No map is faithful

Every account of map projections opens by saying that a sphere cannot be flattened without distortion. It is almost always asserted, occasionally illustrated with an orange peel, and very rarely explained — which leaves the reader with no way to tell whether it is a deep fact or a practical difficulty that a sufficiently clever cartographer might one day get around.

It is a theorem, and the computation follows.

Curvature, computed two ways #

A surface has a quantity at each point called its Gaussian curvature, written $K$. There are two ways to get at it.

The first is extrinsic. Look at how the surface bends in the space around it: find the directions in which it curves most and least, multiply those two curvatures together, and that product is $K$. This route needs the surrounding space, because it needs a normal direction pointing out of the surface.

The second is intrinsic. Measure lengths and angles inside the surface, build the first fundamental form — the three functions $E$, $F$ and $G$ that say how far a small step in each coordinate actually goes — and compute

$$K = -\frac{1}{2\sqrt{EG}}\left[\frac{\partial}{\partial u}\!\left(\frac{G_u}{\sqrt{EG}}\right) + \frac{\partial}{\partial v}\!\left(\frac{E_v}{\sqrt{EG}}\right)\right]$$

Nothing on the right refers to a third dimension. A two-dimensional creature confined to the surface, with a ruler and no concept of an outside, could carry out every measurement this formula needs.

That the two routes give the same number is Gauss’s Theorema Egregium — the remarkable theorem, and he chose the adjective himself. Both are implemented in this site’s library and required to agree on every build; across all six surfaces above they agree to better than $2\times10^{-5}$.

Why that settles it #

An isometry is a map that preserves distances. If distances are preserved then all the measurements feeding the intrinsic formula are preserved, so $K$ is preserved.

A sphere of radius $R$ has $K = 1/R^2$ everywhere, which is positive. A plane has $K = 0$ everywhere. They are different numbers, so no isometry exists between any patch of the sphere and any patch of the plane.

That is the whole argument. Two lines, and it does not depend on how the map is constructed, how clever the projection is, or how small the region — a postage-stamp piece of sphere still has $K = 1/R^2$ and still cannot be laid flat without stretching something.

This is why the impossibility is a different kind of statement from the practical difficulties elsewhere on this site. Which properties to trade off is a design question; that something must be traded is not.

What can be flattened #

The theorem cuts the other way too, and the consequence is more surprising than the prohibition.

A cylinder is curved. It curves in one direction and is straight in the other, so one of its two principal curvatures is zero, so their product is zero. $K = 0$, and a cylinder is intrinsically flat.

The same is true of a cone. Both can be cut from a flat sheet and rolled up with nothing stretched, which is why paper cones and paper tubes exist and paper spheres do not. Surfaces of this kind are called developable, and the computation separates them from the others cleanly: the site’s library measures the plane, cylinder and cone at zero to within $10^{-9}$, and the sphere at exactly 1 in its own units — a separation of nine orders of magnitude.

This has a direct bearing on the traditional vocabulary of projections. Cylindrical and conic projections are named for developable surfaces, and the names suggest the sphere is being wrapped in something that unrolls faithfully. It is not. The sphere is not a cylinder; projecting it onto one is not an isometry of anything; and the fact that the cylinder later unrolls perfectly does not repair the damage done in getting the sphere onto it.

Curvature that changes sign #

Not every surface has one curvature.

The sign has a meaning that survives without any reference to a surrounding space. Positive curvature means triangles have angles summing to more than 180°, and circles have circumference less than $2\pi r$. Negative means the opposite. Zero means Euclid.

Which gives the two-dimensional creature a way to find out where it lives: draw a large triangle, add the angles, compare with 180°. The excess is the integral of $K$ over the triangle, which is the Gauss–Bonnet theorem, and it is why the intrinsic route is not a mathematical curiosity but a genuinely available measurement.

The same argument, in the other direction #

There is a converse worth stating, because it explains why so many different projections exist rather than one best compromise.

Two surfaces with the same curvature everywhere are locally isometric. A cylinder and a plane both have $K = 0$, and a patch of one really can be laid on the other exactly. So the obstruction is entirely the curvature and nothing else — there is no additional condition, no residual difficulty beyond it.

Which means the impossibility is exactly as large as the curvature and no larger. A small region of sphere has a small total curvature, and can be mapped with small distortion; that is why a street map is fine and a world map is not. The distortion in any map is bounded below by the curvature it has to absorb, and a projection’s job is to decide where to put it.

