What is claimed about maps, and what the measurement returned

Cartography is argued about more than almost any other technical subject and measured in almost none of those arguments. A projection is called conformal because that is its name. An equal-area projection is called honest because equal-area sounds like honest. Mercator is called a distortion of the world without anybody saying which of the three distortions is meant, or how large it is, or what the alternative costs.

This collection's founding rule is the reply to that: no property is printed until it has been computed from the projection's own four partial derivatives. The rule produces verdicts, and until this page those verdicts were spread over 319 essays with no way to reach them — so a reader arriving carrying one of these claims had no route to the essay that takes it apart.

There are 50 of them below. Each carries the claim as it is usually heard, a verdict, what was found instead, and — the line that matters — the quantity that settles it, computed while this page is built from the same libraries the figures are drawn from. Not one of these numbers is typed in. If any of them stopped agreeing with the essay it points at, the build would stop rather than serve the page.

Only one of the five verdicts is wrong, and that is the finding rather than a hedge. Almost nothing repeated about maps is simply false. The interesting failures are a true statement about a name taken for one about a surface; a theorem carried past the hypotheses it was proved under; a real effect that is correct and three orders of magnitude smaller than the one the same sentence ignores; and — the category this subject produces more of than any other — a statement about the graticule that reads as a statement about the map. Telling those four apart is most of the work, and calling them all myths is how the subject stays unmeasured.

The claim is correct about what the projection is called and false about the surface it draws. This is the category the site was built to answer: every one of these is settled by four partial derivatives, and none of them had been.

“Web Mercator is conformal — it is Mercator, and Mercator is the conformal one.”

Web Mercator applies the spherical Mercator formulae to ellipsoidal latitudes, which are not the latitudes those formulae were derived for. The result is a projection that is conformal nowhere except on the equator, and the site's machinery found it without being told to look — the assertion that prints the word simply refused to pass.

What settles it: 0.3848° of maximum angular deformation against 1.2e-6° for spherical Mercator — a factor of 324,165, and 3,848 times the tolerance the word is granted at. Argued in Web Mercator is not conformal, on the audit ladder.

“The plate carrée is unprojected — it is the raw latitude and longitude, with no projection applied.”

Plotting latitude against longitude on squared paper is a projection with a name, a construction and a distortion, and the fact that it takes no arithmetic does not exempt it from having one. It is neither conformal nor equal-area anywhere off the equator, which is the worst of both and is why nobody chooses it for a map that has to be read.

What settles it: at 60° the parallel is stretched 2.00× while the meridian is held at 1.00×, so the areal factor is 2.00 and the maximum angular deformation is 38.94° — not zero, which conformal requires, and not one, which equal-area requires. Argued in The plate carrée, the projection nobody chooses, on the audit ladder.

“An equal-area projection shows the world as it really is.”

It shows areas as they really are, exactly, and buys that with shape. Equal-area is one of the three things a map can be wrong about, and a projection that fixes it perfectly has spent everything it had on that one — which is a defensible choice and is not the same as being undistorted.

What settles it: Gall–Peters holds area to 1.7e-11 — exact to arithmetic noise — while deforming angles by up to 146.4° at 84° of latitude. Argued in The projection that shows true size, on the audit ladder.

“Mercator inflates the north because of who drew it.”

The inflation is not a decision anybody took. Requiring a constant compass bearing to be a straight line forces the parallel scale to sec φ, conformality then forces the meridian scale to match it, and the areal factor is the product — so the exaggeration is the same quantity as the property the projection was built for. It cannot be reduced without giving up the rhumb line, and it is a consequence rather than an intent.

What settles it: a ten-degree cell at 70° comes out 3.85× the area of the same cell on the equator, and at 60° 2.36× — each of them sec²φ, to the digit, and neither adjustable. Argued in Why Mercator exists, on the paths ladder.

“An equal-area projection preserves area — that is what the name means.”

