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

Astronomic latitude — where it appears

The latitude a star sight returns, measured from the direction of the plumb line rather than from an ellipsoid's normal. Its difference from the geodetic latitude is the deflection of the vertical, which is the slope of the geoid and is produced by lateral variation in mass.

Named by 2 essays across one field — 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 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.

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.

datums · Height

Named alongside it

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

Deflection of the verticalEquipotentialGeoidOrthometric heightSurface normalToleranceAuxiliary latitudeClosed formDatumGeodetic latitudeGeopotential numberGradient

All concepts