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

Normal gravity — where it appears

The gravity field of the reference ellipsoid, computed in closed form from four defining constants. It varies by about half a per cent from equator to pole, and everything called an anomaly is a difference from it.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

Three surfaces, and the two heights between them. The ellipsoid, the geoid and the ground, with the two heights a coordinate can carry. Ellipsoidal height h is what a satellite fix returns and is measured from a surface defined by four constants. Orthometric height H is what a level and a staff measure and is referred to the geoid — the equipotential surface that best fits mean sea level. They differ by the separation N, drawn here as 45 metres because that is a stated input rather than a computed one: a geoid model is a data product with a truncation degree in it, and this site computes rather than downloads. The arithmetic h = H + N is exact whatever N is.

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.

datums · Height 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 Why a levelled height is not a distance. The correction between raw levelling and orthometric height, for lines at 200, 500, 1000, 2000 metres above the geoid running north from 50°. It is not instrument error: level surfaces converge towards the pole, so a run that stays on one of them gains height relative to another. A line 2000 metres up reaches 647 millimetres over 400 kilometres. The dashed line is 10 millimetres, which is about what a first-order levelling network closes to over that distance — so this is not a refinement, it is the larger of the two numbers.

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.

datums · Height 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 One level surface, 1000 m up at the equator, in two height systems. A single equipotential surface — the shape a body of water takes — with the number each height system gives it, all the way from the equator to the pole. The orthometric height, which is the distance up the plumb line and therefore a length, falls by 5.28 metres along it, because level surfaces converge polewards. The dynamic height, which is the geopotential number divided by one constant gravity value, is flat to 0.000 millimetres — it is the same number everywhere on the surface, and it is not a distance from anything.

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.

datums · Height The height of a 4000-metre summit, against the density assumed beneath it. The geopotential number is 39204 m² s⁻² and is not in doubt. Turning it into a length divides it by the mean gravity along the plumb line, which is inside the mountain — and reconstructing that from the gravity measured at the surface needs a density. Taking the rock to be 2400 rather than the 2670 it actually is puts the summit 185 mm low; taking it to be 2900 puts it 157 mm high. Skipping the reduction entirely puts it 691 mm high, which is why the reduction exists.

The line a height is measured along

Five rungs have argued about the surface a height is measured *from* and every one of them took the line it is measured *along* to be straight and known. It is neither: through a stated buried mass a plumb line arrives 47 millimetres from the point below the summit, and the height it gives depends on the density of rock nobody has seen — 342 millimetres of spread at 4,000 metres and 1.37 metres at 8,000.

datums · Height Round a triangle the geoid's deflection has a flux and no circulation. A fifty-kilometre triangle on a stated patch of geoid whose undulations follow Kaula's spectrum between 10 and 400 km, with the deflection of the vertical drawn as strokes down the geoid's slope. Beside each side, the deflection's component across it and along it, each integrated along the side, in metres: across −0.689, −0.538, +0.499; along −0.235, +0.275, −0.040. The sum across is −0.727 m, which over the Earth's radius is −0.0235″ of the triangle's excess. The sum along is +0.000003 m — the geoid's height carried round the loop and returned to where it started.

A levelling loop reads the circulation, and the geoid has none

A triangle of angles reads the geoid as the flux of the deflection of the vertical through its sides. A levelling loop is the other integral round the same curve, the circulation, and at the geoid that one is identically zero, because the deflection is the slope of a height. What a levelling loop run on the ground misses by is the Jacobian of gravity and height over the loop: 31 mm round a fifty-kilometre loop in a stated mountain range, none of it from the free-air gradient that changes gravity by 1,333 mGal there, uncorrelated with what the triangle reads, and two and a half times the noise of the levelling where the triangle's reading is a fiftieth of its own.

impossibility · Curvature

Named alongside it

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

EquipotentialGeoidOrthometric heightVerificationGeopotential numberLevellingToleranceVertical datumEllipsoidWGS84AssertionConservation

All concepts