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

Gaussian curvature, and what it forbids

surface-curvature computes it two ways on six surfaces and requires the answers to agree, which is Theorema Egregium exercised on every build. developable is the zero-curvature case, curvature-field the case that changes sign, curvature-integral and triangle-excess the integrated forms, holonomy-split the direction carried round a loop, same-curvature two surfaces that share a number and not a shape, and plane-survey the practical question of how small a region has to be before the curvature stops mattering. The twenty-one distance- modes make the same argument with a ruler instead of a derivative: a set of distances measured on the sphere is the distance set of no flat picture in any dimension — distance-determinant and distance-spectrum are the test, distance-diameter the square law, distance-counting the 2n − 3 a flat picture can hold, and distance-curved and distance-curvatureaxis the one page that holds them all, which is the sphere at its own radius.

surface-curvature computes it two ways on six surfaces and requires the answers to agree, which is Theorema Egregium exercised on every build. developable is the zero-curvature case, curvature-field the case that changes sign, curvature-integral and triangle-excess the integrated forms, holonomy-split the direction carried round a loop, same-curvature two surfaces that share a number and not a shape, and plane-survey the practical question of how small a region has to be before the curvature stops mattering. The twenty-one distance- modes make the same argument with a ruler instead of a derivative: a set of distances measured on the sphere is the distance set of no flat picture in any dimension — distance-determinant and distance-spectrum are the test, distance-diameter the square law, distance-counting the 2n − 3 a flat picture can hold, and distance-curved and distance-curvatureaxis the one page that holds them all, which is the sphere at its own radius.

Every one of these is the same generator answering a different question, which is why they share a file, a set of colour roles and a set of assertions. Changing it changes all 29.

22 essays call it. Every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

What it draws

One picture per question the family answers. Each has a page of its own.

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.

surface-curvature

Six surfaces and their Gaussian curvature A cylinder unrolls exactly; a sphere does not. Both pictures show the same grid. On the left it is wrapped round a cylinder of radius 1, 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.

developable

A cylinder unrolls exactly; a sphere does not Gaussian curvature across a torus. Curvature 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.

curvature-field

Gaussian curvature across a torus The angles of a triangle, and the curvature inside it. A triangle on the sphere with sides that are great-circle arcs. Its three angles sum to 212.26°, overshooting the flat 180° by 32.26°. Integrating the curvature over the interior gives 0.56306 against an excess of 0.56306 — the same number by two routes that share no arithmetic. That is Gauss–Bonnet, and it is how a flatlander measures the curvature of a world it cannot step outside.

triangle-excess

The angles of a triangle, and the curvature inside it Total curvature is a counting number in disguise. The Gaussian curvature of each surface, integrated over the whole of it. Gauss–Bonnet fixes the answer at 2πχ, where χ is the Euler characteristic — vertices minus edges plus faces, a quantity with no geometry in it at all. The sphere gives 12.5664 against 4π = 12.5664. The torus gives -2.4e-11 against zero, and it does so by cancellation: its outer half is positively curved and its inner half negatively, in exactly equal measure.

curvature-integral

Total curvature is a counting number in disguise What a loop round a parallel turns a direction by, and what it encloses. Two quantities against the latitude of the loop. The turning — the angle a transported direction comes back rotated by, relative to the local north — is 2π sin φ, measured. The curvature the loop encloses is the area of the cap above it, 2π(1 − sin φ). They sum to 2π at every latitude, drawn as the flat line across the top, and neither is the other: at 40° the turning is 4.039 radians and the enclosed curvature 2.244. The difference is the winding of "north" itself around the pole, which is exactly one revolution however small the loop.

holonomy-split

What a loop round a parallel turns a direction by, and what it encloses 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.

same-curvature

Three surfaces with the same curvature, one of which is a sphere How large a patch can be treated as flat, at 10 parts per million. The smallest scale distortion any map of a circular patch can have, against the radius of the patch, on logarithmic axes. The line is straight with a slope of 2.00: the error grows as the SQUARE of the size, so a patch ten times wider is a hundred times worse. A tolerance of 10 ppm is reached at a radius of 40.3 km — 81 km across — and that is the number behind the boundary between plane surveying and geodesy.

plane-survey

How large a patch can be treated as flat, at 10 parts per million 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.

distance-quadmap

The four places, and every distance between them The best flat picture of them, and what it still gets wrong. The four places placed on a plane so that the WORST of the six distances is as nearly right as any flat arrangement can make it. There is no projection here and no map: the dots carry no coordinates, only separations. Each edge is labelled with how far its drawn length is from the truth once the picture has been given its one free scale. The best any arrangement achieves is 0.629% on the worst edge — London to Tokyo — and no rearrangement lowers it, because the six numbers are not the distances of four points in a plane at all.

