Tolerance — where it appears
Named by 144 essays across 9 fields — each of them below, with the objects they name alongside it.
A screen map is a pyramid of tiles
The scheme every slippy map runs on is a coordinate system with three integers and one projection, and almost all of it is forced. A square world is what makes the quadtree work, the levels are exact powers of two, and the published resolution — 156,543 metres per pixel at zoom zero — is a distance on the ground at exactly one latitude.
The seven parameters, and what each one does
A datum transformation is published as three metres, three arcseconds and one part per million, in three different units — which hides the fact that all seven are the same size of thing. Converted to ground displacement, the scale term beats two of the translations.
The scale of a screen map is not one number
A zoom level prints one scale for the whole world, and the map is at that scale along exactly one line. At 60° north the picture labelled 1:136,495 is a 1:68,765 map — and the factor is not quite sec φ either, because the projection puts a geodetic latitude into a spherical formula.
The aspect is a free choice
A projection's distortion pattern is fixed relative to its own axis, and where that axis points is entirely up to the cartographer. Rotating it is the cheapest available improvement and it is the one most often left unmade.
The scale factor of a line
A scale factor is a property of a point and a distance is not measured at a point. The three rules in use for averaging one along a line differ by a hundred metres on a 362-kilometre baseline, and which of them is adequate is decided by the line's direction rather than its length.
A scale bar is right in one place
The bar in the corner of a world map is a picture of a distance, and it is a true picture along one line. On Mercator it reads 500 kilometres for a thousand at 60° north — and on an equal-area map it reads 500 one way and 2,000 the other, so the projection recommended for measuring is the one on which no single correction exists.
The scale factor was chosen
A grid's scale factor is not a constant of nature. Britain's 0.9996012717 is the reciprocal square root of the worst distortion a tangent projection would have had over the country, it halves the worst-case error exactly, and the halving is the most that construction can ever buy.
Where a fit leaves residuals
Seven parameters can carry a rigid motion and a size exactly. A triangulation network is neither, so the best possible transformation between two datums leaves metres on the table — in a pattern, not as noise — and which seven parameters come out depends on where the markers were.
Measuring instead of naming
A projection is called conformal because that is its name. Running the definition as a computation over several hundred points takes about twenty lines, catches a projection the whole internet uses, and is almost never done.
The pixel is a place with a size
Drawing a coordinate rounds it to a pixel, which moves it by up to half a diagonal — 16.8 metres at zoom 12. Whether two points 43 metres apart appear as two dots is not a property of the data at all: they merge for 12 per cent of the positions the tile grid could take, and the closed form predicts 12.4.
Molodensky's shortcut
The exact way to shift a datum needs an iteration nobody wanted to run on a 1950s machine, so a direct formula was derived instead. It lands within centimetres — and the abridged version everybody quotes is a different formula, worse by a factor of fifty.
The units are part of the coordinate
Two feet were in use in the United States until 2022, differing by exactly two parts per million. A plan drawn in the wrong one has every dimension right to a fifth of a millimetre and sits three-quarters of a metre from where it belongs, which is why it passes every check anybody runs.
When a formula is not enough
National mapping agencies distribute datum shifts as tables of numbers on a grid rather than as parameters. What decides the spacing that table needs is not the size of the shift — a hundred-metre shift tabulates coarsely, and a thirty-centimetre ripple riding on it does not.
A tile is drawn without its neighbours
Independence is what makes the scheme scale: one request touches one square of the world. It also means a feature crossing nine tiles is nine features, each labelled at its own centroid — up to 1,863 kilometres from the whole one — and the displacement is bounded by the size of the feature rather than the size of a tile, so it grows as the reader zooms out.
Designing a grid for one region
A country with a grid of its own beats the international zone that would otherwise carry it by a factor of 1.51. That is worth having and it is much less than the argument is usually made to sound, and the difference between the two numbers is the whole case for and against national grids.
How small is flat enough
A builder works in plane coordinates and a national mapping agency does not, and the line between them is not a convention. The unavoidable error of treating a patch of the Earth as flat grows as the square of its size, and the size at any stated tolerance is a number.
Where two zones meet
A twenty-kilometre baseline across a zone boundary is computed correctly in either zone — the two answers differ by nanometres — and taking one end from each returns 392 kilometres. Both zones are right and the seam between them is not a small error but a category one.
Flying a curve in straight legs
Nobody steers a great circle, because a great circle requires the heading to change continuously. What is actually flown is a handful of constant-heading legs between waypoints on it, and the gap between plan and curve falls as the square of the number of legs.
The epoch is part of the coordinate
A latitude and a longitude with no date on them are incomplete, because the ground they refer to is moving at between ten and seventy millimetres a year. Australia crosses an ordinary survey tolerance in under two years and moved 1.8 metres between two versions of its own datum.
A grid scaled to the ground is not a map
Construction sites work in coordinates where a tape reading equals a computed distance, which is achieved by multiplying the national grid by a constant. The result is exact at one point, degrades east–west forty times faster than north–south, and is not a map projection at all.
Height above what?
A satellite reports a height above a mathematical surface. A level and a staff report a height above the surface water settles on. The two disagree by tens of metres, both are correct, and only one of them decides which way a pipe drains.
The indicatrix is a limit
Tissot's ellipse describes an infinitesimal circle, and every published one is drawn finite. The error of the description falls as the radius rather than as its square — halving the circle halves the lie — and on the Robinson projection at a table entry it does not fall at all: sixty metres and six kilometres are both 1.17 per cent wrong.
