Cartographic Projection
every map is wrong, and by exactly how much
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The thread: A theorem, not a limitation — page 3

No map is faithful, and this is not a shortcoming awaiting a cleverer cartographer. Gaussian curvature is intrinsic, a sphere has some and a plane has none, and that settles it. Essays 49 to 72 of 74.
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. The impossibility

How wrong a flat picture has to be

The rung below proves no flat picture of four places is exact and leaves the size of the failure to a determinant nobody can read. Measured directly, the least error falls as the square of how much sphere the places span — fitted exponent 2.0088 — and the same exponent comes back from five different arrangements while the constant in front of it moves by a factor of seventeen.

 The pass between the basins, measured rather than bracketed. The height of the lowest path from one near-optimal basin to the other, for regions of growing size. It is found by sorting the score surface and joining cells in order, so it is the minimax path's own height rather than the level at which a bisection stops finding two pieces. The fitted exponent is -1.60 at a coefficient of determination of 0.913. The previous rung's level-set measurement gave −1.33 and the argument predicts −1, so measuring the height directly moves the answer FURTHER from the prediction rather than towards it. What each projection optimises

The height of the pass between two basins

The previous rung recorded a shortfall: the fracture threshold falls as the −1.33 power where the argument predicts −1, and the difference was supposed to be the height of the pass. It is not. Measuring the pass directly gives −1.60, which is further from the prediction, and the two basins turn out to be exact symmetric copies with nothing to decompose.

 The corner against where along the seam the feature crosses. A feature crossing the seam of a cube at right angles, drawn on each face's own map and unfolded, with the crossing moved along the edge. Every curve starts at zero, because the face's mirror through the edge's own midpoint reverses the along-edge direction and forces the shear term in the Jacobian to be odd. Away from it the gnomonic — the map with no corner at all in the rung below — reaches 44.4°, three times the equal-area map's and far past the conformal one's. The edge's half-length is 35.3°, so the right-hand end is still well inside it. The families

The corner is not at the midpoint

The rung below measured the corner a feature gets crossing a polyhedral seam, and found none at all under the gnomonic face map. It crossed at the edge's own midpoint every time — the one point on the edge where a face's own mirror symmetry forces the corner to vanish. Two fifths of the way to the vertex the gnomonic gives 20.15°, the equal-area map 7.28° and the conformal map under a degree, which reverses the ordering entirely.

 A conformal map of Vesta, in the coordinates that make one possible. The body's own isothermal coordinate, used as the map — which is Mercator's construction carried out on a body that has no axis to be Mercator about. The lines are Jacobi's ellipsoidal coordinates, which are the surface's lines of curvature, at 30° and 20° spacing; the two quadratures that turn them into an isothermal pair are one-dimensional. The measured angular deformation away from the marked points is 1.1e-6°, which is this site's noise floor, and the areal factor spans a factor of 162 — conformal, and emphatically not equal-area. The four marks are the umbilics, where the coordinate collapses and the map has nothing to say. What the numbers refer to

A conformal map of a body with three axes

This site built an equal-area map of a triaxial body and wrote down what it could not do: the conformal one, which needs an isothermal coordinate that a surface with no axis of revolution was said not to have. It has one, Jacobi found it in 1839 for a different reason, and it takes two one-dimensional integrals.

 Two members, their average, and the family's best answer to it. Two equal-area conics at cone constants 0.25 and 0.85, the average of the two, and the member of the family nearest that average — at 0.540, which is not the parameter midpoint. The average is not a conic at all: its parallels are still arcs but they are arcs of circles about different centres, so no single cone constant reproduces it and the residual is 0.1722. The families

A family is not closed under averaging

Nine rungs treat a family as a set of maps with a parameter running through it, and this collection's own compromise projections are averages of members. An average of two members is not a member: two conics far apart average to something 31 per cent of their own separation outside the family — and averaging within a family makes the map worse every time, while averaging across two makes it 16 per cent better.

 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. The impossibility

Five distances of six, and never more

A flat picture of n places has 2n − 3 free numbers and n(n−1)/2 distances, so it can be exactly right about 2n − 3 of them and no more — five of six for four places, twenty-nine of a hundred and twenty for sixteen, and 3.98 per cent for a hundred. The bound is reached by a construction with no fitting in it, and an azimuthal equidistant map spends only n − 1.