How much distortion is forced #

The theorem says distortion is unavoidable. It also says how much, which is less often mentioned and more useful.

Total curvature over a region is the integral of $K$ over it, and for a sphere that is just the area divided by $R^2$. A region covering a fraction $f$ of the sphere carries total curvature $4\pi f$, and that is the amount any flat map of it must absorb.

Which gives a scale rule with real content. A city block is about $10^{-10}$ of the Earth’s surface, so the distortion any projection must impose on it is of that order — utterly negligible, and the reason surveyors treat small areas as flat without apology. A country might be $10^{-3}$, where the distortion is small and worth managing. A hemisphere is $0.5$, where it is unavoidable and large.

The plate carrée measured along a meridian is exact at the equator and nowhere else, with both distortions growing steadily with latitude — an unmanaged version of the growth every projection has to exhibit somewhere.

So the familiar advice that projection choice matters more for larger areas is not a rule of thumb. It is the curvature integral, and the transition from “does not matter” to “dominates everything” happens over about six orders of magnitude of area.

The bound, made exact #

For one case the forced distortion can be written down exactly rather than bounded, and it is the case that matters most in practice.

Take a spherical cap of angular radius $\rho$ and ask for the conformal map of it whose scale varies least. Chebyshev’s criterion picks it out — by symmetry it is the stereographic projection centred on the cap — and the ratio between the scale at the rim and the scale at the centre is

$$\frac{k(\rho)}{k(0)} = \sec^2\frac{\rho}{2}$$

That is not the distortion of a particular projection. It is the smallest any conformal projection of that cap can achieve, and every other one is worse.

The numbers say why the subject behaves as it does. A cap of radius 0.01° — about a kilometre — is forced to vary by 7.6 parts per billion, which no survey instrument has ever resolved. At 1° it is 76 parts per million, half a metre in seven kilometres, which a modern survey does resolve. At 10° it is 0.77%. At 45° it is 17%. Over a hemisphere the scale must double.

The bound grows as $\rho^2/4$ for small caps, which is to say proportionally to area, which is to say proportionally to the total curvature enclosed. The theorem and the engineering rule of thumb are the same statement.

What an isometry would have to be #

It is worth being concrete about the object the theorem rules out, because the impossibility is easier to feel with a specific target.

A faithful map would preserve every distance between every pair of points. That is much stronger than preserving areas or angles: it would mean a ruler laid on the map and a tape measure laid on the ground always agreed, everywhere, in every direction.

Such a map exists between a cylinder and a plane. Cut a paper tube and flatten it, and every distance measured along the paper is unchanged — the paper knows nothing happened. That is what an isometry feels like from inside, and it is available because both surfaces have $K = 0$.

Between a sphere and a plane it is not available, and the failure is not gradual. There is no nearly isometric map that gets closer with cleverness; the obstruction is a number that does not match, and numbers either match or they do not.

An isometry would draw every indicatrix on that map as an identical circle of the same size as the sphere’s own. None of them is, and the theorem above says none of them could be.

The orange peel, and why it is a bad demonstration #

The standard illustration is to peel an orange and try to flatten the skin. It tears, and the tearing is offered as proof.

It does demonstrate something real and it demonstrates it for the wrong reason. Orange peel is thick, stiff and inelastic, and a thin flexible membrane would behave differently — which invites the thought that a better material would work.

It would not, and the theorem is why. The obstruction is not the material’s stiffness but the surface’s curvature, and it applies to an idealised membrane with no thickness and no resistance to bending. That is the content of an intrinsic quantity: it is a property of the geometry rather than of anything made of it.

A better demonstration is the one in the developable figure above — a cylinder that does flatten perfectly, made of the same kind of paper that fails on a sphere. The contrast is where the argument lives.

The theorem’s reach #

Gauss’s result is not confined to cartography, and knowing where else it bites makes the map case feel less like a special difficulty.

Any attempt to flatten a curved surface meets it. Sheet metal is formed into cylinders and cones and stamped into anything else, because stamping stretches. A tailor’s pattern is cut in flat pieces and the curvature of a shoulder is built from darts and seams — which are tears, arranged deliberately. A cardboard box is a developable surface, and so is every packaging form that folds rather than moulds.

