Where the surface curves the other way

Twelve rungs of this ladder argue the impossibility on a sphere, an ellipsoid, a body with three axes, a surface of revolution built from a stated curvature. Every one of those surfaces has positive Gaussian curvature everywhere.

The surface a map of the ground actually shows is not one of them.

The quantity, and where it comes from #

A terrain is a height above a plane — which is what a topographic surface is, locally, and is not what a datum is. For a graph z = h(x, y) the Gaussian curvature is

$$K = \frac{h_{xx}h_{yy} - h_{xy}^2}{(1 + h_x^2 + h_y^2)^2}.$$

The numerator is the determinant of the Hessian, so the sign of the curvature is the sign of that determinant: positive at a summit, positive at a hollow, negative at a pass. That is exactly the classification the second derivative cannot make on a sphere, used here for the one thing it does correctly.

The terrain is stated — a sum of Gaussian caps at named places, widths and heights — rather than taken from an elevation model. An elevation model has a resolution, and a second derivative computed from one is mostly a measurement of that resolution. This is the coastline decision applied one derivative further out, where it binds harder.

Both signs, and the second one is the common one #

Even a single smooth hill is mostly saddle. That row is the one worth stopping on, because it is the simplest possible relief and the intuition about it is wrong. A Gaussian cap is convex only near its summit; over the whole of its skirt one principal curvature is convex and the other is concave, and the skirt is the larger part of it. Eighty-nine per cent.

So negative curvature is not a feature of unusually rugged ground. It is what ground is.

The share is taken over the curved part of the window rather than over the window, and the distinction is the difference between a measurement and an artefact. Far from every cap the surface is a plane, its curvature is zero, and the sign of a number that is zero is the last bits of a subtraction. Counting those put the single summit at 96.5 per cent negative, all of it flat ground, and the floor — a thousandth of the window’s own peak — is what removes them.

The size, which is the other half #

Twenty-two thousand times the Earth’s own curvature.

That number is what licenses the whole of this rung. Every argument on this ladder about the impossibility of a faithful map is an argument about the curvature of the surface being mapped, and if the terrain’s curvature were a small perturbation of the sphere’s it could be neglected. It is not a perturbation; the sphere’s own curvature is invisible beside it, and the control row makes that concrete — a cap wide enough to look like a sphere comes back at 1.6 times the Earth’s, and everything with relief in it is four or five orders of magnitude above.

How small is flat enough answers the question for a sphere and gets 1,274 kilometres at a two-centimetre tolerance, which is why a plane survey works over a county. The same argument on a surface of curvature K gives a radius going as |K|^−1/2, so the answer scales as the square root of the ratio of the two curvatures — and on this terrain it is 8.6 kilometres.

That is a real limit and it applies to a real operation: fitting a plane to the ground rather than to the datum surface, which is what a local site grid does. Eight kilometres is a large site and a small county, so the limit sits exactly where a surveyor’s judgement about it would be least reliable.

The curvature this ladder has been using #

It is worth putting the two on the same page, because the ladder has spent a great deal of care on a variation of 1.35 per cent in a number that the ground routinely multiplies by twenty thousand.

That is not a criticism of the earlier rungs. They are about the surface a datum is, and a datum is smooth by definition — the whole point of an ellipsoid is that it is the shape without the ground on it. What this rung adds is that the surface a map depicts, once anybody draws relief on it, is a different surface with a different curvature and a different sign, and the two have never been compared.

The phrase to take from that picture is a negative sea. The positive parts are the summits and the hollows, which are isolated; the negative part is everything between them, which is connected. On a landscape, convexity is the exception and it is where the names are.

What the theorem says about mixed signs #

The impossibility argument this ladder rests on is Gauss’s Theorema Egregium — curvature is intrinsic, a plane has none, so no isometry exists onto anything that has some. It does not care about the sign, and a surface with mixed curvature is unmappable for exactly the same reason a sphere is.

