The floors that unroll

Which surfaces can be laid flat on a table without stretching, tearing or wrinkling anything? The question sounds like it should be answered by looking, and looking gets it wrong in both directions.

A ridged floor — curving strongly one way, flat the other — unrolls exactly. A dished floor, which curves much less and looks gentler, cannot be unrolled by any means at all. A sheet of paper rolls into a cylinder and into a cone and never into a sphere — which is why no picture surface keeps everything and why six flat faces are used instead of one curved one — which is why an orange peel will not lie flat and a paper cup will.

The number that decides it was found by Gauss and he called the result remarkable, which for a mathematician writing in Latin was strong language. This essay measures it on four floors and then uses it, because the answer decides whether a whole class of question is well posed.

Curvature that survives bending #

Take a surface and ask how curved it is at a point. There are two answers and they are different objects.

The extrinsic answer describes how the surface sits in space: which way it bends, and how sharply, in each direction through the point. Roll a sheet of paper and this changes everywhere.

The intrinsic answer describes what a creature living in the surface could measure — distances along it, angles between paths in it, the circumference of a circle of given radius drawn on it. Roll a sheet of paper and none of that changes at all, because nothing measured inside the surface has moved.

Gaussian curvature is intrinsic. That is the remarkable theorem: a quantity defined from how the surface bends in space turns out to depend only on measurements made inside it, so no bending changes it.

The consequence is immediate and it is the whole classification. A flat sheet has Gaussian curvature zero everywhere. Bending cannot change curvature. So any surface that can be got from a flat sheet by bending has curvature zero everywhere — and any surface with curvature anywhere cannot be got from a flat sheet, and cannot be laid onto one, by any method whatever.

That last clause is what makes this an obstruction rather than a difficulty. It is not that a clever enough flattening would do better; it is that there is no flattening, in the way that there is no largest prime.

Four floors, measured #

For a surface given as a height $y = h(x, z)$, the Gaussian curvature has a short form:

$$K \;=\; \frac{h_{xx}h_{zz} - h_{xz}^2}{\left(1 + h_x^2 + h_z^2\right)^2}.$$

The numerator is the determinant of the second-derivative matrix, and everything hangs on it. If the surface curves in one direction and is flat in the other, one of the second derivatives is zero and the determinant is zero — however sharply the first direction curves.

The four:

A flat floor: every derivative zero, $K = 0$. Trivially developable, and its development is the identity.

A ridge, $h = kx^2$: curves in $x$, flat in $z$, so $h_{zz} = 0$ and $K = 0$. Developable, and this is the case worth the surprise — the ridge is visibly, strongly curved.

A step, two horizontal planes with a jump: flat on both sides, $K = 0$ away from the seam. Developable piecewise, which is a different and weaker statement, and the seam is where every claim about it stops.

A dish, $h = k(x^2 + z^2)$: curves the same way in both directions, both second derivatives positive, $K > 0$. Not developable, at 0.0144 per square metre.

The derivatives here are taken by central differences rather than written out, and the reason is the step. A jump has no second derivative at all, and a file that wrote analytic derivatives would have had to special-case it or quietly leave it out — and leaving it out is how a family of surfaces silently becomes a family of smooth surfaces without anybody deciding that.

The ridge, unrolled exactly #

Saying a surface is developable is one thing; producing the development and measuring it is another, and the second is what makes the claim a number.

For the ridge the development is arc length along the parabolic section, with the flat direction left alone:

$$u(x) \;=\; \int_0^x \sqrt{1 + 4k^2t^2}\,\mathrm{d}t \;=\; \tfrac{1}{2}x\sqrt{1+4k^2x^2} \;+\; \tfrac{1}{4k}\operatorname{asinh}(2kx), \qquad v = z.$$

Written in closed form rather than integrated numerically, because it is the thing being claimed exact and a quadrature’s error would be indistinguishable from the surface failing to develop.

Measuring the claim needs a definition of “stretches nothing”, and the local one is the right one. At each point, compare the differential of the flattening with the surface’s own metric; the two numbers that come out are how much the most- and least-stretched directions are scaled. An isometry has both exactly 1.

Across the patch the ridge’s worst departure from 1 is 7.4e-9. That is arithmetic. The map does not nearly preserve lengths; it preserves them.

