What can be unrolled

A cardboard tube can be slit lengthwise and pressed flat, and nothing tears, wrinkles or stretches. A paper cone can be made from a flat disc with a wedge removed. A ball cannot be made from flat paper by any method, and cannot be flattened by any method either.

The property separating the first two from the third has a name and a number.

Two kinds of curvature #

A surface bends by different amounts in different directions. At any point there is a direction of greatest bending and one of least, and those two numbers are the principal curvatures, $\kappa_1$ and $\kappa_2$.

Their product is the Gaussian curvature $K = \kappa_1\kappa_2$, and it is the one that cannot be changed by bending.

Their average is the mean curvature, which measures how the surface sits in space and changes freely when the surface is bent.

A cylinder curves in one direction and is straight along its axis, so $\kappa_2 = 0$ and therefore $K = 0$ whatever $\kappa_1$ does. The cylinder is as bent as one likes in space and intrinsically indistinguishable from a plane.

That distinction — bent in space, flat inside — is the whole of the idea, and it is why the everyday word “curved” is not precise enough to do any work here.

The three developable surfaces #

Surfaces with $K = 0$ everywhere are called developable, and up to a technical qualification there are only three kinds:

Cylinders, generalised: take any curve and sweep a straight line along it in a fixed direction.

Cones: take any curve and sweep a straight line through a fixed point.

Tangent developables: sweep the tangent lines of a space curve.

Every one is a ruled surface — through each point there is a straight line lying entirely in the surface — and it is that line that stays straight while the surface flattens. The flattening rotates the surface about each ruling in turn, which changes how it sits in space and none of the distances within it.

Which is why sheet metal is formed into cylinders and cones and pressed into anything else, and why a paper map of the world has to be printed as gores.

The separation is not marginal #

It would be reasonable to suspect that “zero curvature” is an idealisation and real surfaces are somewhere on a continuum.

They are not, in the cases that matter. The site’s library measures the plane, cylinder and cone at zero to within $10^{-9}$ — arithmetic noise — and the sphere at exactly 1 in its own units. That is a separation of nine orders of magnitude, and the gate asserts it: the sphere’s smallest measured curvature must exceed the cylinder’s largest by a factor of at least a thousand.

So developability is a genuine dichotomy rather than a threshold somebody chose. A surface either has a direction in which it does not bend, or it does not.

Why this matters for projections #

The traditional families of map projection are named after developable surfaces. Cylindrical projections are described as wrapping a cylinder round the globe; conic projections as fitting a cone over it; azimuthal ones as laying a plane against it.

The picture is memorable, and it invites a conclusion that is false. If the sphere is being wrapped in something that unrolls perfectly, why is there any distortion?

Because the sphere is not the cylinder. Getting the sphere’s surface onto the cylinder is where the distortion happens, and that step is not an isometry of anything — points are pushed outward along some rule, and the rule stretches. The cylinder’s later unrolling is faithful and there is nothing left to save by then.

This is worth being blunt about because the wrapping picture is how the subject is almost always introduced. The developable surface contributes nothing to the projection’s accuracy. It contributes the shape of the result — a rectangle, a fan, a disc — and the family name is a statement about output shape, not about fidelity.

Where the fiction shows #

Many projections in the “cylindrical” family are not obtainable by any geometric projection onto a cylinder at all.

Mercator is the clearest case. Its parallel spacing is $\ln\tan(\pi/4 + \varphi/2)$, which is not the result of projecting rays from anywhere onto anything — it is the function that makes the vertical stretch match the horizontal stretch, derived from the conformality requirement rather than from a construction. There is no light source that produces it.

Gall–Peters, Lambert’s cylindrical equal-area and the rest are similar: they are defined by the property they must have, and the cylinder is a description of the output rather than a recipe.

The genuinely geometric cylindrical projections are a small subset, and are mostly of historical interest.

Curvature that varies #

One more case, because it shows the concept is local rather than global.

Developability is a condition at every point. A torus has points where $K = 0$ and is not developable, because the condition fails elsewhere. The two circles where it vanishes are genuinely flat lines on a curved surface, which is a strange enough object to be worth noticing.

The same reasoning explains why a small piece of sphere can be approximately flattened while no piece can be flattened exactly. The total curvature over a small patch is small, so the distortion needed is small, and that is why a street map is fine and a world map is not.

