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

Meridian arc — where it appears

The distance from the equator to a given latitude along a meridian, which on an ellipsoid is an elliptic integral. It is computed here twice — by series and by quadrature — so that agreement between the two is evidence rather than restatement.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

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 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 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 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.

families · Ellipsoid 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.

wrong · Ellipsoid 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.

families · Ellipsoid 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.

wrong · Ellipsoid

Named alongside it

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

EllipsoidFlatteningTransverse MercatorConvergenceTruncationAuxiliary latitudeConformalityDatumGeodetic latitudeNumerical integrationQuadratureRadius of curvature

All concepts