Cartographic Projection
every map is wrong, and by exactly how much
The impossibility

What can be unrolled

A cylinder is obviously curved and is intrinsically flat, so it lays out on a table with nothing stretched. A sphere is not. The distinction is exactly zero Gaussian curvature, and it is why paper tubes exist and paper globes do not.
15 min read 8 figures A theorem, not a limitationComputed, not quoted

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.

A cylinder unrolls exactly; a sphere does notBoth pictures show the same grid. On the left it is wrapped round a cylinder, on the right it is laid flat, and every distance in the grid is the same in both — the circumference is 2π and so is the width of the rectangle, checked to 10⁻⁹. This is possible because a cylinder has zero Gaussian curvature. No corresponding picture exists for a sphere.wrappedunrolled — every distance unchanged→K = 0 on bothcircumference 2π = width 2π
Fig. 1 The same grid, wrapped round a cylinder and laid flat. Every distance is identical in both — the circumference is 2π and so is the width of the rectangle, checked to a part in a billion. The cylinder is curved in space and flat in the only sense that matters here.

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, κ1\kappa_1 and κ2\kappa_2.

Their product is the Gaussian curvature K=κ1κ2K = \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 κ2=0\kappa_2 = 0 and therefore K=0K = 0 whatever κ1\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=0K = 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.

Six surfaces and their Gaussian curvatureCurvature computed at the centre of each surface, by both available routes — from the way the surface sits in space, and from distances measured inside it alone. The two agree, which is Gauss's theorem. A surface with K = 0 can be unrolled flat without stretching; the cylinder and the cone can, and the sphere cannot.planeK = 0 — unrolls flatcylinderK = 0 — unrolls flatconeK = 0 — unrolls flatsphereK = 1.00torusK = 1.79pseudosphereK = -1.00K computed at each centretwo routes, one answer
Fig. 2 Six surfaces with their computed curvature. The plane, cylinder and cone measure zero to within 10⁻⁹ and unroll exactly. The sphere measures 1 in its own units, the torus varies in sign, and the pseudosphere is constant and negative. None of the last three can be flattened.

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 10−910^{-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.

6 projections of the same sphereThe same graticule under equirectangular, mercator, gallPeters, sinusoidal, mollweide, hammer. Every one of them is a faithful drawing of the same object and no two agree, because each has chosen a different thing to preserve.EquirectangularMercatorGall–PetersSinusoidalMollweideHammersame sphere, same graticuleno two agree
Fig. 3 Six projections, three of them cylindrical by construction. They are not equally good and are not equally anything — the family says how the graticule is laid out and almost nothing about what is preserved.

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⁡(π/4+φ/2)\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.

Gaussian curvature across a torusCurvature along a cross-section of a torus. curvature of both signs — positive outside, negative inside, zero on two circles. Where the curve crosses zero the surface is momentarily flat in the intrinsic sense, and a strip along that circle could be unrolled without stretching.0+4.2position around the cross-sectionpositive: dome · negative: saddleon a torus
Fig. 4 Gaussian curvature across a torus. Positive on the outside, negative on the inside, and exactly zero on the two circles between. A narrow strip along one of those circles could be unrolled without stretching, and no larger piece could.

Developability is a condition at every point. A torus has points where K=0K = 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 KK, so a surface with K≠0K \ne 0 cannot be bent into a plane with K=0K = 0.

Everything in this essay is that one sentence, applied.

Gaussian curvature across a pseudosphereCurvature along a cross-section of a pseudosphere. constant negative curvature — the surface that is everywhere a saddle. Where the curve crosses zero the surface is momentarily flat in the intrinsic sense, and a strip along that circle could be unrolled without stretching.0+1.0position around the cross-sectionpositive: dome · negative: saddleon a pseudosphere
Fig. 5 The pseudosphere, with constant negative curvature. It cannot be flattened either — negative curvature is as much an obstruction as positive. A saddle has too much surface near a point rather than too little, and flattening it would require compression rather than tearing.

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 rr drawn on it has circumference less than 2πr2\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 2πr2\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.

SinusoidalThe graticule of the Sinusoidal projection at 30° of longitude and 15° of latitude. equal-area, and the parallels keep their true length. It is equal-area.equal-areadrawn in Sinusoidal
Fig. 6 The sinusoidal projection, which is what a single uninterrupted gore-like construction looks like. It is equal-area everywhere and the shape distortion at the outer edges is severe — which is why the interrupted versions cut it into lobes and discard the worst parts.

The number of gores

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

A gore covering Δλ\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.

SinusoidalThe graticule of the Sinusoidal projection at 30° of longitude and 15° of latitude. equal-area, and the parallels keep their true length. It is equal-area.equal-areadrawn in Sinusoidal
Fig. 7 A single sinusoidal lobe covering the whole world is what one very wide gore would look like. The shape distortion at the outer edges is severe, which 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=0K = 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.

Gaussian curvature across a torusCurvature along a cross-section of a torus. curvature of both signs — positive outside, negative inside, zero on two circles. Where the curve crosses zero the surface is momentarily flat in the intrinsic sense, and a strip along that circle could be unrolled without stretching.0+4.2position around the cross-sectionpositive: dome · negative: saddleon a torus
Fig. 8 Curvature varying across a torus, passing through zero twice. The two circles where it vanishes are ruled and locally flat, and a narrow strip along either could be unrolled — a reminder that developability is a condition at each point rather than a property of a whole surface.

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 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 2×10−52\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 10−910^{-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 KK 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.

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.

Where this goes next

planeK = 0 — unrolls flatcylinderK = 0 — unrolls flatconeK = 0 — unrolls flatsphereK = 1.00torusK = 1.79pseudosphereK = -1.00K computed at each centretwo routes, one answer No map is faithful Not "no map yet", and not "no map at page size". Gaussian curvature can be computed from inside a surface, a sphere has some and a plane has none, and that closes the question permanently. conformalequal-areacompromiseclaim failscylindricalMercatorMercatorLambertGall–PetersBehrmannEquirectangularMillerWebpseudocylindricalSinusoidalMollweideEckertRobinsonpseudoazimuthalHammerWinkelazimuthalStereographicLambertOrthographicGnomonicAzimuthalconicLambertAlbersrows: how it is builtcolumns: what it preserves Cylinders, cones and planes The standard taxonomy sorts projections by the shape of paper they were notionally rolled from. It is memorable, it is how everyone is taught, and it says almost nothing about the properties anyone actually chooses on. 10⁻⁶10⁻⁴10⁻²110010⁻¹¹10⁻⁸10⁻⁵10⁻²1010000maximum angular deformation / degreesmaximum areal errornothing here, everMercatorStereographicWeb MercatorGall–PetersMollweideEquirectangulartolerances: 0.0001° and 0.00000121 projections measured The trade-off is two lines Conformal means the two principal scales are equal. Equal-area means their product is one. Both at once forces both to one, which is an isometry, which the curvature forbids. That is the entire argument.