A degree is not a unit of length
"Within 0.01 degrees" is a condition anybody can write and no instrument can measure. On the ground it is an ellipse — 1,106 metres north–south and 558 east–west at 60° — and even on the equator it is not a circle, because the meridian's radius of curvature is smaller than the parallel's by the flattening.
How many sheets an atlas needs
A tolerance on the scale error inverts, through Chebyshev's bound, into a sheet radius — and a covering problem turns the radius into a count. One part in a thousand costs 1,210 sheets of 403 kilometres radius, the count goes as the reciprocal of the tolerance exactly, and the projection multiplies it by anything from one to fifty-six.
Curvature that varies from place to place
The impossibility this site is built on was argued on a sphere, where the curvature is one number. On the Earth it varies by 1.35 per cent, on Jupiter by 31, on Vesta by a factor of 2.7 — and on a body with three axes it varies along a parallel, which no formula in latitude can express.
What a tape measures
Four steps stand between an instrument reading and a coordinate, and their sizes are not in the order anybody expects. On a 77-kilometre line the slope reduction is 120 metres, the grid reduction 30, and a step nobody names — using the height above sea level where the height above the ellipsoid is wanted — is 583 millimetres.
Computing an area needs a surface
A shoelace over a ring of coordinates returns a number whatever the coordinates are. For one twenty-by-ten-degree cell it returns 1.75 times the true area in degrees, 3.06 in a conformal plane, and exactly the closed form in an equal-area one — and the closed form itself is 0.45 per cent out, because the sphere is a model too.
A traverse must close
The check every survey specification leans on cannot see the error this whole field is about. Scale every leg of a closed figure by four hundred parts per million and it closes to the last bit of a double, while every dimension in it is wrong.
A levelled height is not a distance
Level surfaces converge towards the poles by five metres in a thousand, so a chain of perfectly executed levelling observations does not sum to a height difference. The correction over four hundred kilometres of northing is larger than the network's own closure.
The plumb line is not the normal
A latitude measured from the stars and a latitude that means a position on the ellipsoid are different angles, because a plumb bob hangs along gravity and gravity is not perpendicular to a mathematical surface. Ten arcseconds of difference is 309 metres of ground.
The two ways to spread a misclosure
Bowditch's rule and the Transit rule take the same closed figure and the same misclosure and disagree about which legs were wrong. Both close it exactly, neither puts the correction on the leg that actually carries the blunder, and no measurement can settle which is right.
A straight segment is a claim about a plane
Two exact endpoints, joined by a straight line in the plane the file is stored in. On the ground the line is 718 kilometres from the route it claims between New York and London, and 2,961 between London and Tokyo. The departure grows as the square of the length — fitted exponent 2.001 — so a stated tolerance costs vertices as a square root.
Setting out runs the chain backwards
Turning a design coordinate into something to observe on the ground means undoing the reduction chain, and undoing a chain reverses the order as well as the operations. Two of the corrections do not commute, and getting them the wrong way round misses by a millimetre at fifty kilometres.
The ground is not the grid
A tape measure on a hillside has to be brought down to the ellipsoid and then out onto the map, and the two corrections have opposite signs. On a grid whose scale factor exceeds one there is exactly one elevation where they cancel — 2,551 metres, for a factor of 1.0004.
The second derivative has its own ranking
Rank eight world maps by how much they stretch and Mercator comes sixth of eight. Rank the same eight by how much they bend and it comes third. The two halves of the second-order score disagree with each other more sharply than either disagrees with the first-order one — Spearman 0.45 against 0.69.
Two projections that cannot be told apart
Over a small enough region every projection is the same picture, so the question is how small. The answer is not one number: two conformal projections need twelve degrees of extent before their graticules can be separated, and two projections with different anisotropy separate below one.
The third coordinate moves too
A datum shift is quoted as a horizontal displacement because horizontal is what people look at. The transformation acts on a three-dimensional point, and its vertical component is between a quarter and a half of the horizontal one — 51 metres, for a British coordinate.
What a face can preserve
The obvious map onto a polyhedron's face is the gnomonic, and it draws every great circle straight while stretching the corners by a factor of 1.50. Replace it with a construction that holds the areal factor at one to a part in a million and the shape error rises from 8.0° to 11.9° — the same trade the whole sphere forces, arriving on a piece of it a twentieth the size.
The second derivative over a region
Flexion has been measured at points and over the whole sphere, and never over a region — which is the only unit anybody chooses a projection for. Doing it finds that the second-order criterion moves the winner in one region of four, and that one projection in the library has no second derivative at all.
Nearest is a question about the metric
Asked in the plane the data is stored in, a nearest-neighbour query returns a different site for 5 of 400 queries over Europe on Web Mercator — and 103 of 400 on the plate carrée, with the wrong answer up to 309 kilometres further away. Every disagreement is a contest closer than the region's own scale spread, and over a city there are none.
A local model has an order
Every georeferencing tool fits a polynomial between two coordinate systems and the choice of degree is usually made by counting control points. What it buys is an order of convergence — 2, 3 and 4, measured — and the first term an affine model cannot hold is the second derivative this ladder has spent five essays on.
The tolerance decides the model
Every correction inverts to a distance — the line length at which it alone exceeds a stated tolerance. The ranking those distances produce is the same at a millimetre and at a decimetre, and it is different for two jobs at the same tolerance on different ground.