What the plan costs #

The interesting comparison is not between the exact development and nothing. It is between the exact development and the flattening everybody actually uses, which is the plan: read the floor’s coordinates off a drawing looking straight down.

That is a flattening. It is just not an isometry.

Two readings of that figure matter and they are different.

On the ridge, the 0.86% is avoidable. An exact development exists; using the plan instead is a choice, and a better choice is available for the cost of one integral.

On the dish, the 1.03% is not. No flattening does better than some amount, because the curvature is an obstruction. The plan is a bad flattening and every other flattening is also bad; what varies is where the badness is put.

That is the distinction the whole essay is for. Two numbers of the same size, one of which is a mistake and one of which is a fact.

The plan’s error is a slope, and it is not the obstruction #

The two numbers in the strain figure are close together and it would be easy to read them as measuring the same thing at two severities. They do not, and separating them takes one derivative.

Reading a floor off its plan compresses lengths by the cosine of the surface’s own tilt, so for a height field $h$ the worst strain is

$$1 - \frac{1}{\sqrt{1+|\nabla h|^{2}}} \;\approx\; \tfrac{1}{2}|\nabla h|^{2}.$$

Half the square of the steepest slope, and nothing else. On this patch the ridge reaches a slope of 0.132, and $0.132^{2}/2 = 0.87\%$ against the 0.86 measured; the dish reaches 0.144 and gives 1.03, which is its number exactly.

Now notice what is absent from that expression. There is no second derivative in it, so the plan’s error has nothing to do with curvature and nothing to do with developability. A perfectly flat floor tilted at 30° has a slope of 0.577 and a plan strain of 15 per cent — twenty times the dish’s — while being a plane, developable exactly, and flattened without error by any map that knows about the tilt. The plan is bad there because it is the wrong flattening, not because the surface resists flattening.

So the figure’s two readings need a third quantity, which is the one the essay’s own theorem supplies. The irreducible distortion is set by the Gaussian curvature, and Gauss’s own circumference formula puts a number on it: a geodesic circle of radius $\rho$ on a surface of curvature $K$ has circumference $2\pi\rho\left(1 - K\rho^{2}/6\right)$, so no flattening of that disc can do better than a strain of about

$$\frac{K\rho^{2}}{6}.$$

For the dish, $K = 0.0144$ per square metre over a patch of radius 1.1 m, that is 0.29 per cent.

Which changes the reading of the figure considerably. The dish’s plan strain is 1.03 per cent and its floor is 0.29, so roughly seven tenths of the dish’s error is avoidable too — not by an exact development, which does not exist, but by a better flattening, and the best available is three and a half times better than the plan. The honest three-way split is: the ridge’s 0.86 per cent is entirely avoidable, the dish’s 1.03 per cent is three quarters avoidable, and only the last 0.29 is the obstruction the theorem is about.

That is worth stating because “not developable” is often heard as “nothing can be done”, and the quantity that says how much can be done is $K\rho^{2}$ — the curvature times the area of the patch, near enough. It falls quadratically as the patch shrinks, which is why a dished floor tiled in small pieces flattens almost perfectly and in one piece does not, and why every atlas is a set of sheets. The same arithmetic decides how finely a curved picture has to be cut before it can be printed, and it is why the cartographic trade-off between angle and area is a trade-off at all rather than a solvable problem: the obstruction is real, it has a size, and the size is small enough to be beaten by subdivision and never by cleverness.

One consequence for this site’s own floors. The shadow measurements are taken on patches a couple of metres across, where $K\rho^{2}/6$ is a few parts in a thousand — so on those floors the intrinsic obstruction is smaller than almost every other error in the arrangement, and the departures being measured are extrinsic ones about how the surface sits in space rather than intrinsic ones about what it is. Two floors can have identical Gaussian curvature and completely different shadows, which is the sense in which the conformal surfaces and the developable ones are answering different questions about the same object.

Where it decides whether a question exists #

The classification is not an aesthetic one. It decides whether a recovery is well posed.

Recovering a design painted on a curved floor means returning the marks to the design’s own flat coordinates — the ones a signwriter measured with a flexible rule. On a ridge those coordinates exist and are unique, so “the flat design that was painted on it” names something definite, and the recovery has a right answer — with a tolerance on how well the floor has to be known attached to it.

