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

Flattening — where it appears

How much an ellipsoid departs from a sphere, as the difference between its axes over the larger of them — about one part in 298 for the Earth. It is about one part in 298 for the Earth and about one part in 169 for Mars, so a spherical formula that is adequate here is not adequate there.

Named by 25 essays across 7 fields — each of them below, with the objects they name alongside it.

The same coordinate on four datums. One pair of numbers — 2.0° west, 54.5° north — read as a coordinate on OSGB36, ED50, NAD27 and on WGS84, and plotted where each reading puts the mark on the ground. The spread runs to 195 metres. The numbers are identical; only what they refer to differs.

What a coordinate refers to

A latitude and a longitude are not a place. They are a claim about a model, and the model has a shape, a position, a set of marked stones and a date — change any one of the four and the ground under the numbers moves by hundreds of metres.

datums · Datum A cylinder unrolls exactly; a sphere does not. Both pictures show the same grid. On the left it is wrapped round a cylinder of radius 1, on the right it is laid flat, and every distance in the grid is the same in both — the circumference is 2π and so is the width of the rectangle, checked to 10⁻⁹. This is possible because a cylinder has zero Gaussian curvature. No corresponding picture exists for a sphere.

What can be unrolled

A cylinder is obviously curved and is intrinsically flat, so it lays out on a table with nothing stretched. A sphere is not. The distinction is exactly zero Gaussian curvature, and it is why paper tubes exist and paper globes do not.

impossibility · Curvature Fitting a sphere to 10° of latitude at 54.5°. The Earth's Gaussian radius of curvature across the band, with two candidate spheres drawn against it. The global mean radius is 6371.0 kilometres and sits 2203 parts per million away from the ground here — 2.2 metres in every kilometre measured. The best local radius is 6385.0 kilometres and has no bias at all by construction, leaving 325 parts per million of spread that no sphere can remove, because the curvature varies across the band and a sphere's does not. The gain is a factor of 6.9.

A datum is fitted to a region

Forty countries adopted forty ellipsoids, and the usual explanation is that measurement was poor. It was not — a figure fitted to one country beats the global one over that country by a factor of seven, and what it removes is a systematic bias of two metres in every kilometre.

datums · Datum 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.

wrong · Ellipsoid 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.

families · Ellipsoid What treating the Earth as a sphere costs, per journey. The ellipsoidal geodesic minus the spherical great circle, in kilometres, for five journeys. The correction is a few tenths of a per cent and it changes sign: a route running east–west at mid latitude is longer on the ellipsoid, and one running along a meridian is shorter, because an oblate body is fatter round the equator and flatter pole to pole. Both distances are computed — Vincenty's iteration against the haversine formula — and the ellipsoidal one is checked against a geodesic obtained by integrating its own differential equation.

Geodesics on the ellipsoid, and why they are hard

The shortest path on a flattened Earth is not a plane curve, has no closed form, and can be longer or shorter than the spherical answer depending on which way it runs. Every practical method is a series or an iteration, and the correction changes sign.

paths · Paths 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.

wrong · Ellipsoid Where the inverse problem stops converging, 3° around the antipode of London. Every target in a 6° square centred on the point diametrically opposite London, shaded by how many iterations Vincenty's inverse formula needed to find the geodesic to it. 3 of 625 — 0% — never converged at all, and the worst success took 113 steps against a handful anywhere else on Earth. The failure is not a defect in the formula: near the antipode the shortest path is nearly ambiguous, and an iteration looking for one answer is being asked which of many.

The route with no shortest path

Between a point and the point diametrically opposite there are infinitely many shortest routes and no shortest route, and the standard formula for the distance between two places stops converging in a neighbourhood of it. The failure is a property of the question rather than a defect in the answer.

paths · Paths The radius of equal curvature, from equator to pole. The radius of the sphere that has the same Gaussian curvature as the ellipsoid does, at each latitude. On WGS84 it runs from 6357 km at the equator to 6400 km at the pole — the polar region is the flattened part of a squashed ball and is therefore the LEAST curved — so K itself varies by 1.35%. A surface whose curvature varies cannot be laid on one whose curvature does not, so no map from the sphere to the ellipsoid is faithful either, and the least a conformal one can vary in scale is 6739 parts per million.