Reprojecting a raster invents values
Moving a picture from one projection to another moves no coordinate — the maps are exact both ways. What is lost is that a target cell's centre does not land on a source cell's centre, so a value has to be made up for it, and the making-up has an order of convergence: 1.00 for nearest, 1.98 for bilinear, 2.93 for a cubic, measured by refining the grid.
The flattening is not a free parameter
An ellipsoid is usually presented as two numbers somebody fitted. One of them is not free — Clairaut's theorem relates the shape of a rotating body to the gravity on it, and the relation holds on WGS84 with a residual of 3.1×10⁻⁵ — which is 2.74 times f², exactly what a first-order theorem is entitled to.
A corridor has a width the page cannot keep
Buffering a line is the most-run operation in spatial analysis and the corridor it produces is a reach set with two failures nobody separates: stroked on the page, one width covers 190 to 471 kilometres of ground along a single route; computed in closed form, the formula stops being the area at a width the route's own length fixes, and eventually claims more ground than the sphere has.
A map with no formula
The solved projection has no name, no formula and no closed-form inverse. It is fourteen numbers — and the rate at which those numbers fall away decides whether a map can be shipped at all: geometrically for a smooth region, and like a power for one with corners.
The figure of the Earth was measured
Two expeditions went to Lapland and Peru to measure the length of a degree of latitude, and the whole signal separating a flattened Earth from a round one is a kilometre in a hundred and eleven. Inverting two arcs amplifies their error by 118 — and a kilometre of error returns a lemon-shaped planet.
A published coordinate is a result
Re-observe a control mark perfectly, with an instrument good to twenty millimetres, and the answer disagrees with the published value by a hundred metres. Neither number is wrong. The difference measures the interval between two definitions.
The aspect has three numbers, not one
The site's aspect search has swung the projection's axis through one plane for a long time, and recorded that a region whose long axis runs diagonally has its optimum somewhere that plane never reaches. Searching the whole sphere of pole positions finds 2.6 times more improvement over Japan and 3.4 over the conterminous United States.
The places where a map is exactly right
Nine essays on this ladder say a map cannot be right everywhere. None asks where it IS right — and the answer is a curve, a pair of curves, or two isolated places, never a patch. Measured across fourteen projections the set's neighbourhood shrinks with an exponent of 0.48, 1.0 or 2.0, and the value the impossibility forbids is 0.
Zoom is a ladder
A tiling scheme exists only at integer zoom levels a factor of two apart, so a request for 1:25,000 is answered with 1:34,124 — 36 per cent coarser, and 86 per cent coarser in area. The mismatch runs from 1/√2 to √2 and repeats identically at every doubling, and nothing anywhere reports it, because the map that arrives is a perfectly good map of something.
The sphere is not the plane at small counts
The site's atlas arithmetic multiplies an ideal sheet count by 2π/√27, the thinnest covering density of the plane. At the four counts whose optimal covering of the sphere is a theorem the plane's number is 21 per cent high at two caps and 9, 5 and 2 per cent low at four, six and twelve — wrong in both directions, and the direction changes with the count.
Two parameter sets, one transformation
Agencies publish seven-parameter datum transformations that differ by hundreds of metres in translation, and the usual reading is that one of them is better. Over the region either was fitted to they are the same transformation: a hundred metres of translation, re-absorbed by the rotations and the scale, moves a British coordinate by 5.6 metres and an Australian one by 193.
When the answer is not in the library
Fitting twenty candidates to a map and ranking the residuals always produces a winner, which makes it a ceremony unless it can also produce a refusal. Held out of its own library, a Mercator map is named as a conformal conic, leaving 0.4 per cent of the map's width unexplained — and the quantity that tells the two situations apart is not the residual but the margin, which is 1.6 when the truth is absent and 10¹³ when it is present.
An area on the grid is not an area on the ground
A grid's scale factor is a property of lengths and what a surveyor sells is an area. Squaring a departure doubles it, so a hectare drawn on the British grid at its central meridian has 10,008 square metres of ground under it — and correcting an area with the line factor instead of its square leaves half the error behind.
What a closed figure cannot see
A closed traverse imposes exactly two conditions on its observations, so everything else is free — and the freedom is not spread evenly. At a twenty-millimetre tolerance the worst-placed station in a seven-station loop hides an angle blunder of 2.1 arcseconds and the station standing seventy-two metres from the close hides 57, because a rotation about a point near the finish moves the finish hardly at all.
A centroid belongs to a plane
Every renderer labels a region at its centroid, and every centroid is a shoelace over coordinates as stored — which is a statement about the plane they are in. Six planes put the middle of one 20° × 20° region up to 273 kilometres apart, the equal-area member is 66 kilometres out, and the disagreement falls as the square of the region's size.
A height that is not a length
Level surfaces converge polewards, so the surface a lake sits on is 5.28 metres lower at the pole than at the equator and a height system that reports lengths says a lake runs downhill. The fix reports a number that is constant on the surface and is not a distance from anything: a hundred-metre climb raises it by 99.73 metres at the equator and 100.26 at the pole.
A coordinate is a number with a width
Every number on this site so far has been exact. A written coordinate is not: five decimal places of a degree is 1.112 metres of latitude everywhere and 1.112 metres of longitude only on the equator, falling to 0.097 at 85°. The same written precision carves the ground into a cell that is square in one place and eleven times longer than it is wide in another.
