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

Theorema Egregium — where it appears

Gauss's theorem that a surface's curvature can be computed from measurements made within the surface alone. It is why no map of a sphere preserves every length, and it is exercised on every build here rather than cited, by computing the curvature twice along independent routes.

Named by 11 essays across 3 fields — each of them below, with the objects they name alongside it.

Six surfaces and their Gaussian curvature. Curvature 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: here the plane, the cylinder and the cone do, and the sphere, the torus and the pseudosphere do not.

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.

impossibility · Curvature The radius of equal curvature, from equator to pole. The radius of the sphere that has the same Gaussian curvature as the ellipsoid does, at each latitude. On WGS84 it runs from 6357 km at the equator to 6400 km at the pole — the polar region is the flattened part of a squashed ball and is therefore the LEAST curved — so K itself varies by 1.35%. A surface whose curvature varies cannot be laid on one whose curvature does not, so no map from the sphere to the ellipsoid is faithful either, and the least a conformal one can vary in scale is 6739 parts per million.

The curvature of the Earth is not one number

An ellipsoid's Gaussian curvature varies by 1.35% from equator to pole, so the sphere is not developable onto the ellipsoid any more than the plane is onto the sphere. The spherical approximation is a projection with an irreducible cost, and the cost is 6,739 parts per million.

impossibility · Curvature A direction carried once round the sphere. A vector transported round a closed loop on the sphere, kept as parallel to itself as the surface allows at every step — the component that leaves the tangent plane is removed and nothing else is done to it. The heavy arrows are its direction at the start and at the finish, drawn from the same point; the light ones are its direction along the way. It comes back turned through 282.0 degrees, which is 0.783 of a revolution, and nothing in the transport turned it. Drawn in an orthographic projection of the embedding, which is itself a map and has its own distortion.

A direction carried round a loop

Carry a bearing round a circuit, keeping it as parallel to itself as the surface allows, and it comes back turned. Round a parallel at 45° the turn is 4.443 radians and the cap enclosed is 1.840, and they sum to exactly one revolution — so the turning is not the curvature, and on a cone it is all of one and none of the other.

impossibility · Curvature How much curvature varies from place to place, body by body. The Gaussian curvature along a meridian, each body against its own smallest value, so the curves are comparable. Written from the bodies actually drawn: Earth varies by 1.0135, Mars varies by 1.0239, Vesta varies by 2.71, Phobos varies by 4.16, from the flattest place on each to the sharpest. A surface of constant curvature is a sphere and nothing else is, so the impossibility argument this field is built on has a local version on every real body: how flat a patch is depends on where the patch is.

Curvature that varies from place to place

The impossibility this site is built on was argued on a sphere, where the curvature is one number. On the Earth it varies by 1.35 per cent, on Jupiter by 31, on Vesta by a factor of 2.7 — and on a body with three axes it varies along a parallel, which no formula in latitude can express.

impossibility · Curvature Three surfaces with the same curvature, one of which is a sphere. Every one of these is a surface of revolution whose Gaussian curvature is 1 at every point, built by solving r″ + r = 0 for the meridian rather than by writing a shape down. The spindle closes to a point with an angle deficit, the sphere closes smoothly, and the bulge does not close at all — it ends in two circular edges. A surveyor confined to a patch of any of them, measuring angles and distances, cannot tell which one it is.

Two surfaces with the same curvature

The ladder's base says curvature is the obstruction to a faithful map. It has never asked whether curvature is the whole obstruction — and it is, locally: there is a whole family of surfaces with the unit sphere's curvature at every point, none of them a sphere, and a geodesic circle drawn on one agrees with the same circle on another to one part in 10¹⁴.

impossibility · Curvature One length, three corrugations. A straight segment shortened by a factor of 0.9, then wiggled across its own direction until its length is back to what it was. The wiggle's amplitude is what buys the length and the number of wiggles is free, so all three curves have exactly the target length while the third stays 16 times closer to the shortened segment than the first. That is the mechanism: the family converges to a map that is not isometric, while every member of it is.

