Curvature that varies from place to place

The argument this whole site rests on is four sentences long. A sphere has Gaussian curvature 1/R², a plane has zero, curvature is intrinsic — preserved by any map that preserves distances — and therefore no map from sphere to plane preserves distances. No map is faithful makes it, and the machinery behind it computes K two independent ways on six surfaces and requires them to agree.

Every one of those six surfaces has a constant curvature or an axis of revolution. That is not a coincidence: it is what makes the arithmetic short. And it means the site has been arguing about curvature for eight essays without ever treating it as what it actually is on a real body, which is a field — a number that depends on where.

What varies, and by how much #

On an ellipsoid of revolution the Gaussian curvature is 1/(MN), the reciprocal of the product of the two principal radii of curvature, and it is a function of latitude alone. The curvature is not one number computes it and reports the terrestrial figure: K varies by 1.35 per cent between equator and pole, and the least curved part of the Earth is the polar region, because that is the flattened part of a squashed ball.

The same expression, evaluated on other bodies, gives a range worth putting in one place:

The units are per square kilometre, which is why the numbers differ by orders of magnitude between bodies: a small body is sharply curved. What is comparable across rows is the ratio.

The last column is the one this rung exists for. On every body of revolution the curvature is constant along a parallel — exactly, by symmetry, and the measurement returns 1.0000 to the arithmetic’s floor. On a triaxial body it is not, and no function of latitude can express it, for the same reason that no auxiliary latitude survives there.

The formula the site had does not apply here #

There are two routes to Gaussian curvature and the site uses both, because requiring them to agree is what turns Gauss’s theorem from a citation into a check.

The extrinsic route reads the surface’s shape in space. For a triaxial ellipsoid it is one of the shortest closed forms in surface theory:

$$K = \frac{p^4}{(abc)^2}$$

with p the distance from the centre to the tangent plane at the point. On a sphere p = R and it collapses to 1/R², which is the first thing to check about it.

The intrinsic route uses the metric alone, which is what makes the theorem remarkable: a flatlander confined to the surface, able to measure only distances within it, can compute the same number. The site’s curvatureIntrinsic does this — and it cannot be used here, and the reason is the point of the rung.

That function is the orthogonal-coordinate form of the intrinsic curvature. It assumes F = 0, which is to say that the coordinate lines cross at right angles. Every surface the site has measured has had an axis of revolution, so F has always been zero and the assumption has never cost anything. On a triaxial body it is false — the departure reaches 1.56° on Vesta — so the general expression is needed, which is Brioschi’s formula: two 3 × 3 determinants over (EG − F²)², built from the first and second derivatives of E, F and G.

Why the theorem is worth re-checking on an awkward surface #

An agreement between two routes is only evidence if the routes are independent, and it is easy for them to stop being independent without anybody noticing. The orthogonal formula and the closed form are both short enough to look obviously right; a check that ran them on a sphere and a cylinder would pass whatever mistakes they shared about the F term, because F is zero there and every term containing it disappears.

Brioschi’s formula on a triaxial body exercises exactly those terms. It uses F, F_u, F_v and F_uv, all of which are identically zero in every previous check the site has run. Getting the same answer as the embedding route to three parts in a million is therefore a stronger statement than the six-surface check was, and it is the reason the numbers in the rest of this essay can be trusted.

The check as it stood before this essay used six surfaces, each with its curvature computed by both routes and required to agree. Every one of them has orthogonal coordinate lines, so the intrinsic route used there is the special case — and the plane, cylinder and cone at $K = 0$ are the reason a map onto them is a different proposition from a map onto anything else.

Every ratio in the table is a power of two axis ratios #

The six-row table reports two columns of measurements, and both columns are closed forms in the body’s own semi-axes. The closed form for the curvature makes it immediate: K = p⁴/(abc)², and p — the distance from the centre to the tangent plane — takes its largest value a at the end of the longest axis and its smallest value c at the end of the shortest. So

$$\frac{K_{\max}}{K_{\min}} = \left(\frac{a}{c}\right)^{4}, \qquad \frac{K_{\max}}{K_{\min}}\bigg|_{\text{along the equator}} = \left(\frac{a}{b}\right)^{4}.$$

Against the measured columns, every row:

Five bodies spanning a factor of three hundred in ratio, two columns, one expression apiece, agreeing to every digit printed.

The along-a-parallel column is the same identity with the middle axis in it, which is why it is exactly one on every body of revolution: a = b there, and a fourth power of one is one. That is not a coincidence to be checked; it is the symmetry, written down.