Inside is a claim about the edges
Whether a point is inside a polygon is not a property of the point and the polygon. It is a property of the plane the edges were understood to be straight in, and between two readings of the same file there is a band of disagreement — 29 per cent of one triangle's area, 4.2 million square kilometres, and the file does not say which reading it means.
How big a triangle it takes
Gauss proved that a surface-dweller can read the curvature off a triangle's angles. Doing it is another matter: a fifty-kilometre triangle has 5.49 seconds of excess, a one-second theodolite gives that excess a standard deviation of 1.73, and reading K to one per cent needs a side of 281 kilometres. Not one of the great surveys built a triangle within a factor of three of that.
The chain the satellite does not have
The four-step reduction from a tape reading to a grid coordinate exists because a tape does not know where it is. A satellite observation does, so the same distance can be got by projecting both ends and subtracting — and the two answers agreeing to a fifth of a millimetre is the first check the chain has ever had.
The same projection on a different body
A projection's distortion does not depend on the body's size at all — only on its flattening — so Web Mercator's 0.3848° of angular deformation is not a fact about Web Mercator. It is 2f radians, and the same mistake made on Mars measures 0.6765°, on Jupiter 7.68°, and on a body scaled ten times larger than Mars the identical number to twelve figures.
The most compact shape depends on the paper
A geodesic disc attains the isoperimetric bound on a sphere — its score is one, exactly, at any radius. Score the same nine regions from their images on ten projections and four of them put something else on top, an oval and its own 45° rotation come out 4.7 per cent apart, and the projection that preserves the order is not the equal-area one.
Two charts are enough, and one is not
The topological minimum for an atlas of the sphere is two sheets, and the counting fixes a price nobody chose: the worst point of any two-chart atlas is 90° from a chart's centre, where a conformal chart's areal factor is exactly 4 and an equal-area one's angular deformation is 38.94°. A national series has a hundred and twenty thousand sheets, and a hundred and twenty thousand minus two of them are bought by accuracy.
The score is not stable at any scale
One boundary, read at eight resolutions from sixteen points to two thousand and forty-eight: the compactness score falls from 0.980 to 0.834 and is still falling. Changing the projection instead moves it by 0.69 per cent. The two decisions are made by the same person on the same afternoon and only one of them is ever reported.
A cell system trades area for shape
A grid can hold every cell to exactly the same area or hold every cell nearly square, and the measurement says it cannot do both: the equal-area cube's areas agree to a part in a thousand and its worst cell is 1.29 times as long as it is wide, while the tangent-warped cube holds shape to 1.19 and lets area vary by 20 per cent.
Sixty zones was a decision about one latitude
Twelve rungs price a grid, a zone, an origin and a reference, and every one of them works inside a single zone. The number of zones has never been asked about: six degrees meets its tolerance at the equator and is loose everywhere else, so a system spending the same tolerance evenly would use 51 zones at the equator and 8 at 82° — and UTM's worst error is 981 parts per million, not the 400 always quoted.
Every reach set ever drawn is too small
An isochrone is drawn by walking out along a finite number of bearings and joining the points, so its vertices are on the true boundary and its edges are chords — which puts the drawn set inside the true one, always, at every count, for any convex reach set. The deficit falls as the square of the count, and a spherical cap loses less than a circle by exactly cos t (1 + cos t)/2.
A query is a disc, and a disc is not a cell
Everything a cell system does is an address lookup except the one question anybody actually asks it: find everything within five kilometres of here. That is a disc, and the number of cells it fetches is not its area divided by a cell's — at the radii a query is really made at, it is three to seventeen times that.
A deflection is the slope of a mass
One arcsecond of deflection of the vertical is 4.85 millimetres of geoid slope in a kilometre, and that exact relation produces no deflection anywhere by itself, because a deflection is the gradient of a geoid and a geoid model is cited rather than computed. State a mass instead, and every quantity is a closed form: a salt dome five kilometres across bends the plumb line by 2.21 arcseconds.
The address is a curve through the sphere
A database does not fetch a set of cells, it reads ranges of identifiers — so the cost of a query is how many runs its cells form, not how many cells it needs. Hilbert order wins that measurement and loses the one usually quoted for it: its neighbouring cells are further apart in identifier than row-major's, on average and at worst.
An edge has no order of convergence
On a smooth field the three resampling kernels converge at orders 1, 2 and 3 and the choice is obvious. Across a discontinuity they converge at 0.78, 0.58 and 0.60 — within a factor of 1.4 of each other, in an order that puts nearest-neighbour first, and a real raster is mostly edges.
The tolerance that decides the verdict
Every verdict handed out about a projection's claimed property rests on a tolerance chosen once, long ago, at sixty times a measured noise floor. Swept, it decides nothing: the population is bimodal, the tolerance sits in a gap 238,000 times wide for conformality and 646 million times wide for equal area, and the verdict with the least room is a passing one whose margin is the arithmetic's rather than the map's.
How wrong a flat picture has to be
A determinant proves no flat picture of four places is exact and leaves the size of the failure where nobody can read it. 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.
A crossing is a chain of decisions
A route is handed back as a curve, and nothing anybody flies is decided once. Re-planning is worth exactly nothing when the forecast turns out right — a sub-path of an optimal path is optimal, and the chain is the plan. When it turns out wrong it recovers 19 per cent of the cost at two decisions and 83 at eighteen, and never all of it.