It preserves area on the body its formula was derived for. Feed the spherical formula the geodetic latitude a coordinate actually carries, and measure against the ellipsoid that latitude refers to, and the areal factor is nowhere one. The failure is invisible to every check anybody runs, because it is a redistribution: the map has almost exactly the right amount of area in it and puts it in the wrong places, so a total comes out right to two parts in a thousand while every cell is out by two thirds of a per cent.

What settles it: 1.00674 at the equator and 0.99332 at 88°, a spread of 1.34 per cent, against 1.1e-10 for the same projection built with the authalic latitude. Argued in Equal-area on the wrong body, on the audit ladder.

“A conformal projection draws shapes correctly.”

Conformal is a statement about angles at a point, and it is exactly true. It says nothing about any curve of finite length: the shortest route between two places is drawn as a curve on every conformal map there is, and the rate it bends is a second derivative that no first-order quantity — not the indicatrix, not the areal factor, not ω — reports. A reader ruling a straight line on a conformal chart is using the property the word does not cover.

What settles it: at 20°E 40°N Mercator's angular deformation is 0.0e+0° — zero to the limit of the arithmetic — while a great circle through the same point is drawn turning at 0.839 radians per radian of arc, which is tan φ exactly. Argued in Tissot stops at the first derivative, on the flexion ladder.

“An equidistant projection preserves distances.”

It preserves distances from a stated point or along a stated family of lines, and from nothing else. The two-point equidistant projection is exact from each of two named places to arithmetic noise, and the distance it draws between two ordinary points is unconstrained — the condition names two places, not a metric.

What settles it: distances from the two centres are right to 2.1e-14 relative, and the worst pair not involving a centre is drawn 761 per cent too long. Argued in A projection written as a condition, on the condition ladder.

“Mercator's average areal distortion is a number — the world map inflates area by so many times on average.”

The areal factor is sec²φ and its area-weighted mean over the sphere is artanh(sin Φ)/sin Φ, which has no limit: every halving of the remaining gap to the pole adds ln 2 and it goes on doing so. A sampler asked for it returns a finite number, and the number is the logarithm of the sampler's own count. The Kavrayskiy number over the same map converges, so one of the two summary figures routinely printed for Mercator is a measurement and the other is a report on whoever computed it.

What settles it: 3.14 over ±85° and 7.04 over ±89.9°, still rising by ln 2 per halving; a cell-centre rule returns 5.67 at 64 samples and 9.14 at 2,048. Argued in A mean that does not exist can still be printed, on the sampling ladder.

“A projection's worst distortion over a region is the largest value a grid of sample points finds.”

A maximum over a sample is a lower bound with a sign, never an estimate that could come out high, and the shortfall is not small. Mercator's areal factor over a band to 60° north is exactly sec²60° = 4 on the band's top edge, which no cell centre ever reaches; a twelve-by-twelve grid returns 3.464. The rate at which the shortfall falls says whether the extreme is on the region's boundary or inside it, and every equal-area projection returns exactly zero because a constant field has no maximum to miss.

What settles it: 3.4639 against an exact 4.0000 — short by 13.4 per cent, and still 1.86 per cent short at ninety-six samples a side. Argued in The worst point is not on the grid, on the sampling ladder.

“Fitting a library of candidate projections to a map's graticule identifies the projection it was drawn in.”

It returns a best fit, and a best fit without a spread is a point estimate rather than a measurement. A control point read off a map has a width, and every candidate whose residual is inside that width has been rejected by nothing — so the honest answer is a set. Over a sixteen-degree region at three parts in a thousand of digitising noise, a conformal conic's graticule admits the conformal conic, the equal-area conic and the polyconic, which preserve three different things.

What settles it: 10 of 20 candidates admitted over a four-degree region at one per cent noise and 3 over fifty degrees; at eight degrees and three parts in a thousand the set is Conformal conic, Equal-area conic, Polyconic. Argued in The answer is a set, on the identify ladder.

Not true, and something computed here says so. The shortest list of the five, which is itself worth noticing — confident falsehood is rarer in this subject than confident imprecision.

“Latitude is the angle at the centre of the Earth between the equatorial plane and the point.”

