Cartographic Projection
every map is wrong, and by exactly how much
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The thread: Measured, not named — page 9

A projection is usually called conformal because that is its name. Here it is called conformal only after the angular deformation has been computed at several hundred points and found to be zero. Essays 193 to 216 of 292.
Give the edge a width and the kernels get their orders back. Every edge this collection has resampled across has been exactly discontinuous, which is not what a sensor produces: a footprint, an atmosphere and a lens all smooth a boundary over a cell or two before anything is resampled. Convolving the edge with a Gaussian of stated width and refitting gives 1.23, 1.97 and 3.60 at one degree of blur, against 0.78, 0.58 and 0.60 with no blur at all. The blur is held fixed in degrees while the grid refines, which is what happens to a real sensor's data as its resolution improves. What a machine does with it

A real edge has a width

Thirteen edges were measured and every one of them was exactly discontinuous, which no sensor has ever produced. Convolving them with a point-spread function of one degree — a cell or two — takes the three kernels from 0.78, 0.58 and 0.60 back to 1.23, 1.97 and 3.60, and takes the edge along a parallel, which converged at −1.49, up to 1.92 for bilinear and 3.73 for cubic.

 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.

 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.

 One pair of numbers, a hundred and twenty places. The easting 412,000 and northing 5,678,000, interpreted in each of the sixty zones and in both hemispheres. The northern candidates are a ring of sixty at 51.247° north, spaced exactly six degrees apart; the southern ones are a second ring at -39.043°, which is not the mirror of the first because the southern convention subtracts the northing from ten million. Every one of the hundred and twenty is a perfectly valid reading of the same two numbers. Grids, and what a survey does

One pair of numbers, a hundred and twenty places

A UTM coordinate is two numbers and a zone. Drop the zone and the numbers are still valid in each of the sixty; drop the hemisphere too and the pair names a hundred and twenty places. They form two rings at one latitude each, spaced exactly six degrees apart, and every one of them has the same grid convergence and the same scale factor — so no further geometric measurement can choose between them.

 Four of the seven on the front can never be first. Every library projection over the whole sphere, with both errors normalised to the table's own range. The seven filled circles are the Pareto front — nothing beats them on both counts. The line through three of them is the lower convex hull, and a weighted sum of the two errors is a straight line in this space, so only a hull vertex can ever come first. Four projections — Web Mercator, Miller cylindrical, Equirectangular, Winkel tripel — sit in the dents, undominated and unchoosable. What is taught wrongly

Which projection a weighting can make best

Rung eight finds the seven world projections nothing beats on both counts and tells a reader with a preference that one of them is their answer. Four of the seven are not: they sit in dents of the front, undominated and unreachable, and no weighting of angle against area can ever put them first. Robinson can be first, on 2.8 per cent of the weight range.

 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.

 Two expansions of one integral. The worst error along the whole meridian, against the number of sine terms kept, for the series in the third flattening and the classical series in e². Both are checked against a Simpson's rule on the defining integral, which shares no algebra with either. At one and two terms they are the same number to four digits and the e² series is fractionally ahead; from the third term the n series pulls away, and at four it is 1175 times more accurate — 7.6e-8 metres against 9.0e-5. What is taught wrongly

Which small quantity the series is in

Every ellipsoidal formula in this collection is a truncated series in the third flattening, inherited from Krüger in 1912 and justified nowhere. Measured against the classical expansion in e², at four sine terms it is 1,175 times more accurate — and at one and two terms it is fractionally worse, which is not what the folklore implies.

 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.

 A transformation is more certain in some places than others. The horizontal position uncertainty a Helmert transformation carries, from the stated widths of its own seven parameters, along three meridians. It runs from 40 mm to 68 mm — a factor of 1.71 — and it falls towards the poles, because the rotation terms act on the distance from the Earth's axis. A single figure quoted for "the accuracy of the transformation" is the value at some latitude nobody wrote down. What the numbers refer to

The seven parameters have their own uncertainty

Nine essays on this ladder print a datum transformation as seven exact numbers. Every published set is the output of a least-squares fit and arrives with standard errors as much a part of the result as the parameters — and pushing those widths through to the ground gives an ellipse, not a number, that is 68 mm across at the equator and 43 mm at 70°.

 One field, one round trip between two cell schemes. Left: a stated field binned into an equal-angle grid of 36 by 18 cells. Right: the same field after being rebinned into an equal-area grid of 30 by 15 offset by six degrees of longitude, and rebinned back. Every step is exact area-weighted averaging, the total is preserved to 2 × 10⁻¹⁶, and the root-mean-square difference between the two pictures is 0.144 on a field whose own standard deviation is 0.370. What a machine does with it

The same data on two grids

Five essays have addressed, queried and ordered cells within one scheme and nobody has moved a number between two. Doing it exactly — area-weighted, both directions — preserves the total to 2 × 10⁻¹⁶ and loses 39 per cent of the field's own standard deviation in a single round trip; six round trips leave 23 per cent of its variance. The quantity that would reveal the damage is the one that never moves.