distance-flat

The best flat picture of them, and what it still gets wrong The spectrum that decides whether a picture exists. The four eigenvalues of the double-centred matrix of squared distances, for the same four places measured two ways. Drawn straight through the rock, the chordal distances give three positive eigenvalues and no negative one, which is Schoenberg's statement that they are the distances of four points in three dimensions — and so they are, because the places are already there. Measured along the surface, one eigenvalue is negative: -2.165e+6 against a largest of 1.879e+8. A negative eigenvalue is not a large error. It is the statement that no Euclidean space of any dimension holds these distances.

distance-spectrum

The spectrum that decides whether a picture exists Every three of them fit, and the four together do not. Each subset of the four places, with the smallest eigenvalue of its own double-centred distance matrix — the quantity that is zero exactly when a flat picture exists and negative when none does. Every one of the four triangles sits at zero to rounding, which is the triangle inequality doing what it always does. The one four-place subset does not, and the bar it draws is the whole of the impossibility this ladder is about. Nothing about the places was chosen to make this happen; it happens to any four points on a sphere that are not on one great circle.

distance-triples

Every three of them fit, and the four together do not The determinant that has to be zero, and is not. The Cayley–Menger determinant of the four places, divided by the sixth power of their diameter so that only the shape is left, against how much of the sphere they span. Zero is what a flat picture requires; the chordal distances give it to rounding at every size, and the geodesic ones never do. The fitted slope is 2.149 on the log axes, so the departure falls as roughly the square of the span — the same exponent every other measurement in this ladder finds, arriving here through a determinant rather than through an optimisation.

distance-determinant

The determinant that has to be zero, and is not The least error a flat picture can have, against how much sphere it spans. The same configuration of places, shrunk about its own centroid so that every bearing is kept and only the span changes, with the least worst-case relative error of the best flat picture at each size. Both axes are logarithmic. The fitted slope over the rows below ninety degrees is 2.0246: the error falls as the SQUARE of the diameter. That is Gauss's theorem arriving as a number for a finite set — curvature is a second derivative, so its first effect on a distance is quadratic in the separation — and it is why a county fits on a sheet and a hemisphere does not.

distance-diameter

The least error a flat picture can have, against how much sphere it spans The same arrangement at three sizes. The best flat picture of London, New York, Tokyo, Sydney, with the configuration shrunk about its own centroid so that every bearing from the centre is unchanged and only how much of the sphere it covers varies. At 152.8° across the worst edge is out by 0.629%; at 39.7° it is 0.0299%; at 9.9° it is 0.0018%. The pictures are indistinguishable to the eye and the numbers under them fall by a factor of sixteen for each factor of four, which is what a square law looks like when it is drawn rather than plotted.

distance-shrink

The same arrangement at three sizes Three ways of scoring the same picture, and one exponent. The same ladder of spans scored three ways: the worst edge of the best possible arrangement, the worst edge of the arrangement classical multidimensional scaling produces, and the root-mean-square error over all the edges. The three curves are parallel and are not the same curve — classical scaling sits a factor of 1.31 above the optimum at every span, and the root-mean-square sits below both because averaging hides the edge the minimax is about. The fitted exponents are 2.0246, 2.0223, 2.0111: the constant in front is a property of the estimator and the exponent is a property of the sphere.

distance-estimators

Three ways of scoring the same picture, and one exponent The exponent belongs to the sphere and the constant to the arrangement. The square law fitted separately on five configurations of places — from five towns inside one country to sixteen cities on every continent — each shrunk through its own ladder of spans. The exponents agree: 2.025, 2.016, 1.999, 2.013, 1.978. The constants in front of them do not, and span a factor of 17. That separation is the point of running five sets rather than one: how badly a particular arrangement of places resists a flat sheet is a fact about the arrangement, and that the resistance falls as the square of the span is a fact about the surface they sit on.

distance-sets

The exponent belongs to the sphere and the constant to the arrangement Every projection against a picture that is not a map. The worst relative distance error over London, New York, Tokyo, Sydney, for each projection in the library and for the best flat arrangement of the same distances. Each azimuthal member is centred on the set's own centroid, which is the fair comparison. The free picture reaches 0.63% and the best projection, Azimuthal equidistant, reaches 8.28% — a ratio of 13.17. The free picture cannot lose, because every projection's own layout was handed to the search as a starting point; what the figure measures is how much the freedom is worth, and it is worth different amounts at different sizes.