That is geocentric latitude, and it is not what any map, chart, datum or satellite receiver means by the word. Geodetic latitude is the angle of the ellipsoid normal, which does not pass through the centre except at the equator and the poles. Every coordinate a reader has ever handled is the second kind.

What settles it: the two disagree by up to 11.55 arcminutes, at 45° of latitude, which is 21.4 km on the ground. Argued in Geodetic against geocentric latitude, on the ellipsoid ladder.

“A conformal projection preserves angles, so a triangle drawn on the map has the angles it has on the Earth.”

Conformal preserves the angle between two curves at the point where they cross, and says nothing whatever about a finite figure. The images of two geodesics leave a vertex at the correct angle and then bend; measuring the triangle with a protractor measures the straight chords instead, and those are a different thing. The spherical excess has to go somewhere, and on a flat page it goes into the difference between the tangent and the chord.

What settles it: on Mercator the tangents at the vertices are right to 2.0e-4°, while the straight-sided triangle on the page is out by 25.0° at its worst vertex and sums to 180° against a true 205°. Argued in Conformal does not mean the angles are right, on the audit ladder.

“Averaging two projections destroys whatever exact property either of them had.”

Half of it is right. Averaging two equal-area projections gives one that is not equal-area, because the condition is a determinant equalling one and is quadratic. Averaging two conformal projections gives a conformal projection, every time — a conformal map of the sphere is a holomorphic function of the isometric coordinate, and holomorphic functions form a vector space. The consequence is that a compromise projection must contain an ingredient that is not conformal, which is a structural fact about the whole class.

What settles it: 2.04e-6° of angular deformation across nine mixtures of conformal pairs — below the 0.0001 tolerance the word is granted at — against 16.2 per cent of areal error for mixtures of equal-area ones. Argued in The average of two projections, on the choosing ladder.

“A grid's scale factor corrects a plan's dimensions, so applying it corrects the plan's areas too.”

An areal scale factor is the square of a linear one, so a departure of 400 parts per million in length is 800 in area. Applying the line factor to an area performs exactly the right operation once instead of twice and leaves half the error behind — with the same sign and half the magnitude, which is the shape that survives every plausibility check.

What settles it: 399 ppm in length and 797 in area on the British grid at 52° north — 8.0 m² on a hectare. Argued in An area on the grid is not an area on the ground, on the grid ladder.

“Every map projection can be read backwards — that is what a map is for.”

A projection is a function and nothing requires it to be injective. Craig's retroazimuthal map satisfies its own condition exactly — the direction to a fixed place is read off with a straight edge — and folds over itself on every meridian, so two places thousands of kilometres apart are drawn at the same point and no inverse exists. It is a map that can be pointed with and cannot be located on.

What settles it: on the Greenwich meridian of the Mecca-centred map, 77.8°S and 48.1°S — 3303 km apart — are drawn at heights differing by 2.2e-8 in map units. Argued in A map that cannot be read backwards, on the condition ladder.

“A hexagonal grid divides the globe into equal hexagons.”

It cannot, and the obstruction is Euler's formula rather than a limitation of any construction. A tiling of the sphere by pentagons and hexagons with three faces at every vertex has exactly twelve pentagons, whatever the hexagon count — and the twelve are measurably smaller than their neighbours, so a count aggregated over such a grid has twelve entries that mean something different from the rest.

What settles it: a tiling with 630 hexagons has 12 pentagons, each 0.50 times the area of an average hexagon, with the cells spanning a factor of 2.49 in area. Argued in Hexagons cannot tile the sphere, on the cells ladder.

“A space-filling curve is used for cell identifiers because it keeps neighbouring cells close in the index.”

The conclusion is right and the reason is not. Measured on the statistic usually quoted — the difference in identifier between cells that share an edge — plain row-major numbering beats Hilbert order, on the mean and at the worst pair. What Hilbert order actually wins is the number of contiguous identifier ranges a two-dimensional query occupies, which is what a database reads.

What settles it: mean gap between neighbours 16.5 for row-major against 19.6 for Hilbert, and worst pair 32 against 853. Argued in The address is a curve through the sphere, on the cells ladder.

