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

The thread: Measured, not named — page 10

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 217 to 240 of 292.
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.

 Seven cuts of the same size, in different places on one body. The body's own colatitude and longitude, with the seven windows drawn on it. Each covers the same surface area to 0.24 per cent — the longitude extent is divided by sin θ and a scale is then solved per window — so their sizes are held and only their positions differ. Each is shaded by the areal spread of the conformal map solved on it, listed beside the grid and running from 1.03 to 1.36. The darker windows are the worse ones, and they are the ones over the body's lobes. What the numbers refer to

Cuts of the same size in different places

Where a body is cut decides how well it can be mapped, by a factor of ten — established with five windows of five different sizes, so *where* and *how much* were confounded and the factor could have been entirely about extent. Held to the same surface area to a quarter of a per cent, the answer survives at a factor of 1.32, and what predicts it is the curvature the window encloses.

 A curve built to have dimension 1.2619. The generator replaces every segment with four of equal length at headings 0, +60.0°, −60.0° and 0. Closing the displacement fixes the length ratio at 0.33335, and four copies at that ratio give a dimension of exactly log 4 / log(2 + 2 cos θ) = 1.2619. Nothing here is measured yet: this is the construction the measurement will be checked against. Drawn at depth 5, which is 1024 segments, with the second-level shape shown faint beneath it. What a machine does with it

A line has a length only at a scale

Every measurement on this site so far has been of a curve given by a formula, sampled as finely as the picture needed. A map is not that: the geometry that reaches the page has been through an algorithm whose job is to throw most of it away. The first thing that goes is the idea that the line had a length.

 One of these settles. The largest departure of the fitted map's boundary scale from constant — the quantity the previous rung showed the solver cannot see — against the number of collocation nodes, for the two placements. The clustered fit reaches 7.030e-5 at forty-eight nodes and returns exactly that at every count above it. The evenly spaced fit does not settle at all: it wanders by a factor of 1.43 across the same range, going up as often as down. Refining an evenly collocated fit is not convergence, and the previous rung's finding that more samples improve the report and not the map is this seen from one side. What each projection optimises

The nodes were evenly spaced

The previous rung showed that refining an evenly collocated fit improves the solver's report and leaves the map alone. Moving the same number of nodes to the Chebyshev positions — crowded toward the corners, where a conformal map of a polygon is singular — makes the fit settle: 7.030 × 10⁻⁵ at forty-eight nodes and exactly that at every count above it, against an even fit that wanders by 43 per cent and never converges at all.

 One quantity, one region, and the exponent left free. The scale departure of six projections over the world, aggregated as a p-norm, against p on a logarithmic axis. At p = 1 the best is Eckert IV; at p = 64 it is Winkel tripel. Nothing about the maps changed between the two ends of the axis — only how much of the region a bad point is allowed to spoil. Drawn in no projection: the axes are an exponent and a score. What is taught wrongly

The average was a choice of norm

Ten projections, one region, one measured quantity, and the only free decision left is how to turn a field into a number. Over the world's scale departure the ordering at the mean and the ordering at the worst case have a rank correlation of −0.04, all ten maps change position, and the exponent that produced each answer is stated nowhere.

 Three named members of the family, and one the family has no name for. The equal-area pseudocylindricals are not a list of maps. They are the solutions of one equation — C(φ)·Y′(φ) = cos φ — in two unknown functions, so one function is free and the named members are points in a space of them. Writing Y as six numbers and searching over them finds a map at 25.03° of mean angular deformation against Eckert IV's 28.08, Mollweide's 31.83 and the sinusoidal's 38.86 — 12.2 per cent better than the best map the family has a name for, and every one of the four is equal-area to the same precision. The families

A family is a function, not a list

The equal-area pseudocylindricals are the solutions of one equation in two unknown functions, so one function is free and the named members are points in a space of them. Writing that function as six numbers and searching over them finds a map at 25.03° of mean angular deformation against Eckert IV's 28.08 — and two numbers are already enough to beat every map the family has a name for.