It also runs the other way. General relativity describes gravity as spacetime curvature, and the reason curvature can be a physical quantity rather than a description of an embedding is exactly the intrinsic property Gauss found. There is no fifth dimension that spacetime curves into; the curvature is measurable from inside, and Gauss’s theorem is why that sentence means anything.

That is the strongest form of the argument and it is worth its own essay: the quantity a projection is fighting is not merely conserved but determined by topology, so it answers to something no amount of geometric cleverness can reach.

What a cartographer does with it #

The practical residue is a way of thinking about the job.

The curvature to be absorbed is fixed by the region. It cannot be reduced, only placed. So the projection’s design question is not how to minimise the distortion but where to put it — and that is a question with an answer, because a map has a subject and the distortion can be pushed away from it.

Standard parallels place the zeros. The aspect rotates the whole pattern. Interruption cuts the region into smaller pieces each carrying less. All three are placement decisions, and none of them is an attempt to beat the theorem.

The argument above is stated for a sphere, whose curvature is one number. The body geodesy actually uses is not, and the theorem applies to it with more force rather than less.

A theorem that forbids zero error says nothing about how large the error is, and the second question has an answer that grows quadratically with the size of the map.

What was computed here #

Every curvature on this page is computed, both ways, at build time.

The extrinsic route differentiates the surface’s parameterisation twice, builds the first and second fundamental forms, and takes $(LN - M^2)/(EG - F^2)$. The intrinsic route builds only the first fundamental form and applies the formula above. Requiring them to agree is the theorem, exercised rather than cited, and it is the check that would catch an error in either.

Three further assertions guard the result. Each surface’s computed curvature must match its known closed form — $1/R^2$ for the sphere, zero for the cylinder and cone, $\cos v/(a(b + a\cos v))$ for the torus. The formula used assumes an orthogonal parameterisation, which is checked rather than assumed, since it would silently return the wrong number otherwise. And the developability test must separate the surfaces rather than passing everything: the gate requires the flat ones to come out flat, the curved ones to come out curved, and the sphere to miss flatness by a factor of at least a thousand.

The cylinder unrolling figure carries its own check: the circumference of the wrapped grid must equal the width of the flat one, to $10^{-9}$. If it did not, the picture would be claiming an isometry that the numbers did not support.

What the pictures cannot show #

The surfaces are drawn in orthographic projection, which is itself a map projection with its own distortion. There is no way round this — the page is flat, and every picture of a curved surface on it has already made the compromise the essay is about. The figures state it where it matters.

More fundamentally, curvature is not a visual property. The pictures show shapes, and the numbers underneath them are the content; a cylinder and a sphere look comparably curved and are utterly different intrinsically. That gap between appearance and quantity is the reason the essay leads with a computation rather than an illustration.

And the intrinsic route cannot be shown at all, only described. Its whole point is that it uses no information from outside the surface, and a drawing of a surface is a drawing from outside.

Surfaces that are not parallel either #

The same non-parallelism turns up one level out and is easy to miss. A surface of constant gravitational potential drawn along a quarter meridian sits a kilometre above the ellipsoid at the equator and 994.7 metres above it at the pole: the surfaces a spirit level follows are not parallel to each other, by five parts in a thousand.

The impossibility this essay derives is about flattening a curved surface onto a plane. There is a second one stacked underneath it, and it is about the third coordinate: the surfaces height is measured from converge polewards, so a chain of perfectly executed levelling observations does not sum to a height difference. A levelled height is not a distance measures the correction, which over four hundred kilometres is larger than the survey’s own closure.

Who found it, and when #

Gauss published the Disquisitiones generales circa superficies curvas in 1827, and the theorem is the centrepiece. He had spent years on the geodetic survey of Hanover, measuring triangles across a landscape, and the practical question of how a curved Earth’s measurements relate to a flat map was directly in front of him.

The Latin title he gave the result — theorema egregium — is usually translated “remarkable theorem”, and the choice of adjective is unusual for a mathematician describing his own work. The remarkable part is not that curvature exists but that it is intrinsic: a quantity apparently about how a surface sits in space turns out to be computable without any space at all.

Riemann generalised the idea in 1854 into what became Riemannian geometry, and that machinery is what general relativity is written in. The line runs directly from a survey of Hanover to the curvature of spacetime.

Where this goes next #

The developable case is what can be unrolled. The consequence for what a projection can preserve is the trade-off is two lines. And for the measurement that turns all of this into numbers about particular maps, Tissot’s indicatrix.