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

Ellipsoid — the series

21 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Why there are two latitudes, at 45°. A meridian section of an ellipsoid with the flattening exaggerated 12× — at the true value of 1/298 this outline would be indistinguishable from a circle. Geodetic latitude is the angle the surface normal makes with the equatorial plane; geocentric latitude is the angle the radius makes with it. The two lines meet the plane at different points and the angles differ by 11.55 arcminutes on the real WGS84 ellipsoid, which is about 21 km of ground distance.

    The Earth is a sphere, and when it is not

    Every essay before this one treated the Earth as a ball, and said so. The flattening is one part in three hundred, which is nothing for a distance, everything for a latitude, and exactly enough to make the most-used projection in the world fail the property in its own name.

    part 1 · wrong
  2. Transverse Mercator. The graticule of the Transverse Mercator projection at 30° of longitude and 15° of latitude. the only projection on which a constant compass bearing is a straight line. It is conformal.

    Transverse Mercator and the series that computes it

    The projection most of the world's survey data lives in has no closed form. What every national grid actually computes is a truncated power series in the flattening, and how far it can be trusted is an engineering parameter rather than a property of the projection.

    part 2 · families
  3. Five latitudes that are not the latitude, on WGS84. Each curve is the amount by which one auxiliary latitude falls below the geodetic latitude a coordinate actually means, in arcminutes. All five vanish at the equator and at the poles and peak near 45°, where the geocentric latitude is 11.55 arcminutes below the geodetic one — about 21.4 km on the ground. The curves never cross, which is forced by the algebra rather than by this ellipsoid's particular flattening.

    Geodetic against geocentric latitude

    Two angles, both called latitude, differing by eleven and a half arcminutes at their worst. One is what every coordinate means and the other is what every spherical formula assumes, and the gap between them is twenty-one kilometres on the ground.

    part 2 · wrong
  4. The sixty zones, each six degrees wide. Every zone is a separate transverse Mercator projection about its own central meridian, so the world is covered by sixty maps rather than one. Zone 31 is picked out, running from 0° to 6° with its axis on 3°. Coordinates do not carry across a zone boundary — a point on either side of one has two entirely different eastings, and nothing in the numbers says which zone they belong to.

    UTM and the zone system

    Sixty separate maps of the world, each six degrees wide, each with a scale factor of 0.9996 chosen so the projection is wrong everywhere and less wrong at the edges. Every constant in the definition is a measured trade rather than a convention.

    part 3 · families
  5. What changing the datum alone does to a coordinate. The distance on the ground between a point as read on its national datum and the same numbers read on WGS84, computed through the published seven-parameter transformation. The shifts run from 49 to 166 metres. For comparison, the scale error a UTM zone introduces at its edge is under a metre per kilometre — so the datum, which is usually left unstated, dominates the projection, which is usually argued about.

    Datum shifts dwarf projection errors

    The projection argument is conducted in parts per million and the datum question is answered in hundreds of metres. A coordinate whose datum is unstated is out by more than any projection choice could ever put it, and almost nobody checks.

    part 3 · wrong
  6. Grid north against true north at 52°, across a 6° zone. seven meridians drawn in the grid's own coordinates. The grid's north is straight up the page everywhere by construction; the meridians are not, and the angle between them is the convergence. It is exactly zero on the central meridian, reaches 2.36° — 142 minutes of arc — at 3° out, and changes sign across the middle. The manual's γ = Δλ sin φ is 3.0″ short of it at the zone edge.

    Grid north is not north

    A grid has one north, parallel everywhere on the sheet by construction. The Earth has a different one at every point. The angle between them reaches two degrees at the edge of a UTM zone, and a straight line on the grid is not a straight line on the ground either.

    part 4 · families
  7. Three curves between 45°N and 55°N, 3026 km apart. The normal section observed from the first point, the normal section observed from the second, and the geodesic, each plotted as its distance to one side of the great-circle chord between the two ends. The two sections are 79.0 metres apart at their widest and the geodesic runs between them, 53.3 metres from the first. In LENGTH they differ by almost nothing — 2.40 millimetres over 3026 kilometres — so an observation that follows the wrong one measures the right distance along the wrong ground. On a sphere all three coincide exactly.

    The normal section is not the geodesic

    A theodolite at A sighted on B swings in one plane and the instrument at B sighted back swings in another, so the two observations trace different curves on the ground — 79 metres apart over 3,026 kilometres — and the shortest path is neither of them. The lengths differ by 2.4 millimetres, so the wrong curve measures the right distance along the wrong ground.

    part 4 · wrong
  8. Clairaut's theorem, and the term it drops. Clairaut's theorem says the flattening of a rotating body plus the flattening of the gravity on it equals five halves of the ratio of centrifugal to gravitational acceleration at the equator. Measured on WGS84: f = 3.3528e-3, f* = 5.3024e-3, and their sum is 8.65525e-3 against the theorem's 8.62447e-3. The residual is 3.08e-5, which is 2.74 times f² — second order, which is what a first-order theorem is entitled to be wrong by.