“Two datum transformations are combined by adding their seven parameters.”

Two affine maps compose into an affine map, and the composed translation is not the sum: it is the second transformation's scale and rotation acting on the first's translation, and a datum translation is hundreds of metres. The sum is a different transformation from the composition, by millimetres, on a chain whose whole purpose is millimetres.

What settles it: adding the parameters lands 7.4 mm from the answer the two transformations actually give, while composing them is exact and the composition's own seven-parameter description is 0.19 mm out. Argued in A chain of transformations does not close, on the datum ladder.

“A conformal map of a triaxial body needs an isothermal coordinate, which such a body does not have.”

This site wrote that sentence down as a recorded shortfall. It is a statement about a coordinate system rather than about a surface: in Jacobi's ellipsoidal coordinates, which are the body's own lines of curvature, the metric separates and an isothermal coordinate follows from two one-dimensional quadratures.

What settles it: Jacobi's coordinate lines cross at 1.8e-8° from square where the parametric ones are 1.56° off, and the map built from them is conformal to 1.1e-6° while spreading areas by a factor of 162. Argued in A conformal map of a body with three axes, on the bodies ladder.

“The reduction chain from a tape reading to a grid distance cannot be checked.”

It could not be, while every available instrument measured lengths and needed the same corrections. A satellite baseline supplies the positions the chain exists to reconstruct, so the same grid distance follows by projecting both ends — an independent route that agrees with the chain and prices the plane arithmetic the chain is usually taught with.

What settles it: over 76.7 km the two routes agree to 0.17 mm, while the textbook slope-then-height reduction lands 462 mm away. Argued in The chain the satellite does not have, on the reduction ladder.

A theorem or a rule of thumb carried past its own hypotheses, usually because it keeps giving very nearly the right answer for a long way beyond them. The dangerous ones do not fail gradually; they reverse.

“UTM's scale factor of 0.9996 reduces the scale error across the zone.”

It does at the equator, and above about 57.5° it makes the map worse. The constant assumes a zone six degrees of longitude wide; a zone is physically narrower the further north it is, so the tangent construction's own error has already fallen below what the secant construction costs. The generator predicts which regime it is in from cos²φ = 400/1382 and asserts the outcome matches, so the check is on where the crossover is rather than on which number happens to be smaller.

What settles it: at the equator the constant takes the worst-case error from 1382 ppm to 981 ppm, and at 60° north it takes it from 343 ppm up to 400 ppm. Argued in UTM and the zone system, on the ellipsoid ladder.

“Molodensky's formulae convert coordinates between datums.”

They convert between datums related by three translations. A datum pair whose published transformation carries rotations and a scale change has four parameters Molodensky does not have, and the formulae accept it silently and return something of the right shape and the wrong value. The library now refuses such a datum by name and keeps the misuse as a deliberate measurement, because the size of it is the argument.

What settles it: applied to OSGB36 it is out by up to 15.3 m — 2,156 times the method's own approximation error of 7.1 mm on a datum it can represent, and the abridged formulae score the same, because both are swamped by the four terms neither carries. Argued in Molodensky's shortcut, on the datum ladder.

“The great circle is the shortest route between two points on the Earth.”

On a sphere, yes, and the Earth is not one. The shortest path on an ellipsoid is a geodesic that is not a plane curve at all, and it does not close on itself. The error is small in percentage terms and is not small in kilometres, which is the combination that keeps the approximation in use and occasionally in trouble.

What settles it: Cape Town to London differs by 34.6 km between the sphere and the ellipsoid — 0.36% of the route. Argued in Geodesics on the ellipsoid, and why they are hard, on the paths ladder.

“Tissot's indicatrix tells you the shape a small region takes on the map.”

It tells you the shape an infinitesimal region takes, and every published indicatrix is drawn finite. The error of the description relative to the ellipse being drawn is first order in the circle's radius rather than second — halving the region halves the lie rather than quartering it — and its coefficient is the rate at which the projection's own scale is changing, which has nothing to do with how distorted the projection is.