The reach set takes the shape of the roads
On open ground a reach set is a disc. On a grid of roads it converges to the convex hull of a diamond and a disc, and once the roads are √2 times faster than the ground between them the ground's speed stops mattering at all. A range ring drawn at road speed then claims 57 per cent more ground than anybody can reach.
A drawn reach set stops at the river
A polygon drawn through a fan of bearings round a reach set is short by a chord deficit that falls as the square of the count — on ground with nothing in the way. Put a river three kilometres off, crossable at bridges, and the polygon misses 30 per cent of the reachable ground at every count from forty-eight bearings to a thousand, and a convergence check passes on 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.
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.
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 rule that keeps a route near land is pinned at both ends
A crossing that may never be more than a stated distance from an airfield is refused below one radius and free above another, and each wall is set by a single feature of the ocean. Between them the price falls continuously, with corners and no jumps — and at the lower wall it falls as the square root of the allowance, so the first kilometre is worth twelve of route.
Four radii of the Earth
The ellipsoid is usually handled with an auxiliary latitude. Every such construction 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, 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.
A crossing bends by a law only a conformal chart can show
A ship leaving pack ice for open water takes its quickest route by bending at the ice edge, so that the sines of its angles to the perpendicular stand in the ratio of its two speeds. The law has nothing in it but angles, so on Mercator it can be checked with a protractor — and on a plate carrée at sixty degrees north the same crossing reads as a ship making ten kilometres an hour in the ice rather than six.
The error belongs to a few of the places
The least error of any flat picture of sixteen cities is 53.3 per cent, and it is not spread over the sixteen. Six pairs hold it there and Sydney is in three of them: leave Sydney out and it falls to 25.5, while leaving out any of five other cities changes it not at all. Choosing which places to draw is worth more than any choice of how to draw them.
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.
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.
A place with a size can be drawn to scale
Points on a sphere have distances no flat picture holds. Places are not points: give every place a radius and the distance between two of them becomes a range, and some flat picture is right about every range once the radius passes a threshold — 38 kilometres for London, New York, Tokyo and Sydney, six metres for five towns in Britain, growing as the cube of the set.
A diversion allowance in a wind is the same circle, moved upwind
A diversion rule written in minutes is turned into a distance by assuming still air. In a uniform wind the places from which an airfield can be reached in sixty minutes are not a stretched region but the still-air circle moved upwind by the wind's run. A 150 km/h northerly makes the still-air route across the North Atlantic need a diversion of 68.7 minutes, a southerly makes it 54.0 — and the least radius at which the ocean can be crossed barely moves.
The answer is a set
Fitting a projection to control points produces 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 first break is mostly its denominator
Three earlier measurements 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 best score the region admits at all rises with exponent 0.92 and carries 65 to 70 per cent of the threshold's movement on three projections, while the pass's own depth falls slowly and steadily, with exponent −0.42 to −0.49.
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.
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.
A flat picture has one direction between two places, and the Earth has two
The bearing from New York to Tokyo and the bearing back are not reverses of each other: they miss by 128 degrees. A flat picture draws one line between the two cities, so no picture containing both can have a bearing error below 64 degrees. On any set of places up to about fifty degrees across, the best flat picture sits exactly on that floor — and away from the poles, so does Mercator.
A jet moves the floor that a uniform wind cannot
A uniform wind leaves the least radius at which the North Atlantic can be crossed almost where it was, because it blows the same way on both halves of every gap between airfields. A jet does not. A southerly of 200 km/h past the southern tip of Greenland lowers that floor by 36.9 kilometres, the same jet 400 kilometres further west raises it by 23.3, and one integral of the wind along each gap predicts every such move to within a few kilometres.
The nodes were evenly spaced
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.
An equidistance line belongs to a surface
A maritime boundary is very often defined as the line equidistant from two coasts — a description with no coordinate in it and no curve to choose between. It has a third ambiguity: equidistant measured how. On the ellipsoid, on the sphere, and with a ruler on three different charts, the same four basepoints give lines up to 39.6 kilometres apart and 2,816 square kilometres of seabed between them.
A line of position is Newton's method, but only on a conformal chart
An altitude of a star puts a ship on a circle five thousand kilometres across, and the intercept method replaces that circle with a straight line. On Mercator the fix those lines give is one step of Newton's method: its error falls as the square of the assumed position's, and re-running it turns 300 kilometres into 24 metres in two steps. On a sheet that does not correct longitude for latitude it keeps a fixed share of every east–west error, one less the cosine of the latitude — 36 per cent at 50° north.
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.
A tripoint defined three times
A tripoint is very often not a coordinate in any treaty. It is a description — the point where the boundary between A and B meets the boundary between B and C — and each of those boundaries is itself a description. So the point is defined three times, once by each pair, and under one convention throughout the three definitions agree to half a micrometre. Under three they enclose 6.69 square kilometres.
The sheet moved before it was measured
Every method of reading a projection off a map takes its control points from the sheet and assumes the sheet is the one 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.
A cocked hat holds the ship one time in four
Three lines of position with any error in them bound a small triangle, and the instinct is that the ship is inside. With independent errors it is inside one time in four, whatever the triangle's shape. A common error in every sight makes that almost always when the stars surround the ship and almost never when they do not. And on a sheet that forgets the latitude, the triangle keeps its size and moves off the ship altogether.
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.