    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.

    part 5 · datums
  9. How badly two arcs determine the flattening. The reciprocal flattening recovered by inverting two measured degree lengths, at 1.5° and 66.33°, against an error introduced into the equatorial one. The exact pair returns 298.26. Ten metres of error — 90 parts per million of a 110-kilometre arc — returns 301.5, and the relative error in the flattening is 118 times the relative error in the arc. At 1000 metres the inversion returns a negative flattening: an Earth longer through the poles than across the equator, which is the answer the Paris Observatory defended for a generation.

    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.

    part 5 · datums
  10. The boundary of the region the series is the map in. The curve where the transverse coordinate reaches 2.918, which is where the terms stop shrinking. It crosses the equator 83.81° from the central meridian and closes towards the poles, because the same longitude is a smaller transverse coordinate at a higher latitude — the boundary is a curve rather than a meridian. The narrow band beside it is a 3° zone, the width national grids actually use, drawn to the same scale: the practical world sits in about a fiftieth of what the series can reach. Drawn in Mollweide.

    Where the series stops being the map

    The transverse Mercator has no closed form on an ellipsoid, so every national grid computes a truncated series. Asking how many terms it needs has an answer everywhere; asking whether more terms always help has an answer only within 83.81° of the central meridian, and the boundary is computed rather than assumed.

    part 6 · families
  11. Going out and coming back, at three orders. A point three degrees from the central meridian, taken forward into the projection and back again with the series truncated at the same order both ways, and the ground distance between where it started and where it returned. The second-order pair is out by a tenth of a metre at some latitudes; the third by half a millimetre; the fourth — the order every national grid formula in ordinary use is written to — by 0.39 mm. The dips are where one of the two series passes through a node, not where the map is better.

    The inverse of the series is not the series of the inverse

    Transverse Mercator on an ellipsoid has no closed form, so every national grid computes it as a truncated series. There are two of them — one out and one back — and they are separate approximations. Composing them does not give the identity: at the order every grid formula in ordinary use is written to, a point three degrees from the central meridian comes back 0.39 mm north of where it started.

    part 7 · families
  12. The equator, with its ellipticity exaggerated forty thousand times. The dashed circle is the equator every projection formula on this site assumes. The solid curve is the equator satellite geodesy reports, drawn with its departure multiplied by 40,000 so that seventy metres on a six-thousand-kilometre radius can be seen at all. The long axis is at 14.9° west and the short one ninety degrees from it, and the difference between them is 70.0 metres — a real quantity, about the height of a twenty-storey building, on a body every geodetic computation treats as a surface of revolution.

    The equator is not a circle either

    Eleven rungs price what pretending the Earth is a sphere costs, and every one of them replaces the sphere with a surface of revolution — a body whose equator is a circle. It is not. The two equatorial radii differ by seventy metres, the two surfaces part by thirty-five, and the auxiliary latitudes every ellipsoidal formula is written in stop existing.

    part 8 · wrong
  13. There is no such thing as the radius of the Earth at a latitude. The two principal radii of curvature of WGS84, against latitude, with the mean radius that every table prints drawn across them. The meridional radius M runs from 6335.44 km at the equator to 6399.59 at the pole; the prime-vertical radius N from 6378.14 to the same value. They differ by 42.70 km at the equator and meet only at the pole, and the single number 6,371 km lies between them at no latitude where either is right.

    The radius of curvature is two numbers

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

    part 9 · datums
  14. Two expansions of one integral. The worst error along the whole meridian, against the number of sine terms kept, for the series in the third flattening and the classical series in e². Both are checked against a Simpson's rule on the defining integral, which shares no algebra with either. At one and two terms they are the same number to four digits and the e² series is fractionally ahead; from the third term the n series pulls away, and at four it is 1175 times more accurate — 7.6e-8 metres against 9.0e-5.

    Which small quantity the series is in

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

    part 10 · wrong
  15. Four radii of the Earth, and one that is a range. The four constants called the mean radius of the Earth, on a scale of kilometres, with the range of the local Gaussian radius √(MN) drawn behind them. Three of the four agree to about a part per million; the rectifying radius is 3560 metres smaller, which is 559 parts per million. The Gaussian radius spans twelve times that range on its own, which is why there is no such thing as the conformal sphere.

    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.

    part 11 · families
  16. Three budgets for one meridian distance, seven orders of magnitude apart. Every quantity that can make a computed distance from the equator to the pole wrong, on a scale of powers of ten. The series in the third flattening, at its worst anywhere on the meridian, is out by 76 nm at four sine terms, 31 µm at three and 22.0 mm at two. The two conventions WGS84 and GRS80 put the pole 82 µm apart. The ellipsoid that best fits the Earth is known only to 15.7 cm from its axis and 563 µm from its flattening, and WGS84 puts the pole 67.2 cm away from it.