What settles it: the ellipse is one per cent wrong beyond 148 km on Mercator and beyond 46 km on the gnomonic projection, at the same point. Argued in The indicatrix is a limit, on the tissot ladder.

“The line between two points on the Earth is the section cut by the plane through them and the vertical.”

There are two such sections and neither is the shortest path. The plane through the first point's normal and the second is not the plane through the second point's normal and the first, because an ellipsoid's normals miss its centre — so two instruments sighting each other trace different curves on the ground, and the geodesic runs between them. Their lengths agree to millimetres, which is why the distinction survives unnoticed: the wrong curve measures the right distance along the wrong ground.

What settles it: 79 m between the two sections over 3026 km, while their lengths differ from the geodesic's by 2.4 mm and 0.7 mm. Argued in The normal section is not the geodesic, on the ellipsoid ladder.

“A closed traverse that meets its closure tolerance has no blunder in it.”

It has no blunder large enough to move the closing point past the tolerance, which is a weaker statement and is unevenly weaker. An angle blunder at a station rotates everything downstream of it about that station, so the misclosure it produces is proportional to that station's distance from the close — and the last station before the close, standing a few tens of metres from the first, moves it hardly at all.

What settles it: at a 20 mm tolerance one station hides 2.1″ and another hides 57″ around one seven-station loop — a factor of 28. Argued in What a closed figure cannot see, on the reduction ladder.

“Whether a point is inside a polygon is a question about the point and the polygon.”

It is a question about the point, the polygon and the plane the polygon's edges were understood to be straight in — and a file records vertices rather than edges. Two correct implementations of point-in-polygon, reading the same coordinates under the two conventions in use, disagree over a band whose width is the edges' own departure from the ground they claim.

What settles it: 29.5 per cent of one triangle — 4,241,237 km² — with the two readings disagreeing about every point in it. Argued in Inside is a claim about the edges, on the dataset ladder.

“Web Mercator's conformality error is a quirk of web mapping.”

It is a quirk of applying a spherical formula to an ellipsoidal latitude, and its size is a function of the body's flattening and of nothing else — not the software, not the radius, not the convention. The same mistake on Mars, Jupiter and Saturn measures 2f radians on each, so the terrestrial number is an instance of a law rather than a property of a rendering choice.

What settles it: 0.3848° on Earth, 0.6765° on Mars and 7.680° on Jupiter — each within 0.3 per cent of 2f radians for the two nearly spherical ones. Argued in The same projection on a different body, on the bodies ladder.

“An equal-area formula is equal-area, so it can be pointed at any body.”

Lambert's cylindrical equal-area formula is exactly equal-area on a sphere, which is the surface it was derived for. Applied to a body with three unequal axes and measured against that body's own first fundamental form, its areal factor is not constant — and the same machinery, given a map built from the body's own strip areas, returns one to a part in a million.

What settles it: on Vesta the formula's areal factor spreads by 1.281 — 28 per cent — against 1.00000 for the map built from that body's own areas. Argued in A map of a body with three axes, on the bodies ladder.

“A body's flattening follows from its rotation, so its shape and its gravity field agree.”

Clairaut's theorem predicts a flattening from a body's dynamical form factor and its spin, and it is a statement about a body in hydrostatic equilibrium. The Earth is one to a part in two thousand. Mars is not, because it carries Tharsis, and its published ellipsoid is therefore not one of its own level surfaces — which is why a Martian elevation needs a convention rather than a measurement.

What settles it: the prediction is out by 0.055 per cent on Earth — 12 m at the pole — and by 12.5 per cent on Mars, which is 2.23 km. Argued in A body with no sea level, on the bodies ladder.

“A conformal map has a scale factor; it varies, and it is finite wherever the map is drawn.”

True of every conformal projection with a smooth domain, and false at the corner of a polyhedral face. The spherical face's corner angle and the flat face's differ, so the map behaves like a fractional power there and its derivative is unbounded — the solid's angle deficit arriving as a singularity of the map rather than as a tear.