A map with no graticule
Identifying a projection starts from 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 river boundary goes where the river goes, or stays where it was
A boundary that follows a river moves when the river moves, and the law of every such boundary turns on one word: whether the river moved gradually or suddenly. Over one meander's life, gradual movement changes the country of about twenty-eight square kilometres of land a loop. A single flood that cuts the loop off strands twenty-six — nearly the same land, arriving as one parcel on the wrong side of the river.
The highest summits stay where they are, and the watershed does not
A treaty that puts a boundary on the highest crests which divide the waters names one line where the water divides on the summits and two where a river has cut back through the range. On a stated range the two part by up to eighteen kilometres. They also differ in kind: an error of ten metres in every height moves the summits three metres and the watershed on the plateau more than a kilometre.
The map that keeps the most ground inside a tolerance
A surveyor does not want the smallest worst case or the smallest mean square. They want as much of the territory as possible within a stated tolerance of true scale and are indifferent to how far outside the rest goes. That is a third criterion, it is answered by a third map, and on a 25° square at half a per cent it keeps 64.1 per cent of the area where the mean-square map keeps 45.4 and Chebyshev's 37.0 — bought by sending a tail out to 5.6 per cent where Chebyshev's worst is 1.5.
The sea is measured from a tide, and the headlands decide how much
A territorial sea is measured from the low-water line, and the low-water line is where a chart's tidal datum meets the shore — so a datum a metre lower moves it twenty metres at a headland and a kilometre across a tidal flat. On a stated coast the twelve-mile limit barely notices the flats: fifty-six square kilometres of shore are uncovered and eleven of sea are gained. A bank that dries at the lower datum adds three hundred and thirty-nine in one step.
An ellipsoid computed to a nanometre is known to a decimetre
The meridian series is exact to seventy-six nanometres on WGS84, and WGS84 is exact by definition: its axis and flattening are conventions with no uncertainty at all. The ellipsoid that actually fits the Earth is a measurement, known to sixteen centimetres from equator to pole, and WGS84 puts the pole sixty-seven centimetres away from it. From its second term, the series is more accurate than anything it is used to compute.
A criterion worth using is one whose answer is not unique
Forty-two starts on one region and one tolerance give forty-two different maps, keeping from 36.98 per cent of the area to 67.50. The two starts anybody would actually use — Chebyshev's map and the mean-square map — end 27.1 points apart, which is more than the criterion buys over either of them. And the spread is widest at exactly the tolerance where the criterion is worth using at all, because both facts are the same fact: an objective with a flat top and a sharp edge has somewhere to hide answers.
The map is finer than the paper it is folded from
Thirteen measurements price a polyhedral map in angles, areas and lengths, and none of them is about paper. A hand-folded net is accurate to about a degree of dihedral angle, which displaces the far edge of a cube's face by 0.955° of arc — 106 km at Earth scale — and that floor does not shrink when the globe is made larger, because a degree is a degree. Against it the conformal face map's seam corner is 0.769° on a cube and 0.0040° on an icosahedron, where it is 165 times finer than the sheet can be folded to.
The geoid has a curvature only if you say where you stopped
Thirteen measurements have priced the curvature of surfaces that are stated — a sphere, an ellipsoid, a triaxial body, a surface with a hole, real relief. The surface a height actually refers to is none of them. It is an equipotential of the Earth's own gravity, it is only ever given as a series stopped at a degree, and its Gaussian curvature carries two more powers of the degree than its height does. So the omitted height converges, the omitted slope diverges as a logarithm, and the omitted curvature diverges as a power: EGM96 carries 0.36 per cent of the sphere's own curvature, EGM2008 carries 2.19, and a one-kilometre model would carry twenty.
The pass that fails first is not the one that was measured
Three attempts to explain why the near-optimal aspect set comes apart when it does have fitted the height of the pass between the two deepest basins. That is not the pass that fails first. At every region size the set is held together by the last near-rival to join the best placement's own basin — a basin between 2 and 19 per cent worse than the best — over ground 1.02 to 2.09 times higher than the deep pair's pass. The two passes fall at nearly one rate, −1.42 and −1.31 powers of the region's span, so what separates them is a factor, not a law.
A bend in the barrier puts the drawn polygon over water
A straight river leaves the near side of a reach set convex, and that is what makes a fan of bearings fail in one direction only: every chord cuts a corner, nothing is claimed that is not there. A meander breaks that: the near side is no longer convex, and a chord between two bearings can cross water. So the drawn polygon now over-claims as well as omitting — by a seventh of a per cent, at every bend measured, and more bearings do not reduce it. The question was whether the two errors partly cancel. They do not, and the reason is not their signs: the over-claim never reaches a hundredth of the omission, because one scales with the whole set and the other only with the river.
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.
An angle is a difference, and the difference doubles the error
Substituting a sphere for the ellipsoid turns every direction at a point, and a surveyor measures angles rather than directions — so the obvious hope is that a turn common to both directions cancels in their difference. It does not cancel, because the turn is not common: it runs as the sine of twice the azimuth, and the largest angle error is exactly twice the largest direction error at a well-shaped corner. A first-order triangulation computed on the geodetic-latitude sphere carries angles nearly nine minutes of arc wrong, and its closure check passes perfectly.