Bending without stretching #

The precise notion behind all of this deserves a name, because it is what separates the two kinds of deformation.

A bending of a surface is a deformation that preserves all distances measured within it. Rolling paper into a tube is a bending; the paper’s fibres do not change length and a line drawn on it keeps its length.

A stretching does not. Pressing paper into a bowl shape requires the fibres to change length, which paper resists by wrinkling and tearing.

Gaussian curvature is invariant under bending and not under stretching, which is exactly why it separates the surfaces that can be flattened from those that cannot. Flattening is a bending, and a bending cannot change $K$, so a surface with $K \ne 0$ cannot be bent into a plane with $K = 0$.

Everything in this essay is that one sentence, applied.

Which way the failure goes #

Positive and negative curvature fail differently, and the difference is visible in what happens when the attempt is made.

Positive curvature — a sphere — has less surface near a point than a plane does. A circle of radius $r$ drawn on it has circumference less than $2\pi r$. Flattening it means the material has to cover more area than it has, so it tears.

Negative curvature — a saddle — has more. Circumference exceeds $2\pi r$, there is surplus material, and flattening it means the material has nowhere to go, so it buckles.

Kale leaves and lettuce edges buckle for exactly this reason: they grow faster at the rim than in the interior, which produces negative curvature, and the excess material ruffles. The same mathematics governs a crocheted hyperbolic plane, which is the standard way of making negative curvature tangible.

Why maps are printed in gores #

The practical consequence for cartography is the globe gore: the lens-shaped strips a paper globe is assembled from.

Each gore is narrow in longitude, so it covers a small fraction of the sphere and carries little total curvature, so it can be printed flat and glued onto a sphere with only slight stretching. Twelve or twenty-four gores, each a few degrees wide, and the accumulated error is below what paper and glue absorb.

That is the same scale argument as everywhere else in this subject: the distortion forced on a region is proportional to the curvature it carries, so small pieces are nearly flat and the whole is not.

Interrupted projections such as Goode’s homolosine are the map-making version of the same idea — cut the map where nobody is looking, usually through oceans, and each piece distorts less than the whole would.

The number of gores #

A practical consequence with an arithmetic answer, since globe gores are the standard demonstration.

A gore covering $\Delta\lambda$ of longitude and the full range of latitude carries total curvature proportional to its area. Halving its width halves the curvature and roughly halves the stretching needed to make it lie on a sphere.

Commercial globes use twelve gores of 30° each, and the residual stretching is within what damp paper and adhesive absorb. Precision globes have used twenty-four. Fewer than about eight and the paper visibly wrinkles at the equator; more than about thirty and the seams become the dominant visual feature.

So the number is a compromise between two failures, and it is settled by materials rather than by geometry — which is the usual situation once the geometry has said what it has to say.

The failure at the wide end is the same picture again: one very wide gore is the whole sinusoidal lobe drawn above, and its shape distortion at the outer edges is exactly why real gores are narrow.

Ruled but not developable #

A distinction worth making, because the two often get conflated.

Every developable surface is ruled — there is a straight line through each point lying in the surface. The converse fails. A hyperbolic paraboloid, the standard saddle, is ruled in two different ways and is not developable: its Gaussian curvature is negative, not zero.

So being made of straight lines is necessary and not sufficient. The rulings also have to be arranged so that the tangent plane is constant along each one, which is the extra condition that forces $K = 0$.

This matters in construction, where ruled surfaces are cheap to build from straight members and developable ones are cheap to clad in flat sheet. A hyperbolic-paraboloid roof can be framed with straight beams and cannot be covered with unstretched sheet, and both facts follow from the same classification.

Where the word is misused #

A small terminological warning, since “developable” appears in contexts where it means something looser.

In computer graphics and manufacturing, a surface is often called developable if it can be flattened approximately, within some tolerance, possibly with a small amount of stretching that the material tolerates. That is a useful engineering notion and it is not the geometric one.

The geometric condition is exact: Gaussian curvature identically zero. A surface with small curvature is not developable, it is nearly developable, and the distinction matters whenever the tolerance is tight or the surface is large.

A sphere is nearly developable over a small patch, which is why local maps work, and it is not developable at all — the curvature it carries is proportional to its area, so the approximation degrades predictably rather than holding until some threshold.