On a dish they do not exist. There is no flat design; there are only flattenings, all of them distorting, none of them distinguished. A recovery computed through some chosen flattening of a dish is a perfectly good calculation that returns the choice rather than the design, and it will do so confidently and without residual.

So the first question to ask about a curved surface, before any rectification, is not how curved it is. It is whether its curvature is zero.

The same theorem, in the other field #

There is a version of this result the curved field has been living with since it was written, and connecting the two is worth a section because they look unrelated.

A map from the sphere of directions to a flat picture is a flattening of a sphere. The sphere has positive curvature. So no picture surface is an isometry of the sphere — no flat picture preserves all angular distances, and every one of them distorts.

That is Gauss’s theorem again, and it is the reason the comparisons in this field never have a winner. A flat plane keeps great circles straight and inflates area beyond all bounds; stereographic keeps every angle and inflates area; equal-area keeps area and bends every line. Each surface gives up something because something has to be given up, and the theorem says so before any of them is examined.

Measuring strain, rather than asserting it #

It is worth saying how “stretches nothing” is turned into a number, because the obvious method does not work and the failure is instructive.

The obvious method is to compare distances. Take points on the surface, measure the distance between each pair along the surface, compare with the distance between their images in the flattening, and report the worst ratio. That is the definition of an isometry and it is the wrong thing to compute, because distances along a surface are geodesic distances, and computing those needs either a solver or a mesh. A mesh introduces its own error of the same order as the effect being measured, so a developable surface would report a small non-zero strain that was entirely about the mesh, and the comparison with a genuinely curved one would be between one real number and one artefact.

The measurement used instead is local and exact. At each point, the surface has a metric — three numbers giving the lengths of and the angle between its own coordinate directions — and the flattening has a differential. Comparing the two gives the two principal stretch factors directly, as the roots of a quadratic, with no geodesics and no mesh anywhere.

That is why the ridge’s strain comes back at $10^{-9}$ rather than at $10^{-3}$: nothing has been discretised. And it is why the dish’s number is trustworthy too, since both are computed by the same code path from the same definition.

The cone, and the case that pays off later #

One developable surface is worth naming separately, because it turns up in an unexpected place.

A cone has zero Gaussian curvature everywhere except at its apex, where it has all of it at once. Cut a cone along a line from the apex and it unrolls to a sector of a disc, exactly, which is why a paper cup can be made from a flat sheet and a paper ball cannot.

That makes a conical mirror an unusually clean object: the metal can be made from a flat sheet, and its surface has an exact flat map. Which does not mean that a picture reflected in a cone is a rolled-up flat picture — the reflection is a different map from the development, and it turns out not to be a projectivity at all. The two facts sit beside each other, and confusing them is easy.

Piecewise, and why it is weaker than it sounds #

The step deserves its own paragraph, because “developable piecewise” reads like a small qualification and is a large one.

Two horizontal planes with a jump between them have zero curvature everywhere except on the seam, and each piece flattens exactly — indeed each piece’s plan is its development, since both pieces are already flat. By every test in this essay the step is the easiest of the four floors.

It is also the one that broke a four-point fit worst when a shadow was cast onto these same four surfaces: 74.95 mm, against 5.67 mm on the dish and 9.07 mm on the ridge. The smooth curved floor was an order of magnitude kinder than the one made of two planes.

The reason is that the questions are different. “Can this be laid flat?” is answered piece by piece and the step passes. “Is the map from this surface to a picture a projectivity?” is answered globally and the step fails hardest, because it is two projectivities glued along a line and there is nothing in a photograph to say where the join is.

So developability is necessary for a well-posed flat recovery and it is nowhere near sufficient. A surface can unroll perfectly and still defeat every method that assumes one map.

What to take away #

Look at the second derivatives, not at the shape. A surface that curves one way and is flat the other is developable however sharply it curves. A surface that curves both ways is not, however gently.

Developable means there is a right answer. A design on a developable surface has flat coordinates, and a recovery of it can be exact. On a curved surface there are no flat coordinates and a recovery returns whichever flattening it was given.

And the plan is a flattening. Reading a curved floor off its plan is not “ignoring the curvature”; it is choosing a particular map, one which happens to be exact when the floor is a plane and is otherwise a per cent or so out. On a developable floor a better one exists for the cost of an integral. On a curved one nothing better exists, and the honest thing is to say which flattening was used.