 The basin has three widths, and they differ by a factor of 4.6. Two sections through the near-optimal basin of the Robinson aspect over Japan, drawn at one scale. Each ellipse is the set of aspects whose score is twice the optimum's, from the objective's own second derivative at the optimum: 32.1°, 16.0°, 6.9° along the three principal directions. A single number for "the width of the basin" is the cube root of their product, 15.3°, and it is not any of them. What each projection optimises

The basins have widths as well as depths

The previous rung measured the height of the pass and recorded a shortfall: shape means widths too. Measured, the basin has three of them — 32°, 16° and 7° at Japan — it gets wider rather than narrower as the region grows, and the exponent it predicts overshoots the measured one by half again.

 A conformal map solved rather than written down. The same patch of parameters on two bodies, mapped to the plane by solving the discrete Cauchy–Riemann equations — one complex equation per triangle, 1568 triangles, least squares, conjugate gradients, and no formula for either surface. Left: a sphere, where the answer is known in closed form and is not used. Right: a body with a bump on it, which has no isothermal coordinate and therefore no closed form at all. The parameter lines cross at right angles in both, to a median of 0.60° and 1.25° of angular deformation. What the numbers refer to

A conformal map of a body that is not a quadric

Jacobi's ellipsoidal coordinates give a triaxial body a conformal map by two quadratures, and this collection wrote down what that argument uses: the surface has to be a quadric. A real body is not. Solving the discrete Cauchy–Riemann equations instead — one complex equation per triangle, two thousand triangles, conjugate gradients — gives a conformal map of a bumped body to a median of 1.10°, converging at first order in the mesh, with the areal factor spreading by 3.07 and refusing to converge at all.

 One of these is a corner. The corner reported at the exact conformal seam, and the corner reported at a cube's gnomonic seam fifteen degrees along its edge, each measured over a shrinking arc. The gnomonic's is 21.4572° at every span from twenty-four degrees down to three — the same number to four decimals — because it is a corner. The conformal one halves whenever the span does, fitted exponent 0.990, because a tangent read from a finite chord of a curved image departs from the true tangent in proportion to the chord. It is not a corner; it is the instrument. The families

The exact map says the seam is smooth

The previous rung could only bound the conformal seam's corner at about two degrees, because the series it was fitted with holds its boundary condition to three parts in a thousand. An exact map exists — the stereographic projection composed with ∫dt/√(1 − t⁴) — and it settles it: the corner falls in exact proportion to the arc the tangent is read over, fitted exponent 0.99, while the gnomonic's 21.4572° is the same to four decimals at every span.

 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 5.1e-15 — 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 10%. 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 152%. The impossibility

The escape is not a dimension

Four places that will not lie in a plane surely lie in a space — and they do not. The double-centred matrix of their great-circle distances has a negative eigenvalue, so no Euclidean space of any dimension holds them, and buying a third dimension improves the picture by nothing whatever, to fifteen decimal places. What does work is a page with curvature, and the curvature is pinned to within two per cent.

 The piece count rises and falls. The number of connected pieces of the near-optimal aspect set for robinson over japan, swept finely through the threshold rather than sampled once below it. It is one piece at a wide threshold, reaches 23 at 1.256, and returns to one as the set shrinks onto the single best aspect. The set first disconnects at 2.244, which is above the peak: the pieces keep multiplying after the first break. This is the sweep the rung below could not afford and it costs one grid, because every threshold reads the same 4992 evaluations. What each projection optimises

The threshold is not a percolation

The rung below found the near-optimal aspect set breaking into twelve pieces rather than two, called the transition a percolation, and recorded that it had not measured the exponent. Swept finely, the piece count rises from one to twenty-three and falls back to one — and refining the grid by a factor of fifteen does not move the peak, while an uncorrelated field on the same lattice grows by a factor of twelve.

 Where the condition holds, and where it was asked to. The boundary scale of a fit collocated at 20 points, drawn all the way round the boundary. The marked points are the ones the condition was imposed at, and the curve passes very near zero at every one of them; between them it does not. The largest departure on the samples is 3.92e-5 and the largest anywhere is 3.27e-4, and the second is the one the map has. What each projection optimises

A condition imposed at points is not a condition

Nine rungs state a condition and solve it, and every solve imposes the condition at a finite set of samples because that is what a linear system is. With barely more equations than unknowns the residual the solver reports is 8.3 times too good — and refining the collocation twentyfold does not improve the map at all, it only makes the report honest.

