Cartographic Projection
every map is wrong, and by exactly how much
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The thread: Measured, not named — page 11

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 241 to 264 of 292.
The same baseline, turned. The part of a 20 km height difference a degree-360 geoid model omits, against the direction the baseline runs, at four anisotropy ratios. A ratio of one is the model rung 9 used and is a flat line — the isotropic covariance cannot depend on a direction, by construction, which is the whole of the objection. At a ratio of two the same baseline omits 165 millimetres along the grain and 241 across it. What the numbers refer to

The correlation is not the same in every direction

Every number in the previous rung came out of Σ cₙ Pₙ(cos ψ) — a covariance that depends on the angular distance and nothing else. Ground has grain: at a modest anisotropy the same 20 km baseline omits 165 millimetres along it and 241 across, and the isotropic answer understates the worse direction by 18.4 per cent.

 Twenty-four versions of one shape, and not one of them gains area. The same closed boundary rotated twenty-four times and simplified at the same tolerance. If the area error were noise the values would straddle zero and their mean would fall towards it; they do not. Every one is negative, the mean is -0.4644 per cent, and the mean is 71 standard errors from zero. A bias of that size cannot be removed by averaging over more boundaries, which is the only defence anybody has against a rounding error. What a machine does with it

A thousand features are wrong in the same direction

The area a simplification costs is unpredictable in sign for one feature. Over a population it is not: twenty-four presentations of one shape all lose area, the mean is seventy standard errors below zero, and no amount of aggregation removes it.

 a transform boundary: what the ground is doing to itself. The same 72 places as the velocity field, with the strain rate computed from the motion's own four partial derivatives rather than from its size. Each cross carries two principal rates: the long stroke is the greater extension, the barred one is shortening. The largest second invariant anywhere here is 241.0 nanostrain/yr, and the arithmetic is the same arithmetic that reads a projection's indicatrix. Drawn in Azimuthal equidistant centred on the window. What the numbers refer to

The ground has an indicatrix too

Two hundred and forty-six essays hold the Earth still while the page is measured. The ground is moving at tens of millimetres a year, and the motion is a map with four partial derivatives — so the same construction that draws Tissot's ellipse draws one for the ground, and the fastest plate in the model returns exactly nothing.

 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.

 One velocity field, three frames. a transform boundary, drawn three times. The ground is the same ground and the arrows are not the same arrows: the fastest velocity anywhere in the 3 panels is 70.6 mm/yr and in the quietest panel the same places are nearly still. A frame is a choice of which rotation to subtract, and there is no measurement that picks one. Every panel has the same strain rate at every point. What the numbers refer to

A velocity needs a frame and a strain rate does not

One place on a plate boundary moves at 5.3, 7.0, 10.2 or 54.6 millimetres a year, and towards the south-east or the north-west, depending on which rotation was subtracted first. Every one of those readings reports the same strain rate, to two parts in a hundred thousand million.

 A 900 km circular accuracy at 55° north, projected. Six thousand ground positions drawn from a circular error of 900 kilometres about one place, each projected in Mercator and plotted as a displacement from the projected place. The curve is the nominal 95 per cent ellipse, computed the standard way — the ground covariance sandwiched between the projection's own derivatives. It holds 93.83 per cent of the points, the cloud is measurably longer than it along its own long axis by 5.52 per cent, and it is not symmetric: the third moment along the page's second axis is 0.727 rather than zero. Measuring distortion

The error ellipse is not an ellipse

Rung two pushed a covariance through a projection with the same matrix sandwich that draws an indicatrix. That is a first-order operation on a map with a second derivative, so the propagated distribution is not the ellipse the sandwich draws — and a nominal 95 per cent ellipse holds 93.06 per cent on one projection and 95.63 on another, in opposite directions, from the same input.

 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.

 The same resolution, the grid moved, and a different answer. The same field at a fixed 18 × 9 division, with the grid slid by fractions of a cell. Nothing is lost — every cell is the same size as before and there are exactly as many of them — and the largest reported value moves over a range of 12.59 per cent. That is more than a whole halving of the resolution costs, which is 10.15 per cent on the same field. The scale effect has an excuse and this one has none. What a machine does with it

The answer depends on the cells it was counted in

Nine essays price the cell as a shape. The number reported out of it is priced nowhere: sliding a grid without changing its resolution moves the largest reported value by 12.6 per cent, which is more than halving the resolution costs.

 How long a network has before it breaks its own tolerance. The years until the worst baseline in a 400-kilometre network exceeds five millimetres plus one part per million of its length, which is an ordinary first-order specification. A boundary zone breaks it in 16 years; a plate interior takes 1819; a rigid plate never does, at any speed, because a rigid body keeps every distance it has. Bars are clipped at five thousand years. What the numbers refer to

A grid stops fitting the ground it was laid on

A hundred-kilometre baseline across a plate boundary changes by 815 millimetres in fifty years, and the same baseline on the fastest plate in the model changes by nothing at all in the same fifty. The interval before an ordinary first-order specification is broken is six years in one place and never in the other.