 The same deflection, from a compact mass and from a broad root. Each curve is the deflection of the vertical across a mass buried 8 km down, with the mass solved so that all of them peak at 10″. The narrow one is the buried sphere the earlier measurement used; the broad ones are crustal roots 40 and 160 km wide, modelled as that mass spread along a line. They agree where it matters most and disagree everywhere else: the signal is 30 km wide for the compact body and 188 km for the widest, which is the difference the plumb line feels as it descends. What the numbers refer to

A mountain is not a buried sphere

The plumb line's drift was measured over a compact buried body and grows as the 0.69 power of the column's height — an exponent that is a statement about how quickly a buried sphere's field weakens with distance rather than about mountains. Spread the same mass into a crustal root and the exponent climbs to 0.84, while the proportionality to the deflection survives exactly.

 The same tolerance, applied in two orders, at 65°. The faint line is the region's boundary as built, 1025 vertices across 400 km of ground. Both pipelines were given the same tolerance of 2000 m on the ground. Simplifying in degrees and then projecting keeps 311 of them; projecting into Mercator and then simplifying keeps 129; doing it on the ground itself, which no pipeline does, keeps 129. The two drawn lines separate by 1883 m, which is 94 per cent of the tolerance that was supposed to bound the whole operation. What a machine does with it

Simplification does not commute with the projection

A pipeline either simplifies the geometry and then projects it, or projects it and then simplifies. Both orders are in use, neither is recorded, and given the same tolerance in ground metres they keep different vertices — 129 of them on the ground, 367 in degree space at 80°, and 459 on an equal-area page.

 Two regions, three answers. Britain and New Zealand, 166° apart, with the pole of the best oblique conic under each of three objectives. Pooling the samples and taking an area-weighted score puts the pole in one place; refusing to let either region be worse than the other puts it somewhere else. The regions are drawn on Mollweide so that equal ground areas are equal page areas. What each projection optimises

The pooled score abandons a region

Fifteen rungs optimise for one region. An atlas is several, and pooling their samples into one area-weighted score is what everybody does — which on Britain and New Zealand serves Britain 1.2 times worse than it could be served alone and New Zealand 125 times worse. The worst-case objective makes them equal at 33 and 59, and the cost of sharing rises with separation from 1.4 to 59.

 The test the ladder asked for, and it refutes the conjecture. How much better the best asymmetric projection is than the best symmetric one, under weightings of four different symmetries, with every symmetric map allowed to re-aim its axis at twelve candidate poles. The conjecture rung eight recorded was that the seven earn their place by PLACING distortion where a symmetric map cannot, so their advantage should collapse under a criterion with no place preference. It does the opposite: the advantage is largest at 1.343 under the uniform weighting and smallest at 1.144 under a band, with the fully asymmetric concentration at 1.204 in between. The winner is named on each row and the map it beat is Equirectangular throughout. The seven are simply better maps. The families

The maps with no family are simply better

Rung eight found seven projections with no continuous symmetry and noticed they are almost exactly the set anybody would choose for a world map, then offered a conjecture with a test attached: their advantage should collapse under a criterion that does not care where anything is. Run, it does the opposite — 1.343 times under a uniform weighting and 1.204 under a concentration. The conjecture is refuted.

 What a sheet does to the points before anybody measures them. The graticule crossings of a map drawn on Conformal conic, with an arrow at each one showing where the same crossing has moved to after the sheet dried — 0.1 per cent along the grain and 0.4 across it, with the grain at 23° to the map's axis, and the displacement magnified 60 times so it can be seen at all. The pattern is a stretch along one direction and a squeeze along the perpendicular, which is what an anisotropic scaling looks like. It is a property of the paper and has nothing to do with the map printed on it. A similarity fit to these points leaves 3.66e-4 of the map's own width unexplained, against 1.22e-10 on the unshrunk sheet. What is taught wrongly

The sheet moved before it was measured

Nine rungs take control points off a map and assume the sheet they came from is the sheet the cartographer drew. Paper shrinks across its grain three times as fast as along it, and on a map whose grain runs along its own axis that shrinkage is EXACTLY a change of standard parallel — one per cent moves the recovered parallel by 0.57 degrees with the residual sitting at the solver's floor.