The fourth power is worth carrying. Curvature is exquisitely sensitive to how far a body is from a sphere: the Earth’s three-tenths of a per cent of flattening becomes 1.35 per cent of curvature variation, and Phobos’s ratio of 1.43 in its axes becomes 4.16 in its curvature. A body a few per cent out of round has a curvature field that is tens of per cent non-uniform, which is the reason the impossibility argument’s local form bites earlier than the shape suggests it should.

And the identities propagate into the rest of the essay, which turns three of its reported numbers into one.

The flat-enough radius varies as 1/√K, so its ratio is (a/c)². For the Earth that is 1.006739, and the essay’s own measurement is a factor of 1.0067 between 492 and 496 kilometres. For Phobos it is 2.0408 against a measured 2.04.

And 6,739 parts per million is the same number again. The least scale variation any conformal map of the Earth’s ellipsoid onto a sphere can have is quoted elsewhere in this collection as 6,739 parts per million; it is (a/b)² − 1 for the terrestrial ellipsoid, to the last digit, which is the flat-enough ratio minus one.

So four quantities that arrive in four different essays by four different routes — the curvature range, its variation along a parallel, the spread in the flat-enough radius, and the floor on a conformal map’s scale variation — are the second and fourth powers of two ratios of semi-axes. Nothing about any of them needs a measurement, and the measurements exist to confirm that the machinery computing the fields is the machinery the closed forms describe.

How small is flat enough answers the practical question this field exists for: over what radius may a survey treat the Earth as a plane? The answer comes from the curvature — the departure of a cap from its own tangent plane grows as K r²/6 — and on the Earth it is about 490 kilometres for one part in a thousand.

On a body whose curvature varies, that radius varies too, as 1/√K.

On the Earth this is a curiosity: 492 kilometres against 496, and no practice changes. The finding is that the curiosity is a consequence of the Earth being nearly spherical rather than a general fact, and that on a body two per cent as large the same question has two answers differing by a factor of two.

There is a second consequence that matters more on Earth, and it is why the terrestrial case is drawn below. The tolerance a survey works to is usually stated as a fraction — a part in ten thousand, say — and the flat-enough radius goes as the square root of that fraction. So a tenfold tighter tolerance is a threefold smaller patch, and a curvature that varies by a per cent moves the boundary by half a per cent. That is why nobody has ever needed this on Earth, and it is worth saying so plainly rather than implying a practical consequence that does not exist.

What was computed, and how #

The closed form is checked against the site’s own ellipsoidal expression. For an oblate body the triaxial formula must return 1/(MN), and it does — which is the reduction check every generalisation on this site is required to pass.

Brioschi’s formula is evaluated by central differences of the exact first fundamental form. E, F and G are closed-form expressions in the two parameters; their first derivatives, the two second derivatives and the mixed one are differenced at a step of 10⁻³ radians. The step was chosen the way the site chooses every differencing step: by requiring the answer to be stable when it is halved.

The curvature range is sampled on a grid of 90 by 90 in the two parameters, and the along a parallel column is the largest ratio over any single row of that grid. Sampling rather than optimising is enough here because the extremes of a triaxial ellipsoid’s curvature are at its axes, which the grid contains.

The flat-enough radius uses the same relation as the earlier essay, so the two are directly comparable: r = √(6ε/K) for a departure of ε. On Earth it returns 492 km at K’s largest and 496 at its smallest, against the 490 km the earlier essay reports from a different sampling of the same relation.

A second place the variation is visible #

There is one surface in the site’s own collection whose curvature varies over a range no planet approaches, and it is worth looking at beside the bodies because it makes the idea of a curvature field concrete without any planetary arithmetic.

A torus makes the same point in one dimension fewer. Its Gaussian curvature is positive on the outside, negative on the inside, and exactly zero on the two circles between — so a strip along one of those circles unrolls without stretching while a strip a little either side of it does not.

Nothing in the solar system does that: an ellipsoid, triaxial or not, is convex and its curvature is positive everywhere. But the torus makes the point that the impossibility argument is about a function, and the function’s sign, range and level sets are all things a map has to contend with. On an ellipsoid the function is positive and varies by a factor of a few; that is the mild case, and it is mild for a reason worth naming — every body in the table is a fluid figure or a fragment of one, and both are convex.

The impossibility argument, restated for a variable curvature #

The original argument compares two constants: a sphere has K = 1/R², a plane has zero, and no isometry exists between surfaces with different curvature. On a body with varying curvature it becomes a statement about functions rather than numbers, and it gets stronger.