What settles it: a cube's spherical face meets its neighbours at 120° against the flat face's 90°, so the scale grows as d^-0.250 into the corner — measured at -0.258. Argued in A conformal map onto a face, on the polyhedral ladder.

“The sheets an atlas needs are its ideal count times the thinnest covering density.”

The density usually quoted is the plane's, proved by Kershner in 1939, and the sphere is not a plane at any finite count. At the four counts whose optimal covering of the sphere is a theorem, the plane's number misses the answer in both directions — high where a cap may be a hemisphere, low where Euler's formula forces twelve pentagons into an otherwise hexagonal arrangement.

What settles it: 2.42 against 2, 3.63 against 4, 5.72 against 6, 11.78 against 12 — worst 21 per cent at 2 caps. Argued in The sphere is not the plane at small counts, on the choosing ladder.

“A query for everything within a radius fetches about as many cells as its area divided by a cell's.”

True when the disc is many cells across and wrong by a factor of several at the sizes a query is actually made at, because every cell the disc's boundary crosses is fetched as well as every cell inside it. The perimeter term exceeds the area term whenever the disc is narrower than about six cells.

What settles it: at 4° the measured cost is 6.0 cells against an area ratio of 1.87 — a factor of 3.2 — and the ratio recovers to 1.18 at 32°. Argued in A query is a disc, and a disc is not a cell, on the cells ladder.

“A cubic resampling kernel is more accurate than a bilinear one, which is more accurate than nearest-neighbour.”

True on a smooth field, where the three converge at orders 1, 2 and 3. Across an edge the three rates collapse to within a factor of 1.4 of one another and the fastest belongs to nearest-neighbour, which does no interpolating at all — and the cubic kernel returns values outside the range of the data it was given, which on a class raster is a class that does not exist.

What settles it: from a two-class raster nearest returns 2 values, bilinear 278 and the cubic kernel 671, with the cubic overshooting the data's own range by 13.1 per cent. Argued in An edge has no order of convergence, on the dataset ladder.

“Reprojecting a file between two projections distorts the shapes in it.”

Reprojection is a map of the plane onto the plane, and between two conformal projections it is a similarity at every point — so it deforms nothing at all. What degrades in a chain of reprojections is a raster's values, which are resampled at every step; a vector coordinate carries no error beyond the arithmetic's own, and a chain that returns to its starting projection is the identity exactly.

What settles it: the transformation between two conformal projections deforms angles by 1.1e-9°, which is the arithmetic's noise floor, against 120.1° when one end is not conformal. Argued in A projection between two projections, on the tissot ladder.

“Another term always makes a truncated series more accurate.”

The Gauss-Krüger series carries cosh(2jη) in its jth term, so the terms grow with distance from the central meridian while the coefficients shrink at a fixed rate. Inside a computed boundary each correction is a fraction of the last; outside it every correction is larger than the last, and no truncation repairs it.

What settles it: the coefficients decay at 1.74 times the third flattening, which puts the boundary at 83.81° from the central meridian — inside it the corrections shrink by 0.080, outside it they grow by 5.59. Argued in Where the series stops being the map, on the ellipsoid ladder.

“The shortest route is the quickest one.”

Exactly true in a medium that does not move, and false as soon as it does. Length and time are the same quantity divided by a constant speed only while the speed over the ground is constant; once a current or a wind is added, the quickest track sails further and arrives sooner, and the return journey takes a different time along a different curve.

What settles it: in a jet peaking at 26 km/h against a craft making 20, the quickest track is 5.1 per cent longer than the great circle and 11.2 per cent quicker. Argued in The quickest route is not the shortest, on the paths ladder.

“Cylindrical near the equator, conic in the middle latitudes, azimuthal at the poles.”

Scored over a population of regions with every family given its own best parameters, the rule is right about two thirds of the time and wrong in one specific way: it keys on latitude, and what decides the answer is shape. For a region as wide as it is tall the azimuthal wins at every latitude from the equator to eighty degrees.

What settles it: right on 19 of 30 regions, and where it is wrong it costs up to 4.49× the best family's distortion, with a mean of 2.25×. Argued in The rule of thumb, scored, on the audit ladder.