 Three face maps on a cube, over a shrinking span. The corner a feature gets crossing the seam of a cube, measured by reading the tangent over an arc and then shrinking the arc by a factor of sixteen. A chord differs from a tangent in proportion to the arc, so a perfectly smooth join reports a corner that HALVES when the span halves — a slope of one on these axes. None of the three lines has a slope of one. The fitted slopes are 0.000, 0.000, -0.018, which is a flat line in each case, and a flat line is a real corner. The three differ in size and not in kind: 20.1513°, 7.2772°, 0.7687°. The families

The span ladder, run on all five

A recorded shortfall said the exact map's span ladder was one loop away from settling the seam on the five Platonic solids. The loop was run and the answer is the other one: the series conformal face map has a corner that does not shrink with the measurement span on any of them — 4.10° on the tetrahedron, 0.77 on the cube, 0.004 on the icosahedron — and it belongs to the truncation rather than to conformality.

 Walking the crossing all the way to a cube's vertex. The corner a feature gets crossing the seam, at crossings that approach the vertex geometrically — the last is 3.5e-4 degrees from it. Every face map is singular at a vertex, so the expectation is that the corner runs away. It does not. The gnomonic settles at 53.1295° and the equal-area map at 12.9656°, and both are finite. The dashed line is the solid's angle deficit, 90.0°, which is what the surface loses at that point and is not what either map's corner reaches. The families

The corner that is the curvature

Every face map is singular at a vertex, so a corner that grows as the crossing walks towards one should run away. It does not: the gnomonic settles at 53.1295° on a cube and the equal-area map at 12.9656, both finite, both reached like the first power of the remaining gap. What does not settle is the deficit beside them — 90 degrees, fixed by Descartes before any projection is chosen.

 The solve's cost is a U in the shape; the curvature is not. Solid: how many conjugate-gradient steps the conformal solve needs, against the window's aspect ratio, at constant surface area. It has a minimum at the square window — 174 steps — and rises to 265 and 275 at the two extremes, which are elongated by the same factor in opposite directions. Dashed: the total Gaussian curvature the window encloses, on its own scale, which falls from 0.311 to -0.012 across the same sweep and is least at one end of it. The cost has its minimum where the window is square and the curvature has its minimum somewhere else, so whatever is making the solve expensive is not what is making the map spread. What the numbers refer to

A long window and a square one

Rung nine held the patch's area and found that where the cut goes still changes the map by a third, with the curvature it encloses predicting the change at r = 0.969. It recorded that it had held area and curvature and not shape. Sweeping the shape at constant area separates two things that had looked like one: the map is curvature and the cost is shape.

 Two features, one shared boundary, simplified apart. Two neighbouring areas whose common boundary is a curve with structure at every scale — a river or a ridge, in effect — each stored with its own copy of that boundary and each simplified on its own at a tolerance of 0.01. The faint outlines are the originals and the solid ones what came back. The two copies of the shared boundary were within 0.01 of each other before the simplification and are not afterwards: 144 probe cells of 40000 now lie inside both features and 0 inside neither. What a machine does with it

A boundary that two features share

Three rungs simplify one curve and price what a tolerance covers. Almost no boundary in a real dataset belongs to one feature: a county's edge is the next county's edge, it is stored twice, and it is simplified twice. What opens between the two answers is a region belonging to both features or to neither, and its area is not bounded by the tolerance.

 One degenerate zero, nudged, becomes two ordinary ones. The direction of steepest ascent within twelve degrees of the north pole, for the sectoral harmonic alone and with two amounts of the tesseral added. On the left is one zero of index −2, a monkey saddle: three ways up and three ways down, and a Hessian that vanishes. On the right are two ordinary saddles of index −1 each, both of which the second-derivative test names correctly. Nothing has been added to the field but a term whose size can be made as small as anyone likes, and the classification changes at every nonzero value of it while the total does not change at all. Measuring distortion

The second derivative cannot classify

Seven essays have used a second derivative to measure a size. The one thing a second derivative is classically used to do is say what KIND of thing is at a point, and on a sphere that use fails: the determinant test totals six where the truth is two, its failure is confined to exactly the field it is asked about, and the count that gets it right never differentiates twice.