 Four radii of the Earth, and one that is a range. The four constants called the mean radius of the Earth, on a scale of kilometres, with the range of the local Gaussian radius √(MN) drawn behind them. Three of the four agree to about a part per million; the rectifying radius is 3560 metres smaller, which is 559 parts per million. The Gaussian radius spans twelve times that range on its own, which is why there is no such thing as the conformal sphere. The families

Four radii of the Earth

Ten rungs handle the ellipsoid with an auxiliary latitude. Every one of those constructions also needs a radius, and the radius that makes each property exact is a different number: the published 6371 km is right for an area to half a part per million and wrong for a meridian distance by 559 — while the radius a conformal map needs is not a constant at all, and spans 6,739.

 One sentence, three readings of it. "a straight line from the initial point on the Rio Grande to a point on the Colorado" — Treaty of Mesilla, 1853. Every curve here answers to those words. the geodesic, the rhumb line, straight on the sheet, drawn between the same two monuments on a conformal conic fitted to the segment itself. The widest pair, the geodesic against the rhumb line, are 7.53 kilometres apart at their worst and enclose 3,925 km². The shading is that ground. It is not an artefact of the drawing: the same figure of the 141st meridian shows one line, because on a meridian every one of these readings is the same curve. Grids, and what a survey does

One sentence, and the ground between its readings

Every land boundary in the world is defined by a sentence, and a sentence naming two monuments does not name a curve. "A straight line" between the Rio Grande and the Colorado admits at least three answers 7.53 kilometres apart at their worst; "the forty-ninth parallel" admits four, 96 kilometres apart, with 130,972 square kilometres between the extremes.

 One unit of a vector tile, in metres of ground. A vector tile's coordinates are integers on a lattice 4096 units across the tile, and the tile halves at every level, so one unit is a distance that halves too: 5.48 m at z10 and 0.086 m at z16, at 55°. It is also a different distance at every latitude, by cos φ, because the tile is in Web Mercator — the same factor that makes a grid metre a different quantity of ground at every latitude, arriving in the file format rather than in the projection. What a machine does with it

A vector tile has an integer grid

Six essays on this ladder treat a vector tile as the thing a raster tile is not: geometry, resolution-free, styled at draw time. Its coordinates are integers on a lattice 4,096 units across a tile, the tile halves at every level, and at 55° north one unit is 88 metres at zoom 6 and 21 millimetres at zoom 18.

 The same patch, pinned six ways. Two vertices have to be held or the conformal energy has a similarity's worth of null space. Which two turns out to decide two of the three numbers reported. The median angular deformation is the same to 8 per cent across all six — that is the map. The areal spread runs from 2.68 to 7.44, so the 3.07 reported for this body was a statement about its corners. And pinning two adjacent vertices, which fixes the similarity through a very short lever, ruins the worst point without touching the median: a badly conditioned constraint pays for its scale in one corner. What the numbers refer to

The map depends on where it was cut

The previous rung solved the discrete conformal equations on a triangulated body and reported an areal spread of 3.07, then recorded that the number might belong to the patch, the boundary and the two pinned vertices rather than to the surface. It belongs to the pins: hold a different pair and it runs from 2.68 to 7.44, while the typical angular deformation does not move at all.

 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 projections that run in a circle. Mercator beats Sinusoidal beats Eckert IV beats Mercator, each on a majority of the same seven criteria over the whole sphere. Every margin is four to three, the narrowest a majority of seven can be, and the criteria that decide each edge are different ones. There is no way to place these three in an order that agrees with all three comparisons, and the obstruction is not a measurement error: every number is exact to the precision the sampler reaches. What is taught wrongly

The ranking is not an order

The previous rung showed that a weighting can make almost any projection best. Remove the weights entirely, let each of the seven criteria vote once, and the answer is worse: over the whole sphere Mercator beats the sinusoidal, the sinusoidal beats Eckert IV, and Eckert IV beats Mercator — four such circles, every margin four to three, with a Condorcet winner sitting above them all.

 The line a commission can actually run. A boundary described as a parallel of latitude and marked by monuments 220 kilometres apart, with the offset exaggerated 700 times so that it can be seen at all. A commission cannot run a parallel: it can set a monument, sight a straight line to the next and clear the trees between, and a straight line between two points of equal latitude is a geodesic, which passes POLEWARD of the parallel everywhere between them. So the marked line lies north of the described one, by 1176.5 metres at the middle of each of its 9 chords, and encloses 1611.2 square kilometres that the words put on the other side. At the 20-kilometre spacing this ladder measures at, the same offset is 9.03 metres. Grids, and what a survey does

The line a commission can actually run

A boundary commission cannot run a parallel of latitude. It can sight a straight line between monuments, and a straight line between two points of equal latitude passes poleward of the parallel — by s² tan φ / 8R, which at a mile of spacing is fifty-eight millimetres and at a hundred kilometres is two hundred and twenty-six metres. The described line and the marked line are different curves, and the marked one governs.