 How aligned a region's ellipses are, for every projection and every region. The resultant length of the doubled indicatrix orientations, over 12 projections and 9 regions. One means every ellipse in the region points the same way; zero means they are spread evenly and cancel. Three cylindrical rows are 1.00 throughout, and everything else varies down the row and across it — which is the answer to the question the number was first stated without: alignment is a property of the pair. Mercator's row is the exception that is not a measurement, because a conformal projection's indicatrix is a circle and a circle has no orientation. Measuring distortion

Whether the ellipses point the same way

The previous rung found that alignment decides whether the average of a region's deformations is above or below the deformation of its average, and stated it at ten projections over one region. Swept over a hundred and eight pairs, the answer is that alignment belongs to the projection 28 per cent, to the region 38, and to neither 33 — and two rows of the table turn out not to be measurements at all.

 Four cities, met — the triangular, x first construction. Every cell drawn here holds the same area of ground and a different area of page: the map has been constructed so that its areal scale factor is the density it was handed, cell by cell, over a contrast of 25.99 to one. The residual against that target is 5.1e-6, measured from the map's own derivatives rather than from the construction. What it cost is the shape: the redistribution alone reaches 139.1° of angular deformation and averages 70.4°, on top of whatever the equal-area projection under it was already doing. Measuring distortion

A map drawn to a density it was handed

Two hundred and twenty-six essays measure distortion after the fact. This one specifies it: a density is handed to a map as a boundary condition, the areal scale factor comes out equal to it to five parts in a million, and every other invariant the site owns becomes the price.

 A ground that is not deforming, read off a Mercator sheet. Every place on this map is carried by a rigid rotation of the whole Earth at forty millimetres a year — the motion that deforms nothing, and that this collection has already shown deforms nothing. The circles are the strain rate a geodesist would report from the grid coordinates alone, up to 17.4 nanostrain/yr. None of it is on the ground. It is the projection's own scale factor changing along the displacement, which is a second derivative of the map arriving in a first-order measurement. What the numbers refer to

The strain a map adds to the ground's

A ground carried rigidly at forty millimetres a year deforms nothing, and read off a Mercator sheet at sixty degrees north it reports 10.9 nanostrain a year. Along a profile through a real boundary the invented part is 0.0 per cent of the answer where the zone is loud and 884 per cent where it is quiet.

 One of these three lines does not slope. Three lengths at each zoom, for a world coordinate near the antimeridian. The pixel halves with every level, as it must. The spacing between representable double-precision values is far below it and halves with it. The spacing between representable SINGLE-precision values does not move at all — 2.0 metres at every zoom, because it is a property of the size of the number and the world does not get smaller. The two cross at zoom 17, and past it a vertex snaps to a lattice coarser than the pixels it is drawn into. What a machine does with it

The renderer runs out of numbers before the zoom does

Eleven essays price the pyramid in exact arithmetic. The pipeline that draws it carries single precision, where a world coordinate near the antimeridian quantises to two metres — at every zoom, because the number does not get smaller when the pixel does. At zoom 22 that is fifty-four pixels.

 A compacting basin, and the tilt it produces. The stated vertical velocity field — a bowl of subsidence 120 km across, with no mass leaving — drawn as circles proportional to the rate, with the tilt of the ground surface as the arrows. The subsidence is largest at the centre and the tilt is exactly zero there, because a smooth bowl has no gradient at its own bottom. The largest tilt is 125.5 nanoradians a year, on a ring at the bowl's own scale length over root two, and it is the quantity a levelling network measures. What the numbers refer to

A vertical rate needs a height system

Four rungs of this anchor measure the two-by-two horizontal tensor, because that is what a tangent chart returns. The larger signal in a subsiding basin is vertical — 126 nanoradians a year of tilt against 0.76 nanostrain a year of horizontal strain — and it is not a measurement at all until the surface it is measured against is named, because that surface is moving too.

 The query, the box it becomes, and the 14 per cent it loses. A query for everything within 400 kilometres of a place at 55° north, drawn in the stored coordinates of Web Mercator. The ring is the true answer's edge; the rectangle is the box the index is given, sized from the scale factor at the query point. The arcs outside the rectangle are true neighbours the index never returns — 13.6 per cent of the rim, reaching 5.0 per cent of the box's own half-width beyond it — and nothing downstream can tell they are missing. What a machine does with it

The query a fast path actually answers

Fifteen rungs price what a stored coordinate means and none asks what it is searched with. No index answers "within two hundred kilometres of here"; an index answers "inside this rectangle of stored coordinates", and the rectangle is built by somebody's arithmetic — which at 55° north silently drops a tenth of the true answer on a conformal projection and three fifths of it on an equal-area one.