What the sign changes is the direction of the failure.

On a sphere, a patch flattened onto a plane has too much material near its edge: the circumference of a geodesic circle is less than 2πr, so laying it flat leaves the rim needing to stretch or the middle needing to crumple. On a saddle it is the other way — the circumference is more than 2πr, so a flattened patch has too little room and the rim has to compress.

On real ground both happen, in adjacent patches, on the same sheet, which is why a paper map cannot be made to fit a hillside even locally and why a topographic sheet is a map of a datum surface rather than of the ground.

The check, and it is Gauss–Bonnet #

A mixed-sign surface is exactly the case where a curvature integral is easy to get wrong, so it is worth checking against a theorem rather than against itself.

Over a disc of the terrain fourteen kilometres across, the integral of the Gaussian curvature comes to 0.00822, and the total turning of its boundary curve — computed from the curve’s own geometry, sharing no arithmetic with the first — comes to 6.28295. Their sum is 6.29117 against 2π = 6.28319.

A gap of 0.13 per cent, on a patch whose curvature changes sign several times inside it. Gauss–Bonnet does not know about the sign and did not need to.

The two terms are also worth reading for their sizes. The turning is 6.283 and the curvature integral is 0.008, so the theorem is very nearly the statement that a closed curve turns through 2π — which it is, on a plane. What the fourteen-kilometre disc of terrain contributes is the thousandth part, and measuring a thousandth part against a whole to a tenth of a per cent is the kind of check that catches a sign error and would not catch a scale error. Both were looked for; the scale is checked by the flat-enough radius above, which involves no integral at all.

That is the same check measuring curvature from inside runs on a sphere and a torus, and running it here is what says the sign changes above are the surface’s rather than a differencing artefact.

What a mapmaker already does about it #

The impossibility is not news to anybody drawing a topographic sheet, and what they do about it is worth naming, because it is a decision that this ladder has been assuming without stating.

A map is a map of the datum surface, not of the ground. Every horizontal coordinate on a topographic sheet is the position of the point projected onto the ellipsoid, and the relief is carried by contours drawn on that projection rather than by any attempt to flatten the terrain. So the map is a map of a smooth convex surface, and the ground’s own curvature — the whole of this rung — never enters the projection at all.

That is exactly right and it has a price. A distance measured on the sheet is a distance on the datum, not on the ground: the ground is not the grid prices the difference, and a slope of six degrees costs half a per cent. The terrain’s curvature is the reason that correction cannot be a constant.

And a slope map is a map of a quantity, not of a surface. A slope is not a shape draws the distinction: a gradient field on a projection is a field of numbers, and it inherits the projection’s distortion rather than the terrain’s curvature. The two failures are separate, and only the first is what this collection has been about.

What a negative curvature does to a scale bar #

The sign has one consequence a reader meets directly, and it is worth stating in the terms a sheet uses.

A scale bar is a claim that a stated page length stands for a stated ground length. On a positively curved patch flattened onto a plane, ground distances near the edge of the patch come out short on the page — the flattening has to stretch the rim, so a fixed page length stands for less ground than the bar says. On a saddle it is the other way: the flattened patch has too little room at the rim, distances are compressed, and the same bar stands for more.

So on real ground the error in a scale bar changes sign between a summit and the pass beside it, over a few kilometres, at a size set by the terrain’s curvature rather than the Earth’s. At the flat-enough radius computed above — 8.6 kilometres for a two-centimetre tolerance — that is exactly the scale at which a walker uses one.

The mapmaker’s answer, again, is that the sheet is a map of the datum surface and the scale bar is a statement about that, with the ground’s own excess left to the reader and to the corrections the ground is not the grid prices. The bar is right about a surface nobody is standing on.

Where the model stops #

The terrain is a graph over a plane, which is right at eighty kilometres and would not be at eight thousand: over a large region the Earth’s own curvature has to be added, and it adds a constant 2.46 × 10⁻¹⁴ to a quantity running to 10⁻⁹, which is why it is left out here and cannot be left out there.