 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.

 How many projections the map could be in. The number of candidates whose residual sits below the measurement noise, against the size of the region, at four noise levels. At one per cent of the map's width — a hand-digitised graticule — a four-degree region admits ten of the twenty candidates and a forty-degree one admits exactly one. Every curve falls, none of them crosses another, and all four end at one: identification works, and what it needs is extent rather than precision. What is taught wrongly

The answer is a set

Eight rungs have produced a best fit — one projection, ranked first, with a margin. A best fit without a spread is not a measurement, and the spread is free: a control point has a width, and every candidate whose residual is inside that width has not been ruled out. At one per cent noise a four-degree region admits ten of twenty candidates and a forty-degree one admits exactly one.

 The threshold, and the two things it is a ratio of. The near-optimal set's fracture threshold on Robinson, against the size of the region, with the two quantities it is a ratio of drawn beside it. The threshold falls with fitted slope -1.293. The best score a region admits at all rises with slope 0.923 — a bigger region is harder to map — and the absolute score of the pass falls with slope -0.370. The first is the sum of the other two by construction, and the arithmetic says which of them is doing the work: the denominator carries 71 per cent of it. What each projection optimises

The first break is mostly its denominator

Three rungs have fitted the near-optimal set's fracture threshold against region size and read the answer as a statement about the landscape. It is a ratio, and separating it takes one multiplication: the pass's own depth is constant to 12 per cent below twenty degrees of span, and the whole of the threshold's movement there is the denominator — the best score the region admits at all — rising with exponent 0.92.

 The drift is a straight line in the deflection. The horizontal distance between where a plumb line hangs at the top of a column of rock and where it hangs at the bottom, against the deflection of the vertical the mass produces at the surface. Four heights of column. Every line is straight through the origin: 12.54 mm of drift per arcsecond of deflection over a 4,000 m line, to two parts in ten thousand across a fortyfold range of deflection. Which is what makes the number transferable — the 47 mm the ladder started from was a statement about one buried sphere, and this is a statement about any mass that produces the same deflection. What the numbers refer to

How far the plumb line bends

The previous rung dropped a plumb line down a four-kilometre column of rock beside one buried mass and found it arrived 47 millimetres from the point below the summit. That is a number about that mass. Parameterising by the deflection of the vertical instead — the quantity surveyors actually measure — gives 12.54 mm per arcsecond, exactly linear across a fortyfold range.

 The piece two schemes share, clipped rather than assumed. A cell of a gnomonic cube, whose four edges are great-circle arcs because a straight line on a gnomonic face is one, against a cell of a longitude–latitude grid, whose north and south edges are parallels and are not. Their overlap is neither a rectangle nor a spherical polygon of any standard kind, and it is 5965687 km² of the cube cell's 5965687 km² — 100.0 per cent. Computing it needs the arc of one boundary intersected with the plane of the other, which is three equations and two roots, and it is exact. What a machine does with it

Cells that are rectangles in no coordinate

The previous rung measured what moving a field between two cell schemes costs, and did it between two schemes whose cells are longitude–latitude rectangles — which is what made every overlap a rectangle with a closed-form area. The schemes anybody actually argues about have cells that are rectangles in no coordinate, and their overlaps have to be clipped.

 What a geoid model leaves out, against the degree it stops at. The RMS of everything above the model's highest degree, from Kaula's rule — the statement that the normalised coefficients at degree n are about 10⁻⁵/n². The line is R × 10⁻⁵ ÷ n, so a model to degree 360 omits 17.7 centimetres and one to 2190 omits 2.9. Every orthometric height derived from such a model carries that as an error, and it is not quoted with the height. What the numbers refer to

The geoid model stops at a degree

Eleven essays treat the geoid as a surface that exists. Every geoid anybody uses is a series truncated at a degree, so every orthometric height derived from one carries an omission error nobody quotes with the height — eighteen centimetres at degree 360 — and the same truncation removes two thirds of the slope, which does not converge at all.

 One meridian, two datums, and the ground between them. "the meridian line of the 141st degree of west longitude" — Anglo-Russian Convention, 1825. The line whose longitude is exactly 141° west, drawn twice: once on nad27 and once on WGS84, with the east–west separation exaggerated 3,000 times. The two are 129.0 metres apart, and remarkably constant — the shift changes by five centimetres over nine degrees of latitude — so the strip between them is a ribbon 129 metres wide and 1039 kilometres long, which is 134.0 square kilometres. The sentence has not changed. The surface the number refers to has. Grids, and what a survey does

A meridian boundary moves when its datum does

The 141st meridian is the one boundary description in this collection with no geometric ambiguity in it: every reading of it is the same curve, exactly. It has a different one. A longitude refers to a datum, the 1825 convention named none, and the line of longitude exactly 141° west sits 129.0 metres apart on NAD27 and WGS84 — a ribbon 1,039 kilometres long and 134 square kilometres in area.

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