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

A deflection of 10 arcseconds, and what it hides

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

It is drawn by height-figure with show: "deflection-geometry" — 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.

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.

 Grid north against true north at 52°, across a 6° zone. seven meridians drawn in the grid's own coordinates. The grid's north is straight up the page everywhere by construction; the meridians are not, and the angle between them is the convergence. It is exactly zero on the central meridian, reaches 2.36° — 142 minutes of arc — at 3° out, and changes sign across the middle. The manual's γ = Δλ sin φ is 3.0″ short of it at the zone edge. The families

Grid north is not north

A grid has one north, parallel everywhere on the sheet by construction. The Earth has a different one at every point. The angle between them reaches two degrees at the edge of a UTM zone, and a straight line on the grid is not a straight line on the ground either.

 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. What the numbers refer to

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.

 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 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. What the numbers refer to

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.

 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. What the numbers refer to

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.

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