The curvature of the Earth is not one number

An ellipsoid's Gaussian curvature varies by 1.35% from equator to pole, so the sphere is not developable onto the ellipsoid any more than the plane is onto the sphere. The spherical approximation is a projection with an irreducible cost, and the cost is 6,739 parts per million.

impossibility · Curvature "Within 0.01°" on the ground, at five latitudes. The same condition drawn at five latitudes, all at one scale. A degree of latitude is a fixed distance — 1106 metres here, varying by less than one per cent from equator to pole — while a degree of longitude collapses as cos φ, from 1113 metres to 289 at 75°. So the "circle" is an ellipse of 3.86:1 there, and it encloses 26 per cent of the ground the same condition covers on the equator. Even on the equator it is not round: M is smaller than N by the flattening, so the shape is 6694 parts per million shorter north–south than east–west.

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.

applied · Dataset Normal gravity, derived from four constants. Gravity on the surface of the level ellipsoid, by Somigliana's closed form, for WGS84. Nothing here is measured: a, f, GM and ω go in and the whole curve comes out, rising 5186 milligal — 0.53 per cent — from equator to pole. The two open marks are the published values of equatorial and polar gravity for WGS84, which the derivation reproduces to ten significant figures rather than borrowing.

The ellipsoid is a level surface

WGS84 publishes two dozen constants and defines four of them. The other twenty are consequences — polar gravity, the potential of the ellipsoid, the coefficient that dominates the Earth's gravity field — and every one comes back here from a, f, GM and ω to the last digit published.

datums · Height Where two equal geodesics meet, from 15° north. A geodesic leaving at azimuth α and its mirror image at −α have the same length wherever they meet, and by symmetry they meet on the antipodal meridian. On a sphere they all meet at one point, the antipode, to 1.4e-7° — the flat line. On the ellipsoid the meeting latitude moves with the azimuth, from -15.563° to -15.009°, so the set of points with two shortest routes is an arc 61 km long, and the distance to it varies by 31 km along its own length.

Where the shortest route stops being the only one

On a sphere there is exactly one point with no shortest route from a given place: the antipode. On the ellipsoid the Earth actually is, that point is an arc — sixty-six kilometres of the antipodal meridian for a point on the equator, half a kilometre for one at 85°, and every point of it reachable by two different geodesics of exactly equal length.

paths · Paths 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.

wrong · Ellipsoid The route at height does not lie above the route on the ground. London to Tokyo, solved at the surface and again at a stated height, with the two ground tracks compared point for point. At a cruising altitude of eleven kilometres the two part by 11.8 metres; the departure is proportional to the height, at a fitted slope of 0.997. On a sphere the same measurement returns 2.7e-9 metres, because the offset of a sphere is a sphere and the two geodesics coincide exactly. The offset of an ellipsoid is not an ellipsoid, and this is what that costs.

The shortest route is not at sea level

Ten rungs route on a surface and nothing is ever flown on one. The offset of a sphere is a sphere, so at altitude the great circle is the great circle. The offset of an ellipsoid is not an ellipsoid — its radii of curvature are M + h and N + h, which belong to no ellipsoid — so the shortest route at cruising height does not lie above the shortest route on the ground.

paths · Paths 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.

datums · Ellipsoid 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.

datums · Ellipsoid

A coordinate on another body

Mars publishes two latitudes for every point and they differ by up to 0.338°, which is twenty kilometres of ground — almost exactly the same distance as the Earth's own 0.192°, because Mars is smaller by nearly the same factor. The libraries that compute either have known both numbers since this collection's early essays and had never been pointed anywhere but here.

datums · Bodies

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.

wrong · Ellipsoid

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.

datums · Bodies

The bound on the body the country is on

The best conformal grid a country could have had was computed against a bound solved on a sphere, with a note saying the flattening was second order and unquantified. It is second order for a named projection — every family's best candidate moves by under 0.7 per cent — and it is 22 per cent for the bound, because an optimal map has already cancelled its own variation and has nothing left to hide a new one in.

practice · Grid

A body with no sea level

On Earth the zero of height is found rather than chosen — water settles onto the equipotential surface by itself. Nowhere else has one, and the difference is measurable: Clairaut's theorem predicts the Earth's flattening from its own gravity field to twelve metres at the pole and misses Mars's by 2.2 kilometres.

datums · Bodies

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.

wrong · Ellipsoid

Four radii of the Earth

Ten rungs handle the ellipsoid with an auxiliary latitude. Every one of those constructions 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.

families · Ellipsoid

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.

datums · Ellipsoid

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.

datums · Datum

Named alongside it

The objects these essays reach for when they reach for this one.

EllipsoidRadius of curvatureVerificationGeodesicConventionDatumNumerical integrationToleranceWGS84Meridian arcSpherical approximationAuxiliary latitude

All concepts