distance-library

Every projection against a picture that is not a map The same places, arranged by a map and arranged by nothing. On the left, London, New York, Tokyo, Sydney where Azimuthal equidistant puts them, centred on the set's own centroid. On the right, the arrangement that makes the worst of the six distances as nearly right as any flat arrangement can. The edges that carry more than half the worst error are drawn heavier. The projection reaches 8.28% and the free picture 0.63%. The free picture is better and is not a map: it has no graticule to hang a coastline on, nothing to say about any place not in the set, and adding a seventeenth city moves every dot in it.

distance-against

The same places, arranged by a map and arranged by nothing What abandoning the projection is worth, and how fast it stops being worth it. For each set of places, the worst-edge error of the best projection in the library divided by the worst-edge error of the best flat arrangement of the same distances. The free picture cannot lose, because every projection's own layout is handed to the search as a starting point. On four world cities the freedom is worth a factor of 13.2; on sixteen it is worth 1.08, which is eight per cent. What collapses it is arithmetic rather than cartography: a picture of n places has 2n − 3 free numbers against n(n−1)/2 distances, so the freedom available per constraint falls as 4/n, and by sixteen places the optimiser has almost nothing left to spend that the projection has not already spent.

distance-advantage

What abandoning the projection is worth, and how fast it stops being worth it The share of its distances a flat picture can hold. A picture of n places on a plane has 2n coordinates and is unchanged by two translations and a rotation, so 2n − 3 numbers in it are genuinely free; each distance held exactly is one equation. So at most 2n − 3 of the n(n−1)/2 distances can be right, whatever the places are and however hard the map maker tries. The share falls as 4/n: five of six for four places, thirteen of twenty-eight for eight, and 197 of 4,950 — 4.0% — for a hundred. The dashed curve is 4/n, which the count approaches from below and never crosses.

distance-counting

The share of its distances a flat picture can hold The distances a construction holds, and the rest it decides. The construction a surveyor would use, on London, New York, Tokyo, Sydney. The first place is put down anywhere, the second at its true distance along an axis, and every further place is trilaterated from those two — which holds 5 distances exactly, drawn heavy, and that is 2n − 3 for n = 4. Nothing is fitted. The remaining 1 distances, drawn thin, are then decided rather than chosen, and on places taken from a sphere they are decided wrongly: the worst is out by 6.87% and the closest to right by 6.867%.

distance-trilateration

The distances a construction holds, and the rest it decides What a map centred on a place spends, and what it leaves unspent. An azimuthal equidistant map centred on London is exactly right about every distance FROM London — measured here at 6.7e-16 on three of them, which is rounding — and about nothing else: the worst of the other 3 is out by 32.3%. Those n − 1 exact distances are the star, and the plane allows 2n − 3. The gap between the two curves is n − 2 distances a flat picture could have held and no projection centred on a place does, because a projection is a rule about the whole sphere and cannot spend its freedom on the places that happen to be in the set.

distance-star

What a map centred on a place spends, and what it leaves unspent The distances a centred map holds, drawn on the page that holds them. London, New York, Tokyo, Sydney on an azimuthal equidistant projection centred on London. The three heavy spokes are exactly right: the projection's radial coordinate IS the angular distance from its centre, so those separations are true to the last bit of the arithmetic — measured at 6.7e-16. Every thin chord is wrong, the worst by 32%, and the reason is that a straight line between two points of this page is not a route on the ground. 3 exact of 6: the map has spent n − 1 of the 2n − 3 a flat picture is allowed.

distance-starmap

The distances a centred map holds, drawn on the page that holds them What each extra dimension buys, which is nothing. The worst relative error of the best k-dimensional picture of London, New York, Tokyo, Sydney, for every k the set allows. Going from a line to a plane buys almost everything — from 180.2% to 1.27%. Going from a plane to a space buys nothing at all: the two numbers agree to every digit, because the third eigenvalue the extra dimension would spend is 2.85e-8 — zero — and the eigenvalue that would help is negative and cannot be spent in a Euclidean space of any dimension. The flat sheet is not the obstruction. Euclid is.

distance-dimensions

What each extra dimension buys, which is nothing How much of the spectrum has the wrong sign. The signature of the double-centred squared-distance matrix for five sets of places, one row each, one cell per eigenvalue, ordered largest first. Filled cells on the left are positive, pale cells are zero, and the cells on the right are negative — the eigenvalues that no Euclidean space of any dimension can hold. Four places give one; sixteen cities give 8 of 16. The chordal distances of every one of these sets give three positive, no negative, at every size, which is the refusal that makes the geodesic column mean something rather than merely being a picture of a spectrum.

