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

Deflection of the vertical — where it appears

The angle between the plumb line and the ellipsoid's normal, which is the slope of the geoid and is produced by lateral variation in mass. It runs to one or two arcseconds over a stable continental interior and to tens in mountainous country, and it bends the plumb line a survey is levelled along.

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

A deflection of 10 arcseconds, and what it hides. The ellipsoid normal and the plumb line at one point, with the geoid tilted against the ellipsoid by 10 arcseconds — drawn 3000× steeper than life, because at true scale the two lines are indistinguishable. The relation is exact and linear: an arcsecond of deflection is the geoid rising 4.85 millimetres in a kilometre, so 10 arcseconds is 48.5 millimetres per kilometre. A star sight measures the plumb line's direction, so astronomic latitude differs from geodetic by exactly this angle — 309 metres of ground, at a point where the coordinate itself is correct.

The plumb line is not the normal

A latitude measured from the stars and a latitude that means a position on the ellipsoid are different angles, because a plumb bob hangs along gravity and gravity is not perpendicular to a mathematical surface. Ten arcseconds of difference is 309 metres of ground.

datums · Height 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 A mass, the geoid it raises, and the plumb lines that lean towards it. A sphere of 5 km radius buried 8 km down, denser than its surroundings by 500 kg per cubic metre — a salt dome, or an ore body. Its mass is 261.8 × 10¹² kg, and outside itself its field is a point mass's exactly, so everything above is a closed form. The geoid rises 22 centimetres over it, drawn at 20,000× the true slope; the plumb line leans by at most 2.21 arcseconds, and it does so 5.7 km to the side rather than above the body, because the deflection is the geoid's SLOPE and a slope is zero at a summit. That is the number this site has been able to relate and unable to compute since its practice phase.

A deflection is the slope of a mass

The site has computed the exact relation between a geoid slope and a deflection of the vertical — one arcsecond is 4.85 millimetres in a kilometre — and computed no deflection anywhere, because a deflection is the gradient of a geoid and the geoid is a citation. State a mass instead, and every quantity is a closed form: a salt dome five kilometres across bends the plumb line by 2.21 arcseconds.

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 The drift is a straight line in the deflection. The horizontal distance between where a plumb line hangs at the top of a column of rock and where it hangs at the bottom, against the deflection of the vertical the mass produces at the surface. Four heights of column. Every line is straight through the origin: 12.54 mm of drift per arcsecond of deflection over a 4,000 m line, to two parts in ten thousand across a fortyfold range of deflection. Which is what makes the number transferable — the 47 mm the ladder started from was a statement about one buried sphere, and this is a statement about any mass that produces the same deflection.

How far the plumb line bends

The previous rung dropped a plumb line down a four-kilometre column of rock beside one buried mass and found it arrived 47 millimetres from the point below the summit. That is a number about that mass. Parameterising by the deflection of the vertical instead — the quantity surveyors actually measure — gives 12.54 mm per arcsecond, exactly linear across a fortyfold range.

datums · Height What a geoid model leaves out, against the degree it stops at. The RMS of everything above the model's highest degree, from Kaula's rule — the statement that the normalised coefficients at degree n are about 10⁻⁵/n². The line is R × 10⁻⁵ ÷ n, so a model to degree 360 omits 17.7 centimetres and one to 2190 omits 2.9. Every orthometric height derived from such a model carries that as an error, and it is not quoted with the height.

The geoid model stops at a degree

Eleven essays treat the geoid as a surface that exists. Every geoid anybody uses is a series truncated at a degree, so every orthometric height derived from one carries an omission error nobody quotes with the height — eighteen centimetres at degree 360 — and the same truncation removes two thirds of the slope, which does not converge at all.

datums · Height The same deflection, from a compact mass and from a broad root. Each curve is the deflection of the vertical across a mass buried 8 km down, with the mass solved so that all of them peak at 10″. The narrow one is the buried sphere the earlier measurement used; the broad ones are crustal roots 40 and 160 km wide, modelled as that mass spread along a line. They agree where it matters most and disagree everywhere else: the signal is 30 km wide for the compact body and 188 km for the widest, which is the difference the plumb line feels as it descends.

A mountain is not a buried sphere

The plumb line's drift was measured over a compact buried body and grows as the 0.69 power of the column's height — an exponent that is a statement about how quickly a buried sphere's field weakens with distance rather than about mountains. Spread the same mass into a crustal root and the exponent climbs to 0.84, while the proportionality to the deflection survives exactly.

datums · Height

Named alongside it

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

GeoidOrthometric heightLevellingEquipotentialGravity anomalyGeopotential numberSurface normalVertical datumAstronomic latitudeAuxiliary latitudeConventionDatum

All concepts