Two surfaces are locally isometric only where their curvatures agree, point for point. A plane has K = 0 everywhere, so a patch of any curved surface fails immediately. But the same argument now also rules out things nobody thought to ask about: a triaxial body cannot be mapped isometrically onto a sphere either, because their curvature functions differ; and it cannot be mapped isometrically onto an ellipsoid of revolution, because no such ellipsoid has a curvature that varies along a parallel.

So the ladder of impossibility has an extra rung on a real body. A map of Vesta is not merely unable to be faithful; it cannot even be reduced to the spherical problem, which is what every terrestrial mapping pipeline does when it treats the Earth as a sphere for a first pass. The Earth is a sphere, and when it is not measures what that substitution costs on Earth, where it is small. On a body with three axes it is not an approximation with a small error; it is a different geometry.

The fourth power also settles the deferred question about the total, without a quadrature. Gauss–Bonnet fixes ∫K dA at 4π for any surface topologically a sphere, so a triaxial ellipsoid’s integral is 4π whatever its axes are — and since the range of K grows as the fourth power of the axis ratio while the total is pinned, a more elongated body concentrates the same 4π into a smaller and smaller region near the ends of its long axis. The total is a counting number and the distribution is a fourth power, which is the sharpest form of the essay’s own closing observation about a polyhedron’s corners: concentrating curvature costs nothing in total and everything locally.

Where the model stops #

The total is still topological, and that has not been re-checked here. Gauss–Bonnet says the integral of K over any closed surface of the same topology is 4π, whatever the curvature does locally, and the site checks that on the sphere and the torus. It must hold on a triaxial ellipsoid too, and the quadrature to demonstrate it is straightforward — it is not done here, and saying so is cheaper than implying it was.

The bodies are their fitted figures. Every number above is a property of a triaxial ellipsoid fitted to an asteroid, not of the asteroid. A real small body has curvature that changes sign — a crater rim is a saddle — and the range would be unbounded rather than a factor of three. That is a different measurement needing a different input, and the fitted figure is the object the coordinates are defined on, which is the object this field is about.

Nothing here optimises. The extremes are found by sampling on a grid, which is adequate for a smooth surface with known symmetry and would not be for a shape model.

The consequence for mapping is stated and not developed. Saying that a triaxial body cannot be reduced to a spherical problem is a statement about isometries, and a mapping pipeline does not need an isometry — it needs an approximation with a stated error. What that error is, for a spherical treatment of a triaxial body, is the measurement the previous rung made for one map and did not make in general: it depends on the projection, the region and which of the two latitude conventions the pipeline believes it has been handed.

What the local variation does not touch is the total. Integrated over the whole of a surface the curvature is fixed by Gauss-Bonnet at $2\pi\chi$: the sphere gives $4\pi$ and the torus gives zero by cancellation between its halves. However the curvature is distributed, the total is a counting number — which is why concentrating all of it at a polyhedron’s corners changes nothing about how much there is.

One caution on the identities. They are exact for the fitted ellipsoid and say nothing about the body: the ratio (a/c)⁴ is a statement about three published numbers, and a real surface with a crater on it has a curvature range that is unbounded rather than a fourth power of anything. What the closed forms buy is that the fitted figure’s behaviour needs no measurement at all, which sharpens what the measurements are for — they check the machinery, and the machinery is what a shape model would have to be handed to.

Who found it, and when #

Gauss published the Theorema Egregium in 1827, in the Disquisitiones generales circa superficies curvas, and the paper’s own worked example is a spheroid — he was writing as a surveyor as much as a geometer, having spent the previous decade on the triangulation of Hanover. Brioschi’s determinant form of the intrinsic curvature dates from 1852 and exists precisely because the orthogonal form is not general.

The curvature of a triaxial ellipsoid as p⁴/(abc)² is older than either and is usually credited to the eighteenth-century work on the ellipsoid’s geometry that Euler began; it is one of those results that is a line of algebra once the right quantity — the distance from the centre to the tangent plane — is noticed.

Where the ladder goes next #

The impossibility field now has its local form: curvature is a field, the trade-off is local, and the flat-enough radius is a function of position. What the field has never had is a way of choosing a map for a region rather than describing what any map costs there.

That is the choosing field’s business, and it has a proved answer for exactly one case — the spherical cap, where Chebyshev’s criterion names the stereographic projection. For every other region the site has stated the criterion and been unable to apply it. The next essay solves it.