“A web map is drawn at a zoom level.”

True of a map lying flat on the screen and false of a tilted one, which is what every major renderer has drawn since about 2015. A pitched camera gives each screen row a different ground resolution and each direction a different one again, so the frame demands a range of levels and an anisotropy that no single integer can express.

What settles it: a flat frame spans 0.00 zoom levels; at 60° of pitch the same frame spans 3.60 up the screen with an anisotropy of 4.49. Argued in A tilted view has no zoom level, on the screen ladder.

“A map's projection can be identified from the picture.”

Only if the map covers enough ground. Fitting a similarity removes everything of first order, so two projections that agree to first order — which is what two conformal projections do, everywhere — leave a residual smaller than a drawn line until the region is ten to twelve degrees across. Every pair that differs in anisotropy separates below one degree.

What settles it: Mercator and the stereographic separate at a half-extent of 12.13°, while Mercator and Miller's separate at 0.75°. Argued in Two projections that cannot be told apart, on the identify ladder.

The effect named is real and the arithmetic is sound. Something else in the same calculation is larger by two or three orders of magnitude and goes unmentioned, so the care spent on the first buys nothing.

“Choosing the right projection is what makes a coordinate accurate.”

Projection error on a well-chosen national grid runs to a few hundred parts per million, which is centimetres over a kilometre. Reading the same latitude and longitude against the wrong datum moves the point by a hundred metres or more, and the two numbers look identical on the page: same format, same precision, same field in the same file. Every hour spent on the projection is spent on the smaller of the two by three orders of magnitude.

What settles it: NAD27 against WGS84 at the same numbers moves the ground point 195 m, while the grid's own scale error across the zone is 400 ppm — about 40 cm in a kilometre, some 487 times smaller. Argued in Datum shifts dwarf projection errors, on the ellipsoid ladder.

“A satellite receiver gives the position, so the coordinate is settled.”

It gives the position at the moment of the observation, on a plate that is moving. A coordinate without an epoch is a measurement without a date, and the ground it names has left. The receiver's own precision is the small term here by a wide margin.

What settles it: the Australia plate carries a marker 65 mm a year, which is 1.62 m over 25 years — against a survey-grade fix good to a centimetre. Argued in The epoch is part of the coordinate, on the datum ladder.

“Height above sea level is a distance, so levelling round a loop must close.”

Levelling measures increments along the plumb line, and the level surfaces the plumb line is normal to are not parallel — they converge polewards, because gravity is stronger there. So a levelled height difference depends on the route taken, by an amount that is invisible on a building site and not invisible on a national network. The instrument is not at fault and reading the misclosure as instrument error is how it gets hidden.

What settles it: a level surface 1,000 m up at 50° north is 82 mm lower after 100 km of northing — 0.82 mm per kilometre, with no error anywhere in the levelling. Argued in A levelled height is not a distance, on the height ladder.

“A datum transformation's seven parameters describe the transformation, so two published sets that differ are two different transformations.”

Over the region either was fitted to they are the same transformation. A network occupying a small part of the Earth does not determine seven independent numbers: a translation can be moved a long way along one direction and re-absorbed by the rotations and the scale, leaving the region's own coordinates almost where they were. What the network's size controls is how well the region hides the difference, and away from the region the two sets part company entirely.

What settles it: 100 m of translation, re-absorbed, moves a British coordinate by 5.6 m and an Australian one by 193 m. Argued in Two parameter sets, one transformation, on the datum ladder.

Meridians and parallels are a choice of parameterisation, not a feature of the surface. A quantity that changes when that choice changes is describing the drawing rather than the projection — and swapping the output axes is enough to prove which is which.

“The scale along the meridian tells you how much the map has been stretched there.”

It tells you how much the map has been stretched along the meridian, and the meridian is a coordinate line somebody chose. Swap the output axes and h and k trade places while nothing about the drawn surface has changed. The quantities that survive that swap — the two principal scales, the areal factor and the maximum angular deformation — are the ones that describe the map, and they are the only ones this site prints.