A real terrain is not smooth. A Gaussian cap has derivatives of every order; a cliff, a scree slope and a river gorge do not, and a second derivative at a discontinuity is not a number. Every published curvature of a real landscape is a curvature of a smoothed landscape, and the smoothing length is the parameter the answer depends on most — which is the reason the terrain here is stated.

And the vertical scale is real rather than exaggerated: nine hundred metres of relief over nine kilometres, which is a slope of about six degrees. Doubling it more than doubles the curvature, since the numerator goes as the square of the height and the denominator only fights back at large slopes.

The generalisation #

A surface’s curvature has a sign, the sign says which way a flattening fails, and the sign of real ground is mostly the one nobody’s intuition uses.

The intuition comes from the sphere, because the sphere is what the subject is about, and a sphere is the one surface where the sign is constant and positive. Everything else — a landscape, a saddle in a potential, a metric with a mixed signature — has both, and every statement of the form “the surface curves away, so the map must stretch” is a statement about half of it.

The practical version is short and is not about maps at all. Check the sign of the determinant before reasoning about the magnitude, because on any surface with structure the determinant changes sign much more often than it changes size.

Who found it, and when #

Gauss’s Theorema Egregium is 1827 and is sign-blind by construction: the curvature is intrinsic whatever it is. Negative curvature entered the subject through the pseudosphere and hyperbolic geometry in the decades after, as an exotic case, and it took a century for anybody to point out that most ordinary surfaces have it over most of themselves.

In geomorphology the two curvatures of a landscape — profile and plan — have been computed from elevation models since the 1970s, and the Gaussian curvature, which is their product in the appropriate frame, is less often reported precisely because its sign changes so often that a map of it looks like noise. What is not standard is the comparison this rung makes: not the terrain’s curvature against zero, but against the curvature of the body the whole subject is about.

What the sign change forbids #

The terrain’s curvature changing sign at short range is the rung’s central measurement, and it rules out a whole class of approach that would otherwise look reasonable.

It forbids a regional curvature. A single number summarising a region’s Gaussian curvature is meaningless when the quantity alternates sign across the region — the mean is near zero, the mean of the magnitude is large, and neither describes anything a map would experience. There is no honest scalar.

It forbids treating the terrain as a perturbation of the reference body. A perturbation argument wants the correction to be small and of one sign; here it is enormous and of both, so the ellipsoid is not an approximation to the ground in any sense the curvature respects. What the ellipsoid approximates is the level surface, which is a different object and is smooth.

And it explains why the whole subject maps the reference body rather than the ground. Not as a simplification, and not because the topography is small, but because the topography’s intrinsic geometry has no structure a projection could be designed against. A surface whose curvature flips sign every few hundred metres admits no useful statement of the form this projection suits this terrain.

Which retroactively justifies a choice every rung of this ladder has made. Mapping a smooth reference surface and treating the ground as data on it is not an idealisation to be apologised for; it is the only arrangement in which the ladder’s questions have answers.

The reason the Gaussian curvature is under-reported in geomorphology is worth one more sentence, because it is a measurement decision rather than an oversight. A quantity whose sign flips at short range produces a map that looks like noise, and a map that looks like noise is discarded — not because the quantity is meaningless but because the display cannot carry it. What this rung does differently is refuse to display it and compare it instead: against the curvature of the reference body, which is a single number, and against which the terrain’s own values are enormous. The comparison survives a quantity that no image of it could.

That is worth stating positively rather than as a concession.

The alternative would be a subject with no results in it.

Where the ladder goes next #

Ten rungs measure the curvature of the surface a map depicts and what it forbids. What none of them measures is the curvature of the surface a map is printed on — a sheet that is bent, rolled or folded is a developable surface with zero Gaussian curvature, so bending it costs nothing, and the one thing that does not survive is a reader’s ability to lay a straight edge on it.