    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.

    part 12 · wrong
  17. Four spheres are wrong about a direction before anything has moved. At 45° of latitude, the worst error over twenty-four starting directions of the azimuth each substituted sphere gives for a geodesic on WGS84, against the distance to its end. Geodetic latitude is wrong by 347″ at one kilometre, and the parametric, authalic and rectifying latitudes by 174″, 116″ and 86.8″ — a half, a third and a quarter of it — flat until a few hundred kilometres. The conformal latitude is right at the start and wrong by 54.3 mas at one kilometre, 5.43″ at 100 and 54.3″ at 1,000; the geocentric latitude starts at 620 mas and joins it. Bessel's sphere, which also corrects the longitude, is exact at every distance to the arithmetic's floor of 1.2e-7″.

    A bearing on a sphere is decided by its latitude, not its radius

    A sphere that stands in for the ellipsoid cannot be wrong about a direction by being the wrong size, because enlarging a sphere moves no angle. What it can be wrong by is the latitude put on it. Geodetic latitude turns every bearing at 45° by up to 347 arcseconds before anything has moved; the conformal latitude starts exact and drifts 54 arcseconds in a thousand kilometres; and only Bessel's sphere, which changes the longitude too, is exact at every distance.

    part 13 · wrong
  18. An angle is a difference of two directions, and the error does not cancel between them. A geodesic triangle with sides of 30 km at 45° N, its first side laid off on a bearing of 45° and the second 60° round from it. Substituting a sphere carrying the geodetic latitude turns each direction at a corner by an amount that runs as the sine of twice its azimuth, so the two directions at A are turned by different amounts and the angle between them is wrong by their difference. Here the worst single direction is out by 345.79″ and the worst angle by 521.09″ — larger, not smaller. Bessel's sphere, which also corrects the longitude, keeps every angle to 6.2e-9″.

    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.

    part 14 · wrong
  19. Where the correction overtakes the sphere it is correcting for. The worst angle error of four substituted spheres against the size of the triangle, with Legendre's spherical excess on the same axes. The two that are wrong at first order in the flattening are flat in the side, because the error belongs to the latitude and not to the distance. The two that are right at first order have no such floor, so what is left is a term rising in proportion to the side. The excess rises as its square, so each pair meets exactly once, and the meeting decides which of the two corrections a computation is entitled to ignore: the geodetic latitude at 490 km, the rectifying latitude at 247 km, the conformal latitude at 23.8 km, the geocentric latitude at 17.6 km.

    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.

    part 15 · wrong
  20. Along a meridian a good sphere's error adds up triangle by triangle; a bad base does not. A strip of thirty-kilometre equilateral triangles running north from 45°, each side computed from the last by Legendre's theorem on a substituted sphere. The scale error of the side just computed, against how many triangles lie between it and the measured base. The conformal sphere adds 7.90 ppm a triangle and reaches 158.1 ppm after 20; the geocentric, 158.9 ppm; the authalic, −52.4 ppm; the rectifying, 0.29 ppm. Shaded: three standard deviations of the scale error one-second angles leave, 3.96 ppm a triangle, growing as the square root of the count. Dashed: a base measured 8 ppm long, which stays 8 ppm at every triangle.

    A strip of triangles carries the sphere's error to the next base

    One triangle cannot tell a substituted sphere from a bad base: both leave the computed side a few parts per million long. A triangulation is not one triangle but a strip, each computed side the next one's base, and along it the two come apart completely. A bad base is one factor, the same at every triangle. A sphere's error is made afresh in each triangle and the strip adds it up, and what it adds up to at the far end is exactly the change in that sphere's own meridian scale between the two ends — 158 parts per million for the conformal sphere over twenty triangles north from 45°, predicted to half a part. Along a parallel it is nothing; at a check base three triangles on it is already three times the noise of one-second angles.

    part 16 · wrong
  21. Adjusted to close on the second base, a sphere's drift of hundreds of ppm becomes a bow of under one. A strip of 40 thirty-kilometre equilateral triangles north from 45°, computed on the conformal sphere, between a base at each end. Unadjusted (dashed), the side error climbs to 315.0 ppm at the second base. Adjusted to close there, each triangle's share of the misclosure equal, what is left (solid) is the sphere's stretch less a straight line between the bases: 0.63 ppm at its largest, at triangle 23. Shaded: three standard deviations of what one-second angles leave in the adjusted strip, 37.6 ppm at the middle and nothing at either base.

    An adjustment hides the sphere only in a strip of equal triangles

    A strip computed on a substituted sphere drifts by the sphere's own stretch, 315 parts per million over forty triangles north from 45°, and an adjustment that closes it on a second base takes almost all of that away: in a strip of equal triangles what is left is the stretch less a straight line, 0.63 parts per million, a third-order bow sixty times below the noise of the old angles. A real chain is not made of equal triangles. The adjustment spreads the misclosure by how uncertain each triangle's angles are, the sphere put it in by how far north each triangle reaches, and the difference is a first-order residual that never clears the noise at any one side and that the old network's own geometry predicts exactly — so a re-observation matched against it reads the sphere at seven standard deviations.

    part 17 · wrong

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