 Four cartograms of one density. The same stated density — four cities — met four different ways, drawn on the same cells. Every panel is a correct cartogram of the same numbers: a region's page area is proportional to its mass in all four. They do not look alike, because the areal scale factor fixes one number per point and a map has four derivatives, so three degrees of freedom per point are left over and each construction spends them differently. Measuring distortion

Every density can be met and none is free

Four maps of the same data, all of them correct, charging between 57.6° and 104.4° of angular deformation for it. There is no such thing as the cartogram of a density — there is an infinite family, and somebody picked a member of it without saying so.

 The sea surface, against the surface heights are measured from. The stated mean dynamic topography — how far the sea stands above the geoid — drawn on Mollweide, with the eight tide gauges marked. It ranges over 2.09 metres, it is smooth, and it is what every national datum's zero is sitting on. Nothing here is a model's output: it is a stated closed form, chosen to have the observed sign structure and the observed size. What the numbers refer to

Every country's zero is a different surface

Ten rungs measure a height against a geoid and treat the geoid as one object. No national datum is on it: each is pinned to the mean sea level at one tide gauge, the sea surface stands up to two metres from the geoid, and the eight European zeros measured here spread over 462 millimetres — a step no levelling can remove and nobody's error.

 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.

 Seven published numbers, twenty-eight actual ones. The correlation matrix of a seven-parameter fit to 64 common points over a region 9° across. Only the diagonal is ever published — seven standard deviations — and the twenty-one off-diagonal entries are not small: the strongest is ty against rx at 0.940. A translation and the rotation that mimics it over a small patch are very nearly the same parameter, so the fit cannot tell them apart and its errors in the two are locked together. What the numbers refer to

The parameters are not independent

Rung seven gives the seven parameters their own uncertainty and stops at seven numbers. There are twenty-eight, and the twenty-one nobody publishes are not small: a translation and the rotation that mimics it correlate at 0.94, the normal matrix has a condition number of 4 × 10¹⁶, and propagating from the diagonal alone overstates the transformation's uncertainty by up to a factor of thirty-six.

 A ruler on a cartogram is not measuring anything. Twelve pairs of places, each measured on the ground and on the page of the triangular, x first cartogram of four cities, with the page scaled so that the median pair reads exactly right — the most generous calibration available. The line is that calibration. The worst pair, Tokyo to Quito, reads 21598 km for a ground distance of 14439 km, an error of 49.6%. The open marks are the same pairs on the equal-area map the cartogram was built from. Measuring distortion

Everything else on the page pays for the areas

A cartogram gets one quantity exactly right and every other reading a page supports is collateral. A ruler on it is out by 35 per cent after the most generous calibration available, and ten of sixty triples of places change which one is in the middle.

 The same labels, placed once and placed per tile. 140 stated label boxes on a 1024-pixel page cut into 256-pixel tiles. The pale boxes are placed the same way by both rules. The dark ones are placed by the tiled renderer and suppressed by the global one — labels that should have lost a collision with something in the next tile and did not, because the tile that drew them could not see it. nine of them, against 104 labels the global rule keeps. What a machine does with it

A label belongs to no tile

Nine rungs price the tile as a piece of geometry. A label is not geometry — it is a page object placed by collision against other page objects, and collision is a global relation while a tile is rendered alone. Cut a page into tiles and 10.3 per cent of the labels are placed differently; the buffer that closes the gap is half a tile at moderate density and a whole one when it is crowded.

 A legend saying "illuminated from 315°" is true at one longitude. A hillshade's azimuth is measured from the top of the sheet, because the shading is computed on the projected raster. The top of the sheet is grid north, so the compass bearing the light comes from is the declared azimuth plus the meridian convergence, and that varies across the sheet. On a conic it swings by 55.3 degrees over eighty degrees of longitude. On a cylindrical projection in its normal aspect it does not swing at all, which is the flat line — the only case the legend is right everywhere. Measuring distortion

The light comes from a page direction

A hillshade's illumination azimuth is declared from the top of the sheet, and the top of the sheet is grid north. On a conic the light therefore swings 52.6° across eighty degrees of longitude, and 72.8 per cent of the sheet is shaded differently from what the legend claims.

 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.

 Three selection rules, and what each one keeps. Keeping one feature in ten from a stated population whose size distribution has a Pareto exponent of a half — the exponent Töpfer's law is a theorem about. Keeping the largest carries 99.99 per cent of the total size and inflates the median feature by a factor of 95. A random sample keeps the median to 1.068 and carries 5.0 per cent of the total. The two rules are right about different things and there is no rule that is right about both, because the total lives in the tail and the median does not. What a machine does with it

Which features survive is not a sample

The rung below answers how many features a scale can carry and treats the population as a number. Which ones survive is a different question: keeping one feature in ten carries 99.99 per cent of the total length and inflates the median feature by a factor of 95, and the shape of the size distribution survives both exactly.

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