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

Why a levelled height is not a distance

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

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.

It is one of 8 figures drawn by the same construction, and the drawing above is this one with nothing changed.

6 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.

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. What the numbers refer to

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.

 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. What the numbers refer to

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.

 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. Without a stated mass the relation between slope and deflection can convert that number but never produce it. What the numbers refer to

A deflection is the slope of a mass

One arcsecond of deflection of the vertical is 4.85 millimetres of geoid slope in a kilometre, and that exact relation produces no deflection anywhere by itself, because a deflection is the gradient of a geoid and a geoid model is cited rather than computed. 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 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 first calculation started from was a statement about one buried sphere, and this is a statement about any mass that produces the same deflection. What the numbers refer to

How far the plumb line bends

An earlier measurement 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.

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