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

Direction decides which rule is adequate, and length does not

Drawn here at the parameters it defaults to, with every essay that calls it.
Direction decides which rule is adequate, and length does not. The three classical rules for the scale factor of a line, on two lines of the British National Grid at right angles, with the error expressed as distance rather than as a ratio. Bars are logarithmic across four decades. The endpoint mean is out by 11 mm along the 556 km north–south line and by 97 m along the 362 km east–west one — worse on the shorter line by a factor of 8735. In a transverse Mercator k varies with the easting and hardly at all with the northing, so an east–west line traverses the whole parabola and a north–south line sits at one point on it. Simpson's rule holds both to under two millimetres.

The three classical rules for the scale factor of a line, on two lines of the British National Grid at right angles, with the error expressed as distance rather than as a ratio. Bars are logarithmic across four decades. The endpoint mean is out by 11 mm along the 556 km north–south line and by 97 m along the 362 km east–west one — worse on the shorter line by a factor of 8735. In a transverse Mercator k varies with the easting and hardly at all with the northing, so an east–west line traverses the whole parabola and a north–south line sits at one point on it. Simpson's rule holds both to under two millimetres.

It is drawn by survey-figure with show: "rule-by-direction" — 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.

2 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.

A scale factor is a property of a point, and a distance is not. The grid scale factor sampled along a 362 km line on the British National Grid at a bearing of 90°, with the three rules in use drawn as the constants they replace it by. In a transverse Mercator k depends on the easting almost alone and goes as its square, so along an east–west line the curve is very nearly a parabola. The endpoint mean is the chord across it and lands 268.6 ppm high, which is 97.3 m over this line. Simpson's rule integrates a parabola exactly and lands 0.004 ppm out — 1.3 mm. Neither number is about the line's length. Grids, and what a survey does

The scale factor of a line

A scale factor is a property of a point and a distance is not measured at a point. The three rules in use for averaging one along a line differ by a hundred metres on a 362-kilometre baseline, and which of them is adequate is decided by the line's direction rather than its length.

 A grid scaled to the ground, and where it stops being one. A surface coordinate system on the British National Grid: the grid multiplied by 1.0004044, the reciprocal of the combined factor at a project origin 120 m above the ellipsoid, so that a grid distance equals a ground distance there. It is exact at the origin by construction and wrong everywhere else, and the two directions are not comparable — the axis is logarithmic across five decades. 60 km due east the set-out distance is out by 5559 mm, and the same distance due north by 20.5 mm. The scale factor of a transverse Mercator depends on the easting and almost not at all on the northing, so a system sold as good "within a few kilometres" has a useful area that is a strip rather than a circle. Grids, and what a survey does

A grid scaled to the ground is not a map

Construction sites work in coordinates where a tape reading equals a computed distance, which is achieved by multiplying the national grid by a constant. The result is exact at one point, degrades east–west forty times faster than north–south, and is not a map projection at all.

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