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

The order the corrections go in, and what it costs

Drawn here at the parameters it defaults to, with every essay that calls it.
The order the corrections go in, and what it costs. Setting out a 51.8 km line on the UTM zone 31N means turning a grid bearing into an azimuth to observe, and there are two corrections: the meridian convergence, 0.8669° here, and the arc-to-chord correction, 6.242″. Each bar carries the lateral offset it produces at the far end of the line, because a bearing error is a number nobody can picture and a sideways miss is the thing that misses. The exact arc-to-chord and the classical formula every manual gives differ by 0.0044″, which is 1.10 mm at the far end — small, real, and the reason the corrections have an order rather than a sum.

Setting out a 51.8 km line on the UTM zone 31N means turning a grid bearing into an azimuth to observe, and there are two corrections: the meridian convergence, 0.8669° here, and the arc-to-chord correction, 6.242″. Each bar carries the lateral offset it produces at the far end of the line, because a bearing error is a number nobody can picture and a sideways miss is the thing that misses. The exact arc-to-chord and the classical formula every manual gives differ by 0.0044″, which is 1.10 mm at the far end — small, real, and the reason the corrections have an order rather than a sum.

It is drawn by survey-figure with show: "setting-out" — one member of a family of 16 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.

3 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.

Four steps between a tape and a drawing. A slope distance of 76.895 km measured at 3.2° on ground 220 m above sea level, reduced to the British National Grid, with every step drawn on a logarithmic scale in millimetres. The slope reduction is the largest by a wide margin at 119.90 m and is the one everybody applies. The grid reduction is 30.23 m. The fourth bar is not a step at all: it is what using the levelled height where the height above the ellipsoid is wanted costs, with a geoid separation of 48.5 m — 583 mm, which is 1.9% of the grid reduction it sits beside and 29 times the tolerance the job closes to. Reading a correction's importance off its share of the chain is how it gets dropped. Grids, and what a survey does

What a tape measures

Four steps stand between an instrument reading and a coordinate, and their sizes are not in the order anybody expects. On a 77-kilometre line the slope reduction is 120 metres, the grid reduction 30, and a step nobody names — using the height above sea level where the height above the ellipsoid is wanted — is 583 millimetres.

 The order the corrections go in, and what it costs. Setting out a 51.8 km line on the UTM zone 31N means turning a grid bearing into an azimuth to observe, and there are two corrections: the meridian convergence, 0.8669° here, and the arc-to-chord correction, 6.242″. Each bar carries the lateral offset it produces at the far end of the line, because a bearing error is a number nobody can picture and a sideways miss is the thing that misses. The exact arc-to-chord and the classical formula every manual gives differ by 0.0044″, which is 1.10 mm at the far end — small, real, and the reason the corrections have an order rather than a sum. Grids, and what a survey does

Setting out runs the chain backwards

Turning a design coordinate into something to observe on the ground means undoing the reduction chain, and undoing a chain reverses the order as well as the operations. Two of the corrections do not commute, and getting them the wrong way round misses by a millimetre at fifty kilometres.

 What a 20 mm closure tolerance lets each station hide. A closed traverse of seven stations in plan, each labelled with the angle blunder that would leave the closure inside a 20 millimetre tolerance. An angle error at a station rotates everything downstream of it about that station, so the closing point moves by the distance from the station to the close — and the last station before the close stands 72 metres from it and can hide 57 arcseconds, against 2.1 at the worst-placed station. The check is not insensitive; it is unevenly sensitive, and nothing in the specification says so. Grids, and what a survey does

What a closed figure cannot see

A closed traverse imposes exactly two conditions on its observations, so everything else is free — and the freedom is not spread evenly. At a twenty-millimetre tolerance the worst-placed station in a seven-station loop hides an angle blunder of 2.1 arcseconds and the station standing seventy-two metres from the close hides 57, because a rotation about a point near the finish moves the finish hardly at all.

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