distance-signature

How much of the spectrum has the wrong sign The page's curvature is not a free parameter. The same trilateration, carried out on a sphere of stated radius instead of on a plane, with the error of the distances the construction decides rather than holds. At the radius the distances were measured on it is 3.3e-16 — exact, by construction, which is the refusal this figure exists to make. Two per cent either side of it the worst decided distance is already out by around 4%. Below 90 per cent of the Earth's radius the construction cannot be completed at all: two circles that must cross do not, and the page is simply too small to hold the places. The flat sheet is the right-hand limit, at 7%.

distance-radius

The page's curvature is not a free parameter Five pages of different curvature, and the one that works. The worst decided distance on a page of each stated radius, for London, New York, Tokyo, Sydney. A page too small is refused rather than fitted — the construction reaches a step where two circles that must cross do not, which is a geometric fact rather than a failure of a solver. A page too large is drawn and wrong. Exactly one radius is right, and it is the radius the distances were measured on, which is the whole of what "the escape is a curved page" amounts to: the page has to be the sphere.

distance-curved

Five pages of different curvature, and the one that works One axis for every page a picture could be drawn on. The three escapes of this rung are one parameter. A hyperbolic page of curvature radius k has Gaussian curvature −1/k², a flat sheet has zero, and a sphere of radius R has +1/R²; the axis runs through all of them in units of the Earth's own curvature, so the Earth sits at +1. The same construction runs at every point — hold 2n − 3 distances exactly, read the rest — and the curve has exactly one zero, at +1, where the error is 3.3e-16. The left branch never dips below the flat sheet's 7%; it approaches it from above as the curvature goes to zero, which is also a check that two constructions written separately with cosines and with hyperbolic cosines agree where they must. Below about +1.2 on the right the sphere is too small to hold the places and the construction refuses rather than fits.

distance-curvatureaxis

One axis for every page a picture could be drawn on

Where it is called

Changing this changes every one of these figures.

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. The impossibility

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.

 A cylinder unrolls exactly; a sphere does not. Both pictures show the same grid. On the left it is wrapped round a cylinder of radius 1, 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. 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.

 The angles of a triangle, and the curvature inside it. A triangle on the sphere with sides that are great-circle arcs. Its three angles sum to 212.26°, overshooting the flat 180° by 32.26°. Integrating the curvature over the interior gives 0.56306 against an excess of 0.56306 — the same number by two routes that share no arithmetic. That is Gauss–Bonnet, and it is how a flatlander measures the curvature of a world it cannot step outside. The impossibility

Measuring curvature from inside

A flatlander with a ruler can find out that its world is round, by drawing a triangle and adding the angles. Gauss–Bonnet turns that into an exact statement, and the total curvature of a closed surface turns out to be a number obtained by counting.

 How large a patch can be treated as flat, at 10 parts per million. The smallest scale distortion any map of a circular patch can have, against the radius of the patch, on logarithmic axes. The line is straight with a slope of 2.00: the error grows as the SQUARE of the size, so a patch ten times wider is a hundred times worse. A tolerance of 10 ppm is reached at a radius of 40.3 km — 81 km across — and that is the number behind the boundary between plane surveying and geodesy. The impossibility

How small is flat enough

A builder works in plane coordinates and a national mapping agency does not, and the line between them is not a convention. The unavoidable error of treating a patch of the Earth as flat grows as the square of its size, and the size at any stated tolerance is a number.

 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 impossibility

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.

 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. The impossibility

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.

 A triangle of 25.0° excess, drawn on Mercator. The three geodesic sides as curves and the three straight sides a ruler draws, with each vertex labelled by how far the ruler's angle is from the true one. The straight-sided triangle's angles sum to exactly 180° because it lies in a plane, and the real one's sum to 205.02°, so the three errors have to account for the whole 25.02° of spherical excess between them — and they do so on this conformal projection exactly as they do on any other. What is taught wrongly

Conformal does not mean the angles are right

A conformal projection preserves angles between curves at a point. Draw a triangle on one with a ruler and its angles are wrong by degrees — and the total error is fixed before the projection is chosen, because a plane triangle sums to 180° and the real one does not.

 The image of a circle on Mercator, against its own indicatrix. A circle of three radii on the ground at 30°, 40°, projected exactly — the solid curve — against the ellipse the indicatrix predicts for it, dashed. The filled dot is the image of the circle's centre and the hollow one is the centre of area of what was actually drawn, which is not the same point. The departure runs from 3.14 per cent at 4° to 15.80 per cent at 16°, so it grows in proportion to the radius rather than to its square: halving the circle halves the relative error and does not quarter it. Measuring distortion

The indicatrix is a limit

Tissot's ellipse describes an infinitesimal circle, and every published one is drawn finite. The error of the description falls as the radius rather than as its square — halving the circle halves the lie — and on the Robinson projection at a table entry it does not fall at all: sixty metres and six kilometres are both 1.17 per cent wrong.

The whole library