A closing observation about materials. Paper, sheet metal and woven cloth all resist stretching and tolerate bending, which is why the developable surfaces are exactly the shapes those materials take without special treatment. Knitted fabric stretches, which is why knitwear takes curved shapes that woven cloth needs darts to achieve — the geometry and the textile behave the same way because they are constrained by the same theorem.

What cannot be unrolled exactly can be unrolled approximately, and the approximation has a size that depends only on how much of the sphere is being flattened.

What was computed here #

Curvature is computed by two independent routes on every build — from the surface’s embedding in space, and from the first fundamental form alone — and the two must agree. That is Gauss’s theorem exercised rather than cited, and across the six surfaces it agrees to better than $2\times10^{-5}$.

The developability test is required to separate the surfaces rather than merely to pass. The gate checks that the flat ones come out flat, the curved ones come out curved, and that both groups are non-empty — a test that classified everything the same way would be worthless and would look identical in the output.

The unrolling figure carries its own arithmetic: the circumference of the wrapped grid must equal the width of the flat one, to $10^{-9}$. Without that check the picture would be asserting an isometry the numbers did not support, which is exactly the kind of claim this site exists to stop making.

What the pictures cannot show #

The surfaces are drawn in orthographic projection, which is a map projection with its own distortion. There is no way around it, and the figures say so.

More to the point, curvature is not visible. A cylinder and a sphere look comparably curved and are utterly different intrinsically, and no drawing can convey the difference — only the numbers can. That gap is why this essay leads with a computation and why the pictures carry their $K$ values as text.

The unrolling animation everyone imagines cannot be drawn either. The figure shows the two end states; the continuous family of isometries between them is real and is a motion, and a still page can only show where it starts and stops.

Who found it, and when #

Euler studied developable surfaces in the 1770s and established that they are exactly the ruled surfaces that can be flattened. Monge developed the theory further in the context of descriptive geometry, which he invented largely to solve fortification problems for the French military.

Gauss supplied the underlying reason in 1827: developability is the vanishing of a curvature that no bending can change. Before that the classification was a collection of facts about particular surfaces; afterwards it was one condition.

The practical knowledge is much older than any of it. Coopers, sailmakers and sheet-metal workers all knew which shapes could be made from flat stock and which required stretching, and the theory arrived to explain a craft that was already correct.

What a craft knew and could not say #

That the practical knowledge is older than the theory is the closing note of the history, and it deserves drawing out, because the relation between the two is not the one the ordering suggests.

The craft knowledge was correct and local. A cooper knows which staves can be cut from flat stock, a sailmaker knows which panels must be shaped, a sheet-metal worker knows which forms need stretching rather than folding. Each of those is the developability condition applied correctly to one family of shapes, learned by apprenticeship and by failure, and reliable within its trade.

What it could not do is transfer. A cooper’s knowledge does not tell a sailmaker anything, because the two crafts share no vocabulary and their shapes look nothing alike — so the same condition was discovered independently many times and each discovery stayed where it was made.

The theory’s contribution is exactly that transfer. One condition, stated once, covering every case any of those trades had learned separately and every case none of them had met. That is what a unification buys, and it is why the theory is worth having even though it corrected nobody: it did not improve any craftsman’s judgement, it made the judgement available to somebody looking at a shape no craft had tried.

And it supplies the negative half, which no craft could. A trade learns which shapes work by finding shapes that work; it has no way to establish that a shape cannot be made, only that nobody has managed it. Gauss’s condition says which are impossible, and an impossibility is the one kind of statement experience cannot produce.

The last of those observations is worth holding onto, because it inverts the order the subject is usually taught in. The craft was correct first: a cooper knows which staves can be cut flat, a sailmaker knows which panels must be broadseamed, a sheet-metal worker knows which shapes need stretching rather than folding — and every one of those judgements is the developability condition, applied reliably, centuries before anybody could state it. What Euler and Gauss supplied was not the knowledge but its unification: one condition covering every case a craftsman had learned separately, and a reason that survives being pointed at a shape nobody has tried.

Where this goes next #

The theorem underneath is no map is faithful. The families named after these surfaces are cylinders, cones and planes. And what the impossibility forces on every projection is the trade-off is two lines.