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

Grid north against true north at 52°, across a 6° zone

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

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.

It is drawn by ellipsoid-figure with show: "grid-convergence" — one member of a family of 12 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.

7 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.

The sixty zones, each six degrees wide. Every zone is a separate transverse Mercator projection about its own central meridian, so the world is covered by sixty maps rather than one. Zone 31 is picked out, running from 0° to 6° with its axis on 3°. Coordinates do not carry across a zone boundary — a point on either side of one has two entirely different eastings, and nothing in the numbers says which zone they belong to. The families

UTM and the zone system

Sixty separate maps of the world, each six degrees wide, each with a scale factor of 0.9996 chosen so the projection is wrong everywhere and less wrong at the edges. Every constant in the definition is a measured trade rather than a convention.

 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.

 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 3-figure grid reference names. The square is the 100-metre cell a 3-figure reference stands for, and the filled dot is the point the reference was made from. A reference is TRUNCATED rather than rounded, so the digits name the square's south-west corner — the open dot — which here is 9.1 metres west and 82.8 metres south of the point. Rounding would have landed on the nearest corner instead, 9.1 metres away east–west. The truncation is deliberate: it is what makes a shorter reference a larger square containing the same point. Grids, and what a survey does

A grid reference names a square

This ladder has priced everything about a grid except how a coordinate on it is written. A grid reference is truncated rather than rounded, so it names the south-west corner of a square rather than a point in it — and a population of references is displaced half a cell each way, which is a bias rather than scatter and does not average out.

 A parallel, and the places with the same northing as its middle. Both curves are drawn on a transverse Mercator zone 6° wide at 45° north. The upper one is the parallel — every place on it is at the same latitude — and it climbs 4379 metres of northing on the way to the zone edge. The lower one is the set of places whose northing equals the parallel's on the central meridian. Between them lies every pair the grid puts in the wrong order, and its widest point is 4.39 kilometres of latitude. Grids, and what a survey does

Further north on the grid is not further north

Thirteen rungs price a grid's origin, scale, units and zones, and every one treats a coordinate as a position. It is also an ordering, and the two disagree: a parallel climbs 4,379 metres of northing on its way to the edge of a UTM zone, so two places 4.4 kilometres apart in latitude can be listed in the wrong order — and the share of pairs it happens to is the convergence, in a different unit.

 One pair of numbers, a hundred and twenty places. The easting 412,000 and northing 5,678,000, interpreted in each of the sixty zones and in both hemispheres. The northern candidates are a ring of sixty at 51.247° north, spaced exactly six degrees apart; the southern ones are a second ring at -39.043°, which is not the mirror of the first because the southern convention subtracts the northing from ten million. Every one of the hundred and twenty is a perfectly valid reading of the same two numbers. Grids, and what a survey does

One pair of numbers, a hundred and twenty places

A UTM coordinate is two numbers and a zone. Drop the zone and the numbers are still valid in each of the sixty; drop the hemisphere too and the pair names a hundred and twenty places. They form two rings at one latitude each, spaced exactly six degrees apart, and every one of them has the same grid convergence and the same scale factor — so no further geometric measurement can choose between them.

 A legend saying "illuminated from 315°" is true at one longitude. A hillshade's azimuth is measured from the top of the sheet, because the shading is computed on the projected raster. The top of the sheet is grid north, so the compass bearing the light comes from is the declared azimuth plus the meridian convergence, and that varies across the sheet. On a conic it swings by 55.3 degrees over eighty degrees of longitude. On a cylindrical projection in its normal aspect it does not swing at all, which is the flat line — the only case the legend is right everywhere. Measuring distortion

The light comes from a page direction

A hillshade's illumination azimuth is declared from the top of the sheet, and the top of the sheet is grid north. On a conic the light therefore swings 52.6° across eighty degrees of longitude, and 72.8 per cent of the sheet is shaded differently from what the legend claims.

The whole library