The pass's depth belongs to the projection, and it is not a constant
Separating the fracture threshold into a pass depth and the best score a region admits raised the question of what the pass depth is. Across three projections and four latitudes it runs over a factor of 4.4, and most of that is the projection: Robinson's stays between 0.018 and 0.021 at every latitude, Mollweide's between 0.040 and 0.079. It is not a constant of anything — it falls as the region grows in ten cases of twelve — and it does not follow how much of the projection the region sees. The denominator, over the same cases, moves by half again.
A streak along the jet moves the floor while the crossing is flown
The diversion floor answers only to the difference between the wind on a gap's two halves, and every case so far produced that difference by moving a jet sideways. A jet carrying a streak — a thousand kilometres of core markedly faster than the rest — produces the same difference with nothing changed across the flow at all, and moves the floor by up to 43 km where the same jet at one speed along its whole length moves it by ten metres. Plotted against the half-integral difference the two geometries fall on one line: the floor cannot tell them apart. And because a streak drifts with the flow, the allowance an operator must satisfy is 30 km higher than the one the departure analysis reported.
A boundary that two features share
Simplifying one curve prices 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.
The correction is the smaller of the two corrections
Legendre's theorem subtracts a third of the spherical excess from each angle of a triangle and solves the rest as a plane figure. It is the correction every nineteenth-century computer applied and the one every textbook explains. Beside the error the choice of substituted sphere leaves in the same triangle it is small: on the thirty-kilometre triangle a first-order chain is made of, omitting it costs 166 millimetres in a computed side and using the geodetic-latitude sphere costs 43 metres. The two terms scale differently, so each sphere has a size at which the correction overtakes it — and for the two spheres a careful computer would have chosen, that size falls inside a real chain.
The seam falls faster than the paper
Divide each face of an icosahedron ν ways and two things shrink at different rates. The fold's error falls as 1/ν, because it is the face's own radius times a degree. The gnomonic seam corner falls as 1/ν², because it needs both a tilt between faces and a distance along their edge. So the gnomonic, 12.8 times the whole assembly budget on twenty faces, goes under it at 720 faces on a 100 mm globe. No globe of any size needs more than 5,120, and what it costs is paper folded out of sight: 5.8 per cent of the sheet at that crossing.
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 second clause decides the half that was already decided
A specification written as a band admits forty-two maps keeping between 37.0 and 67.5 per cent of a region. Adding a tie-break — among those, take the one whose worst departure is least — is a linear programme, and it works: it makes each of the forty-two definite. It also merges three neighbours at the bottom into one, leaves 29.1 of the 30.5 points of spread in area kept, and names a single map only at the tolerance where every map keeps everything and the first clause has stopped asking for anything at all.
The error ellipse is not an ellipse
A positional covariance pushed through a projection is 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.
A copy eased into place loses its seams before its source
A stretch of coast lifted from another sheet and dropped in rigidly leaves two sharp corners where it joins, and both of them are findable. A compiler does not work that way: they pin the ends and ease the middle until it looks continuous. Easing halves the corner within five per cent of the outline and the seams survive it — until forty per cent, where the located seam jumps from four per cent of the coast out to twenty. The source projection survives twice as far, to seventy-five. The piece goes on naming where it came from after the method has lost track of which piece it is.
Written in numbers, the archipelago rule still leaves eighteen
The article on archipelagic baselines gives its tolerances as numbers: no segment over a hundred nautical miles except three per cent of them, which may reach 125, and water between one and nine times the land. On a stated archipelago 8,799 closed systems keep the main islands inside, the ratio passes all of them and the length test eighteen, and the eighteen still differ by 27,585 square kilometres. Thin the land and the ratio does the deciding; the lower bound decides nothing; and the three-per-cent exception can be used only by a system padded to thirty-four segments.
One exponent names one map, and at tight bands a better one than the band usually finds
A specification written as a band keeps a count of area and throws the rest of the scale field away, which is why it had forty-two answers and why a tie-break could not name one. Written instead as the mean of |ln k| raised to a stated power, it is convex for every power from one up, and has one answer. At a band of a tenth of a per cent that single answer keeps more of the region than forty-one of the band's forty-two, and at half a per cent fewer than nine of them: the band's best answer leads it by 14.7 points there, and the band's own search started from it reaches within two points of that best. Only as the band approaches the least worst case does a large power take over.
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.
A forecast's error in where the streak is becomes a margin
A jet streak drifting along a gap moves the diversion floor by up to forty kilometres during a crossing, and an operator knows where the streak is only from a forecast. Planned to the worst over the forecast's own crossing window, the floor needs a margin of about eight kilometres for every hundred of error in the streak's position, and stops paying for itself against simply ignoring the streak near an error of two hundred and fifty. Planned to the departure analysis alone, it is worse than ignoring the streak at any error at all.
The archipelago rule moves water between regimes, not between states
The eighteen baseline systems the archipelago article allows on a stated archipelago differ by 27,585 square kilometres of water inside the lines. Measured outward, as the Convention measures every zone, the difference almost vanishes: whatever a system takes inside it gives up from the zone outside, and water inside plus zone outside differs by 993 square kilometres, seven hundredths of a per cent. Against a straight neighbouring coast on any side, the equidistance line does not move at all, because a baseline's nearest point to a straight coast is always an island. Only a neighbour facing the middle of a gap feels the choice — 2,112 square kilometres at thirty kilometres off, 688 at 250.