 The picture is kept, at four tolerances. One closed curve of 3001 vertices, simplified at four tolerances. Douglas–Peucker's promise holds in every panel: no discarded vertex is further than ε from the line drawn in its place, measured at 0.1158 against 0.128 in the last. The picture survives. The enclosed area does not: it falls by 5.43 per cent, and it falls rather than wandering, because cutting a corner takes area off and never puts it back. What a machine does with it

A tolerance is a promise about the picture

Douglas–Peucker guarantees exactly one thing: no vertex it discarded is further than ε from the line drawn in its place. It says nothing about the enclosed area, nothing about which side of the boundary a point ends up on, and nothing about whether the curve still fails to cross itself — and all three are what the geometry is usually being asked.

 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.

 The whole of what an unlabelled map gives you. The outline of Japan as drawn on Conformal conic, delivered as an ordered list of page positions with nothing attached to any of them. No latitude, no longitude, no scale, no north. The rung's question is whether a projection can be recovered from that, and it can: the correspondence between the ink and the ground is found by sweeping the starting point round the curve and both directions, and the true candidate comes back with a residual of 2.88e-14 against the runner-up's 4.57e-4. What is taught wrongly

A map with no graticule

Ten rungs are handed control points, and a great many maps have none. Handed an outline with no labels on it at all, the method still works — and works better: the correspondence between ink and ground is recoverable exactly, because a similarity preserves ratios of arc length, and the margin on clean observations is 1.6 × 10¹⁰ against a graticule's 9.9 × 10⁶. What breaks it is noise, at three parts in a thousand.

 A ring round the pole at 80°, and the two pieces it makes. the boundary of a small polar cap — and of everything else. A closed curve divides a sphere into two pieces and neither of them is the outside: one is 4 thousand square kilometres and the other is 506 thousand, a ratio of 130.6 to one, and the coordinates are the same either way. The two colours are the two pieces, sampled at points rather than shaded, because shading one of them would already be the decision this figure is about. What a machine does with it

A polygon on a sphere has no outside

Seven essays have treated a stored ring as a boundary between inside and outside. A closed curve on a sphere divides it into two pieces and neither of them is the outside, so every polygon in every file depends on a convention that no coordinate carries — and the two conventions in common use disagree by a factor of fourteen on any ring that contains a pole.

 What a seven-parameter fit's residual is made of. A published transformation accuracy is the root-mean-square residual at the common points, and here it is 1.51 metres. Almost all of it — 1.51 — is the network's own distortion, which a rigid motion and a scale cannot follow and which is present at every point of the country whether it was used in the fit or not. The transformation's own error, measured as the disagreement between the fitted parameters and the true ones over a clean grid, is 0.091 metres: 6 per cent of the quoted figure. The last bar is the control — the same fit with the distortion switched off, at 2.9e-4 metres. What the numbers refer to

What another common point buys

Rung three finds that a seven-parameter datum fit leaves a pattern rather than noise. Six per cent of the residual it reports is the transformation's own error and the other ninety-four is distortion no seven parameters can follow — so adding common points improves a term that was already small and cannot touch the one that is quoted.

 What simplifying a boundary does to the number stored beside it. One region, simplified at five tolerances, with the error in the two quantities a consumer computes from the pair. If the density was stored, the total it implies moves by exactly the area's error — -1.55 per cent at the loosest tolerance. If the total was stored, the density it implies moves the other way by the same amount. Nothing in the file says which of the two was measured and which is being derived, and the simplification is normally done by a tool that never opens the attribute table. What a machine does with it

The attribute is a claim about the geometry

Fourteen essays price what a stored coordinate means and not one asks what the number stored beside it means. A rate is a quantity divided by an area, the area belongs to the geometry, and no format records which area — so a simplification that moves the outline by nothing visible moves the implied total by 1.55 per cent, an unweighted average of densities is 4.09 per cent out, and a choropleth gives a polar square kilometre fifteen times the ink of an equatorial one.