What settles it: at 60° on the plate carrée h is 1.000 and k is 2.000; transposing the output exchanges them, while a = 2.000, b = 1.000, the areal factor 2.000 and ω = 38.94° are unmoved. Argued in What survives a change of coordinates, on the tissot ladder.

“The grid's northing lines point north.”

They point north on the central meridian of the zone and nowhere else. Everywhere else the meridian's image is a curve and the grid line is straight, and the angle between them — the convergence — is what a surveyor has to apply before a bearing computed on the grid can be set out on the ground. It is a property of the coordinate system and is often mistaken for a property of the compass, which has a different error of its own.

What settles it: 2.46° at 3° east and 55° north, and 5.09° at the western edge of the same zone at 58° — both far larger than the bearings a survey is expected to close on. Argued in Grid north is not north, on the ellipsoid ladder.

“A screen map at zoom z is at the scale the interface prints for zoom z.”

Twice over. The printed denominator is the equatorial one and the map is at it along exactly one line; and the pyramid exists only at integer levels a factor of two apart, so the scale a reader asked for is not on the ladder and the nearest rung is served instead. The second of those is exactly determined by three known integers and is reported nowhere.

What settles it: a request for 1:25,000 is served at zoom 14, which is 1:34,124 — 36 per cent coarser, and 86 per cent coarser in area. Argued in Zoom is a ladder, on the screen ladder.

“Latitude means the same thing on every body.”

Two conventions are in current use on Mars and they differ by up to a third of a degree; on a triaxial body such as Vesta the planetographic one is not even a function of position, because the surface normal swings as one walks round at constant planetocentric latitude. The word names a construction on a reference figure, and the figure differs from body to body.

What settles it: Mars's two latitude conventions differ by up to 0.338° at 45°, which is 20 km on the ground — against 0.192° and 21 km for the Earth. Argued in A coordinate on another body, on the bodies ladder.

“Gall-Peters and Behrmann are different projections.”

Every cylindrical equal-area projection is an anisotropic scaling of every other — one number, the standard parallel, and the map is stretched by the ratio of two cosines. A fit that allows a reproduction to have stretched one axis therefore cannot separate them at all, at any size, and the difference it removes is the whole of what distinguishes them.

What settles it: fitted under a similarity the pair differ by 6.0 per cent of the map's width; fitted under an affine transformation, by 7.1e-16 — a ratio of 8e+13. Argued in What a careless copy hides, on the identify ladder.

“Each resampling kernel has a convergence order across an edge.”

The order is a property of the kernel and the field together. Across thirteen edges — six orientations, a curve and a corner — every kernel's fitted order spans a range wide enough to contain the ranking between kernels, and for an edge lying along a parallel the error does not fall with refinement at all.

What settles it: nearest-neighbour spans -1.49 to 0.86 and the cubic kernel 0.19 to 0.67; on an aligned edge the error swings over a factor of 34 as the grid is refined, fitting an order of -1.49. Argued in One edge is not an edge, on the dataset ladder.

Nothing here is refuted by assertion. A claim earns a row only when the essay holding it computes something that could have come out the other way — which is why several famous complaints this collection could join in with are missing. Disapproval is not a measurement, and a page of confident corrections with no arithmetic under them would be the same genre as the claims it corrects.

Nothing here is about a map's politics. That argument is real, it is older than most of the projections in it, and it is not settled by derivatives. What can be settled is what each projection does to angles, to areas and to distances, which is the input that argument almost never has. The Mercator row above says what the inflation is and where it comes from; it says nothing about what anyone should conclude from that.

And no claim here rests on a dataset. Every number above comes from a closed form or from a computation on one — a projection's derivatives, a cell of latitude and longitude whose area is exact, a published datum's seven parameters, a normal gravity field derived from four constants. A coastline file has a generalisation level, so an area read off one is partly a measurement of the vendor's choices and the reader cannot tell how much. That decision is argued at the projection that shows true size rather than assumed here.

What is taught wrongly · Measured, not named · Computed, not quoted · All essays