 Töpfer's square root is one line of a family. The fraction of features surviving to a smaller scale, for four stated populations whose size distributions differ only in their exponent. Every one is a straight line on these axes, and the slope of each is its own exponent: 0.3, 0.5, 0.8, 1.2. Töpfer's radical law is the line at 0.5 — the square root — and it is exact for that population and for no other. The law is not a rule of thumb with exceptions; it is a theorem with a hypothesis nobody states. What a machine does with it

How many features a scale can carry

Töpfer's radical law is quoted everywhere as a rule of thumb. It is not one: it is a theorem about a size distribution with a Pareto exponent of exactly one half, exact to 1.8 per cent for that population and out by 99.4 per cent for a lognormal one.

 A parent and its children, twice. An aperture-7 hexagonal hierarchy beside a square one. The heavy outline is the parent and the light ones are its children. On the right every child is wholly inside and the four of them tile the parent exactly. On the left the child lattice is turned by 19.107° relative to the parent's, only the central child is wholly inside, and 7.14% of the parent is covered by no child of its own. The two families have exactly the same total area — a hexagon cannot be tiled by smaller hexagons at any ratio at all, which is why the mismatch is a construction rather than an approximation. What a machine does with it

A cell's children do not fit inside it

Ten rungs price one cell system at one resolution, and every one of them is used hierarchically. A hexagonal hierarchy does not nest: at the aperture-seven scheme the discrete global grids use, one fourteenth of a parent is covered by no child of its own, exactly, and each of the six ring children is eleven twelfths inside.

 The straight line between two correct maps passes through an incorrect one. Both ends of this sequence are exact cartograms of four cities: the triangular, x first construction at s = 0 and the triangular, y first at s = 1, each meeting the density to arithmetic noise. The panels between them are the straight-line blend of the two, which is what an animation between two maps computes. The cells drawn solid have turned inside out — their signed area is negative, so the map has folded over itself there and two places on the sphere are drawn at one place on the page. Measuring distortion

A map that meets its target can fold

The flow a diffusion cartogram integrates is a diffeomorphism at every instant and cannot fold. Every discretisation of it can, and the point at which one does is a root of a quadratic — written down rather than searched for.

 A section through a body with a neck, and the rays that leave it twice. An equatorial section of a stated contact binary — two lobes of radius 1 centred at ±1.5, joined by a neck of radius 0.35, with the origin in the neck. The lines are rays from the origin and the marks are where each one crosses the surface. two of the 25 drawn cross more than once, so along those directions there is no such thing as "the" radius, and a longitude and a latitude do not name a place. What the numbers refer to

A ray from the centre hits the surface twice

Eleven rungs map bodies that are lumpy, triaxial and turning at a drifting rate, and every one assumes the surface is star-shaped about the centre — which is what makes a longitude and a latitude a coordinate at all. A contact binary is not: on a stated body with a neck a third of a lobe wide, 10.9 per cent of the sky has no single radius, and the shape model everybody publishes fills the neck in and adds 1.67 per cent of the volume.

 How far the two routes end up apart. The distance between a line simplified directly at the final tolerance and the same line simplified through two intermediate products, as a multiple of the final tolerance, with a stated extra step applied to each intermediate. With nothing in between the two are the same line to the last bit, because Douglas–Peucker's outputs are nested. Rounding the intermediate to half the tolerance, smoothing it for legibility, or running it through a moving average each break that, and the last of them puts the final product 0.142 away from where a direct route would have put it — nine times the tolerance the product is published under. What a machine does with it

Two routes to one scale

A national series is cascaded — the million is derived from the quarter-million, which was derived from the fifty — and the folklore is that the errors accumulate. They do not: Douglas–Peucker and Visvalingam both cascade to the same line the direct route produces, bit for bit, because both output a sublevel set of a per-vertex number. What breaks it is anything else in the chain, and a moving average puts the product nine tolerances away.

 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. What the numbers refer to

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.

 Two maps of one density: one costs nothing and one costs forty-four degrees. Both drawings meet the same areal request — an exponential ramp of contrast 79.8 to one, whose logarithm is harmonic — to arithmetic noise. The first is a conformal map written down in closed form, the conformal map that meets an exponential ramp, log-harmonic, whose angular deformation is 2.4e-8 degrees. The second is the triangular construction the previous rungs use, at 43.8° on the same request. A grid of squares is drawn through each: the first keeps every angle and the second does not. Measuring distortion

The cheapest map that meets its areas

An earlier essay bracketed a cartogram's least cost between a construction charging eighty degrees and a bound valid only for symmetric densities, and recorded the gap as a shortfall. One request settles it: a density of contrast eighty whose least cost is exactly zero, met by a map written down in closed form, while the standard construction charges 43.8° for it.

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