Impossible in two derivatives, possible in one

The impossibility this whole collection rests on computes a second derivative, so it is a statement about maps that have two. Take one away and it is false: a corrugation restores an exact length while converging to the map that does not, and iterating it gives a flattening whose derivative converges and whose curvature runs to half a million.

impossibility · Curvature Everywhere Albers is exactly right, and the band round it. The set on which both principal scale factors are one — the only ground where a ruler on this map, at the map's own stated scale, measures the true distance in every direction. It is the parallels at 20.000° and 60.000°, drawn as a curve, with the band within 0.01 of true scale shaded round it. That band is 2.673 per cent of the sphere, and it narrows as ε as the tolerance tightens. The curve at its centre has no width at all, and no tolerance makes it have one.

The places where a map is exactly right

Nine essays on this ladder say a map cannot be right everywhere. None asks where it IS right — and the answer is a curve, a pair of curves, or two isolated places, never a patch. Measured across fourteen projections the set's neighbourhood shrinks with an exponent of 0.48, 1.0 or 2.0, and the value the impossibility forbids is 0.

impossibility · Curvature Which scale fields a map could have, and which are only wishes. Liouville's equation — the Laplacian of log k equals 1/k² on the page — is the whole condition for a conformal map of a unit sphere to have a stated scale factor. The first two rows are the scale fields of real projections and they satisfy it to the differencing step. The rest are requests a designer might write, and every one of them fails — except one, which turns out to be a projection somebody already found. Asking for no distortion anywhere fails by exactly one, which is the curvature of the sphere.

Not every distortion can be asked for

Six essays have written projections as conditions and asked how much freedom a condition leaves. The reverse question has never been put: a cartographer knows what distortion they want, so can they ask for it? For a conformal map the answer is a single equation, it is the Theorema Egregium in disguise, and asking for no distortion anywhere fails it by exactly the curvature of the sphere.

choosing · Condition The four places, and every distance between them. London, New York, Tokyo, Sydney on a Mollweide projection, with every great circle between a pair drawn. The six ground distances run from 5,570 km to 16,994 km, and they are the whole of the input to every figure in this ladder — no coordinate, no projection and no coastline enters any of them. The arcs are drawn only to say which pair each number belongs to; on this page they are curves, and on the ground they are the shortest routes.

Four cities that cannot be drawn to scale

Every impossibility in this field so far has been about a surface. This one is about four numbers: London, New York, Tokyo and Sydney have six distances between them, and no four dots on any sheet of paper have those six separations. The test is a determinant Cayley wrote down in 1841, and it comes out −3.34 × 10²³ where zero is required.

impossibility · Embedding The part of a request no conformal map can supply — scale falling with distance from the centre. The difference between the requested scale field and the nearest achievable one, over the patch, with the sign shown by colour. The RMS is 1.81e-1 in the logarithm of the scale and the worst single point is 4.91e-1. This is not an error: it is the part of the request that no conformal map of any kind can grant, and its SHAPE is the answer — it says where the request was impossible, which a single number cannot.

The nearest map to an impossible request

Rung seven found that not every distortion can be asked for. It never asked what happens when one is asked for anyway — and the answer has a shape: the achievable fields are the solutions of an elliptic equation, a request is a point off that set, and the nearest point leaves a residual whose floor is the curvature rather than the size of the ask.

choosing · Condition A surface whose curvature changes sign, and integrates to nothing. a wide ring of major radius 3 and minor radius 1, shaded by its Gaussian curvature. The outer half is positively curved like a sphere, at up to 0.250; the inner half is saddle-shaped and negative, down to -0.500; and the two circles between them, drawn as lines, are exactly flat. The integral of the curvature over the whole surface is zero, which is 2π times the Euler characteristic — and every impossibility in this collection rests on that number being 2 rather than 0.

On a body with a hole, north can be up everywhere

Twelve rungs vary the body's shape and none varies its topology, which is what every impossibility here actually rests on. A torus has a nowhere-zero tangent field, a total curvature of zero rather than 4π, and a conformal world map with no cut and no singular point — and it still cannot be flattened, for the one reason that survives.

datums · Bodies

Named alongside it

The objects these essays reach for when they reach for this one.

Gaussian curvatureIsometryIntrinsic geometryDevelopable surfaceClosed formConformalityDifferential equationFirst fundamental formScale factorTopologyVerificationBoundary-value

All concepts