fitted at 60° north has less curvature mismatch to accommodate than one fitted at the equator, by about half a per cent.

Neither observation changes a decision. Both change what the numbers in the earlier essays are approximations to, which is the sort of bookkeeping a fourth rung is for.

The measurement that has to fail #

The sphere is in the library with a flattening of exactly zero, and its curvature must come out constant to a part in $10^{11}$.

Without that half, a bug that made $M$ and $N$ equal at every latitude — the easiest possible slip, since the two formulae differ only in an exponent — would produce a constant curvature on the ellipsoid too, and the interesting claim would silently become false while the check went on passing. The sphere is the control.

The monotonicity is asserted as well, at every degree of latitude: $K$ must fall from equator to pole without a single reversal. That is what caught the direction error, and it is stronger than checking the two endpoints, which a formula with the wrong sign somewhere in the middle would satisfy.

Curvature from inside, on a body that is not a sphere #

Measuring curvature from inside makes the case that Gaussian curvature is intrinsic — computable from the metric alone, with no reference to how the surface sits in space — and checks it by computing $K$ two ways for six surfaces.

The ellipsoid extends that argument in a way a sphere cannot, because on a sphere the intrinsic and extrinsic routes agree on a constant and any bug that produced a constant would pass. Here the two routes must agree on a function, and reproducing the 1.35% variation from the first fundamental form alone is a much stronger statement of the Theorema Egregium than reproducing a number.

It also gives the curvature ladder a genuine second case: total curvature integrated over the whole ellipsoid must still come to $4\pi$, because that is a topological statement and the topology has not changed, even though the integrand now varies from place to place.

The practical form of the same point: a surveyor confined to the surface, measuring only angles and distances on the ground, could in principle detect the difference between the equator and the pole. The triangle whose excess would reveal it is large — the effect is 1.35% of a quantity that is itself parts per million at survey scale — but it is a measurement rather than an inference from the shape seen from space, which is what “intrinsic” means.

The auxiliary latitudes are the practical form of the same fact: each makes one property of the ellipsoid behave spherically, and each is a different angle.

What was computed here #

$M$ and $N$ were evaluated at every degree of latitude from their defining formulae on each of five reference ellipsoids, and $K = 1/(MN)$ from them. The spread is asserted against the closed form $(1-e^2)^{-2}$ to within $10^{-6}$ per cent, and the monotone fall from equator to pole is checked at all ninety-one samples.

The sphere in the same library, with flattening exactly zero, must return a spread below $10^{-9}$ per cent, which is the control that makes the ellipsoid’s variation a measurement rather than an artefact.

The conformal bound is the identity $K' = K/\lambda^2$ evaluated at the two extremes of the profile.

What the pictures cannot show #

The figure plots $\sqrt{MN}$ rather than $K$, because a quantity near 6,371 km can be read and $2.47\times10^{-14}\,\mathrm{m^{-2}}$ cannot. The two carry the same information and the axis says which is drawn, but the fractional spread is halved by the square root: 1.35% in $K$ appears as 0.67% in the radius.

Nor does the figure show the bound it argues for. 6,739 ppm is a statement about all conformal maps between two surfaces, and no drawing of either surface contains it.

The number that is quoted instead #

“The radius of the Earth is 6,371 km” is the figure in every textbook, and it is worth saying exactly what it is a radius of.

It is the mean radius $R_1 = (2a + b)/3$, an arithmetic average of the three semi-axes, chosen because it makes the sphere’s total surface area a reasonable compromise. It is not the radius that matches the curvature anywhere: the Gaussian radius runs from 6,357 to 6,400 km and passes through 6,371 km at about 21° latitude, so a sphere of the quoted radius has the right curvature along two small circles and the wrong curvature everywhere else.

Other single-number radii are in use and disagree with it. The authalic radius — the sphere of equal surface area — is 6,371.007 km. The volumetric radius is 6,371.001 km. The radius that makes the meridian arc come out right from pole to equator is 6,367.449 km, four kilometres smaller, because that quantity depends on $M$ rather than on $\sqrt{MN}$.

Which one to use is settled by what is being computed, and the differences among them — a few kilometres in six thousand, a few hundred parts per million — sit precisely in the band this essay is about. A number quoted without saying which average it is has an error bar of the same size as the effect.

What a survey does about it #

That is the practical shape of this essay’s claim. The variation cannot be removed, only re-centred, and re-centring it is what forty national ellipsoids were for. A datum is fitted to a region measures what the re-centring bought and what it cost.

The two radii, in an everyday quantity #

The curvature’s variation is 1.35 per cent and this essay’s argument is about what that does to the impossibility. The same two radii decide something a great deal more ordinary, and by a great deal more than one per cent.

A degree of latitude is M(φ) times π/180 and a degree of longitude is N(φ) cos φ times π/180, so a tolerance written in degrees is an ellipse whose aspect runs from 0.9933 at the equator to 3.86 at 75°. The 0.9933 is the part this essay’s machinery explains: at the equator M is smaller than N by 1 − e², which is why the meridian is the most sharply turning part of the ellipsoid exactly where the outline looks flattest.

Two dimensions up the same variation does something similar and larger. A spherical closed form for the area of a cell is 0.45 per cent from the ellipsoid’s own integral over a mid-latitude cell and 0.79 per cent over a polar one: the sphere has the right total area and distributes it wrongly, which is the two radii again in a quantity nobody thinks of as curvature.

Who found it, and when #

The two radii of curvature of an ellipsoid are eighteenth-century geodesy — they are what the arc measurements of the 1730s in Lapland and Peru were designed to distinguish, since the whole question of whether the Earth is oblate or prolate is the question of whether $M$ grows or shrinks towards the pole. Maupertuis’s expedition returned the answer in 1737.

That their product is the Gaussian curvature had to wait for Gauss in 1827, and the consequence drawn here — that the sphere is therefore not developable onto the ellipsoid — is the Theorema Egregium applied to the case the theorem’s own author spent his working life surveying.

Which radius to use, when a sphere has to be used anyway #

The spherical approximation survives, and a practitioner using it has to pick a radius. The variation measured here says that the choice is not free and that there is no single right answer — there is a right answer per quantity.

For an area, use the authalic radius. It is defined as the radius of the sphere with the ellipsoid’s total surface area, so it makes the global total exact by construction. What it does not do is distribute that area correctly: the essay’s own figure has the spherical closed form 0.45 per cent from the ellipsoidal integral over a mid-latitude cell and 0.79 per cent over a polar one, with the errors of opposite sign so that the whole comes out right. A radius that fixes the total is not a radius that fixes any particular region.

For a distance, use the rectifying radius, which reproduces the meridian’s total length. The same caveat applies in the same form: the total is right and the parts are not, since the meridian’s own curvature varies along it.

For local geometry, use the Gaussian mean $\sqrt{MN}$ at the latitude in question. This is the only one of the three that is not a global compromise — it is the radius of the sphere with the ellipsoid’s actual curvature there — and it is the right choice for anything computed over a patch rather than over the world.

The rule underneath is that each radius makes exactly one thing exact, and that thing is named in the radius’s definition. Using an authalic radius for a distance, or a mean radius for an area, is not a small error compounded — it is a quantity that was never being preserved.

Where this goes next #

The impossibility is now stated for the body geodesy actually uses. The remaining questions on this ladder are about what happens to specific quantities when the body is taken seriously, and the sharpest of them is what a grid’s north actually points at.