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

One level surface, 1000 m up at the equator, in two height systems

Drawn here at the parameters it defaults to, with every essay that calls it.
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 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.

It is drawn by height-figure with show: "height-systems" — one member of a family of 8 figures that share a generator, so the drawing above is what that generator returns when it is asked for this one and given nothing else.

4 essays call it. Every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this changes every one of these figures.

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. What the numbers refer to

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.

 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. What the numbers refer to

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.

 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. What the numbers refer to

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.

 The sea surface, against the surface heights are measured from. The stated mean dynamic topography — how far the sea stands above the geoid — drawn on Mollweide, with the eight tide gauges marked. It ranges over 2.09 metres, it is smooth, and it is what every national datum's zero is sitting on. Nothing here is a model's output: it is a stated closed form, chosen to have the observed sign structure and the observed size. What the numbers refer to

Every country's zero is a different surface

Ten rungs measure a height against a geoid and treat the geoid as one object. No national datum is on it: each is pinned to the mean sea level at one tide gauge, the sea surface stands up to two metres from the geoid, and the eight European zeros measured here spread over 462 millimetres — a step no levelling can remove and nobody's error.

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