The band names the exponent for a compact region, and not for a thin one
A tolerance band is best served by the mean departure when it is tight and by the least worst case when it is wide. Read as a fraction of the region's own least worst departure, the band decides which, and the region's size drops out: a square, a triangle or a hexagon at 10°, 25° or 40° switches from one to the other at nine tenths, and a surveyor told only that loses at most two points of area on every compact region but the degenerate cap. A thin ellipse does not switch. From six tenths upward it is served best by exponents in between, and any rule with two exponents in it gives up five to seven points there, wherever its switch is put.
Two maps and the line between them do the work of every exponent
A conformal map's log scale is linear in its coefficients, so every blend of the map that minimises the mean departure and the least worst map is itself a conformal map, available the moment the two are solved. Searched along that one line for each band, the blend keeps within 0.17 points of the best of twelve exponents on every compact region and beats all twelve by up to 1.35 on the square; it trails by 1.5 at most on the thinnest ellipse and 0.74 on four regions with a bay, where the rule of switching between the two ends gave up as much as 7.1. The family's middle members lie up to two thirds of their own departure off the line. The line does not reproduce them; it does their work.
An ensemble carries its own margin, if its spread is honest
A fixed margin of about eight kilometres for every hundred of forecast error covers the diversion floor in nineteen crossings of twenty, but it needs the error's size and it is spent on every crossing alike. Nineteen forecasts of the same crossing need neither. If the truth is one more draw from the spread they come from, the highest of them covers the floor met in nineteen crossings in twenty by rank alone — whatever the error, the streak or the drift — and it holds a fifth to two fifths less in hand than the fixed margin, because it spends nothing on the crossings the streak cannot reach. What it cannot do is notice that its own spread is too small: at half the true spread it covers 88 in a hundred.
Three maps make a plane, and the band's best map lies outside their triangle
Every conformal map whose ln k is a fixed blend of three solved maps — the p = 1 map, the least worst map and the mean-square map — is itself a conformal map, so three solved maps are a plane of them. Inside their triangle the third map recovers most of what the line between the first two gave up on thin regions: 0.63 points short of the best exponent on the thinnest ellipse, where the line was 1.48 short. But the band's best map in the plane is not between the three. On 172 of 182 regions and bands it lies outside the triangle, and it keeps up to twenty points more than any exponent; started from it, a search over all twenty-five coefficients adds about a point on four regions of five and seven on a hexagon. Its price is a worst departure up to nearly five times the least worst.
A cap on the worst departure keeps the thin regions' gain and gives back the square's
The conformal map that keeps the most of a region inside a band can sit far out in the plane of three solved maps and let its edges go to nearly five times the least worst departure. Capped at twice the least worst, the plane still matches or beats the best exponent on 69 region-and-band pairs of 91: on the thin ellipses it keeps the whole of its gain, because that gain sits in a small pocket just past the exponents' own worst departure and arrives all at once; on the square and the bays it gives most of it back, because there the gain is a ramp that runs out to four times the least worst. Under any one cap the plane still beats the exponents that cap admits, by up to eighteen points — and it can name maps that no weighting of area against worst departure would ever choose.
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 tolerance in map units is not a tolerance
A snapping tolerance is a number, and the number is in whatever units the file is in. Five map units on Web Mercator is 4.97 metres of ground at the equator and 0.87 at eighty degrees — so a rule that merges two features three metres apart merges them everywhere below 52.8° north and refuses everywhere above it, in one pass, over one dataset, with nothing recording where the boundary is.
The height a coordinate does not carry
Ten rungs treat a datum shift as a map from one pair of angles to another. It is not one: the transformation runs through Cartesian coordinates, so the answer depends on the height — by 7.7 to 121 millimetres per kilometre depending on the datum and the place, which puts Lhasa 432 millimetres from where a height of zero would have said.
A length measured from noisy points is too long
A distance is a square root, a square root is concave, and the average of the distances is not the distance between the averages. The gap is a bias with one sign: 37 microns on a ten-metre baseline with twenty-millimetre marks, following σ² over the length across two decades, and it adds rather than cancelling — so the same boundary is 1.5 parts per million longer when it is measured in more pieces.
The worst point is not on the grid
Every maximum distortion this collection has printed is a maximum over a sample, and a maximum over a sample is a lower bound. On Mercator over a sixty-degree band a twelve-by-twelve grid reports 3.464 where the answer is exactly 4, and the shortfall does not go away with refinement so much as decay at a rate that says where the extreme is hiding.
Which of these numbers are the sampler's
Four rungs have shown that a sampled maximum understates, a sampled mean can be a report on the sampler, no arrangement of points is neutral, and refining until the answer settles proves nothing. So the collection re-measured itself. The means move by at most 1.1 per cent, the worst points by up to 19, and the rankings — which is what the essays actually argue with — do not move at all.
The area is unbiased and the perimeter is not
A boundary measured from noisy vertices comes out long, always, by σ²/d on every leg. The area enclosed by the same vertices comes out exactly right, because a shoelace is bilinear and the cross terms vanish. So densifying a boundary makes its area five times more precise and its perimeter three thousand times more wrong, and every compactness score computed from it falls short.
Rounding is not noise
Eight rungs treat a coordinate's error as noise that averages down. A published coordinate has a second error that does not: it is deterministic, it is shared between every point in the same cell, and at five centimetres apart ninety-three per cent of pairs come out as the same place — which no amount of independent noise can produce.
Named alongside it
The objects these essays reach for when they reach for this one.
VerificationConventionScale factorClosed formPurposeEstimatorNational GridBoundaryConformalityRealisationTrade-offArea