 Two source geometries of 96 cells each, rebinned to the same three targets. Both curves start from a source of 96 cells and rebin to targets of 32, 128, 512 cells, so the count ratio is identical along them and the only difference is the shape of the source cells: gnomonic squares on a cube against rectangles in longitude and latitude. The ratio dominates — both curves fall by more than half across the range — and the shapes still separate by 25 points at the middle target. The cube loses less, because its cells are all much the same size and the lon/lat source's collapse towards the poles. What a machine does with it

The same number of cells, in two shapes

Moving a field between two cell schemes loses 18 per cent of it per cell in one geometry and 39 in another, and the earlier measurement could not say whether that was the shape of the cells or the ratio of their sizes, because changing the schemes changed both. Holding the counts settles it: the count ratio decides most of the loss, and the shape is still worth a quarter of the field.

 An error in a rotation rate is a longitude error that never stops growing. Where the prime meridian of each body has got to, if its published rotation rate is wrong by one unit in its last published decimal. Every line is straight through the origin, because the error is the rate error multiplied by the elapsed time and nothing else — there is no date after which it settles. Jupiter reaches 2279 metres at the equator after a century; Mars, whose rate is published to twelve decimals rather than seven, reaches 0.0011 metres over the same interval. What the numbers refer to

A longitude that drifts with the rotation rate

Ten essays here map bodies whose shape is the problem. A longitude is not about shape: it is a landmark plus an extrapolation over however many days have passed, and an error in the last published decimal of a rotation rate is a coordinate error that grows without bound in time.

 What reading the seven parameters under the wrong convention costs. The distance between the two published conventions' answers for the same parameter set, at the worst point of the world. The difference is 2s(r × X) exactly — twice the rotation, crossed into the position — so a parameter set with no rotations is immune and one with a large rotation is not: DHDN reaches 152.3 metres and NAD27, whose published transformation is three translations and nothing else, reaches zero. The note beside each bar is the size of that datum's rotation. What the numbers refer to

The rotation has two sign conventions

Eleven essays price a datum transformation's parameters, their fit, their residuals and their uncertainty. None asks what the numbers mean: three of the seven are published under two conventions whose rotations differ in sign, and reading one set with the other formula costs exactly twice the rotation — 55 metres on OSGB36 and 152 on DHDN.

 What a rebinning loses depends on where the target's edges are. Two grids of fixed counts, fixed shapes and fixed resolution, with the target slid across the source from perfect alignment to a full cell. Nothing about either grid changes except where its boundaries fall. The loss runs from 27.5 per cent at zero to 56.3 at half a cell — a factor of 2.04 — and the longitude-only curve returns to its starting value at a full cell to six decimal places, which is the periodicity check. A cell boundary that coincides with a target boundary loses nothing, and a grid comparison that does not say where its boundaries are has left that out. What a machine does with it

When the edges do not line up

Rung eight held the cell counts equal so that shape could be compared without the count ratio drowning it, and recorded a doubt: a longitude–latitude source shares its boundaries with a longitude–latitude target wherever their counts share a factor. The mechanism is real and worth a factor of two. It was not what the published number was made of.

 There is no such thing as the radius of the Earth at a latitude. The two principal radii of curvature of WGS84, against latitude, with the mean radius that every table prints drawn across them. The meridional radius M runs from 6335.44 km at the equator to 6399.59 at the pole; the prime-vertical radius N from 6378.14 to the same value. They differ by 42.70 km at the equator and meet only at the pole, and the single number 6,371 km lies between them at no latitude where either is right. What the numbers refer to

The radius of curvature is two numbers

Twelve essays work on the ellipsoid and every one of them takes a radius when it needs one. There are two at every point, they differ by 42.70 kilometres at the equator, and the single number every table prints is out by 5,583 parts per million on a line running north.

 The same 2-pixel road at three latitudes, zoom 5. The dark bar is the mark as drawn — 2 pixels, identical in all three panels, because that is what the stylesheet says. The pale band behind it is the ground that mark covers, drawn to one common ground scale: 9.78 kilometres at the equator, 6.92 at 45° and 1.70 at 80°. The reader sees the dark bar and is being told about the pale one. What a machine does with it

The road is drawn two pixels wide

Seven rungs measure what a screen map does to position. Nothing on a map is a point: every mark has a width, the width is chosen in pixels, and a two-pixel road covers 9.78 kilometres of ground at the equator and 1.70 at 80° north. That is a generalisation applied at a strength varying by a factor of six across one sheet, by a stylesheet with no latitude in it.

 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.

All threads