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

The ellipsoid, and the grid laid on it

latitude-ladder and ellipse-normals are the several latitudes of one point; curvature-latitude is the radius that varies with them; series-error is what truncating the series costs; utm-zones, grid-scale, grid-convergence, convergence-profile and arc-to-chord are the grid's own quantities; datum-shifts is what changing the ellipsoid moves; equal-area-latitude and section-trio are the auxiliary latitudes and the normal sections.

latitude-ladder and ellipse-normals are the several latitudes of one point; curvature-latitude is the radius that varies with them; series-error is what truncating the series costs; utm-zones, grid-scale, grid-convergence, convergence-profile and arc-to-chord are the grid's own quantities; datum-shifts is what changing the ellipsoid moves; equal-area-latitude and section-trio are the auxiliary latitudes and the normal sections.

Every one of these is the same generator answering a different question, which is why they share a file, a set of colour roles and a set of assertions. Changing it changes all 12.

49 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.

What it draws

One picture per question the family answers. Each has a page of its own.

Five latitudes that are not the latitude, on WGS84. Each curve is the amount by which one auxiliary latitude falls below the geodetic latitude a coordinate actually means, in arcminutes. All five vanish at the equator and at the poles and peak near 45°, where the geocentric latitude is 11.55 arcminutes below the geodetic one — about 21.4 km on the ground. The curves never cross, which is forced by the algebra rather than by this ellipsoid's particular flattening.

latitude-ladder

Five latitudes that are not the latitude, on WGS84 Why there are two latitudes, at 45°. A meridian section of an ellipsoid with the flattening exaggerated 12× — at the true value of 1/298 this outline would be indistinguishable from a circle. Geodetic latitude is the angle the surface normal makes with the equatorial plane; geocentric latitude is the angle the radius makes with it. The two lines meet the plane at different points and the angles differ by 11.55 arcminutes on the real WGS84 ellipsoid, which is about 21 km of ground distance.

ellipse-normals

Why there are two latitudes, at 45° What each term of the Krüger series is worth, 3° off the central meridian. The worst error of the truncated series against an independently computed reference, in metres, on a logarithmic scale. Each additional term gains between two and three decimal orders, so the fourth-order formula every national grid is written to sits at 1.7e-5 m — far below anything the survey it serves can measure. The projection as specified is exact; the projection as computed is this good.

series-error

What each term of the Krüger series is worth, 3° off the central meridian 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.

utm-zones

The sixty zones, each six degrees wide What 0.9996 buys, across one zone at 0°. The point scale factor from the central meridian to the zone edge, measured from the projection's own derivatives. Without the constant the map is exact in the middle and 1382 parts per million too large at the edge. With it the map is 400 parts per million too small in the middle, reaches true scale at 1.75°, and is 981 parts per million out at the edge — a smaller worst case bought by being wrong everywhere.

grid-scale

What 0.9996 buys, across one zone at 0° What changing the datum alone does to a coordinate. The distance on the ground between a point as read on its national datum and the same numbers read on WGS84, computed through the published seven-parameter transformation. The shifts run from 49 to 166 metres. For comparison, the scale error a UTM zone introduces at its edge is under a metre per kilometre — so the datum, which is usually left unstated, dominates the projection, which is usually argued about.

datum-shifts

What changing the datum alone does to a coordinate Four ways to build the same equal-area projection. The cylindrical equal-area projection's own areal scale factor, measured from its Jacobian against the metric of the body it is drawn for. On a sphere with the spherical formula it is one everywhere, which is the control. Feed the same formula the geodetic latitude a coordinate actually carries and measure against the ellipsoid it refers to, and it is 1.00674 on the equator and 0.99332 at 88° — a spread of 1.34 per cent, on a projection whose entire purpose is that there is no spread. Rescaling to make the totals agree does not repair it. The authalic northing q/2 does, exactly.

equal-area-latitude

Four ways to build the same equal-area projection Three curves between 45°N and 55°N, 3026 km apart. The normal section observed from the first point, the normal section observed from the second, and the geodesic, each plotted as its distance to one side of the great-circle chord between the two ends. The two sections are 79.0 metres apart at their widest and the geodesic runs between them, 53.3 metres from the first. In LENGTH they differ by almost nothing — 2.40 millimetres over 3026 kilometres — so an observation that follows the wrong one measures the right distance along the wrong ground. On a sphere all three coincide exactly.

section-trio

Three curves between 45°N and 55°N, 3026 km apart 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.

grid-convergence

Grid north against true north at 52°, across a 6° zone The convergence across a zone, at 20°, 45°, 60°, 75°. Grid north's departure from true north against distance from the central meridian, at four latitudes. Each curve is zero in the middle and antisymmetric about it, and steepens towards the pole because the convergence carries a factor of sin φ. At 75° the edge of a 6° zone is already 2.90° out, which is why a bearing plotted from a map and a bearing observed from the sun disagree by more the further north the work is.

convergence-profile

The convergence across a zone, at 20°, 45°, 60°, 75° The arc-to-chord correction on 0.6° lines at 52° north. six lines of the same length and bearing, placed at different distances from the central meridian, with the angle between the projected geodesic and its chord plotted for each. The correction reaches 28.9″ at the edge of the zone and changes sign across the middle, which is what makes it geometry rather than a fudge factor. The hollow marks are the classical closed form, which agrees to 0.016″.

arc-to-chord

The arc-to-chord correction on 0.6° lines at 52° north The radius of equal curvature, from equator to pole. The radius of the sphere that has the same Gaussian curvature as the ellipsoid does, at each latitude. On WGS84 it runs from 6357 km at the equator to 6400 km at the pole — the polar region is the flattened part of a squashed ball and is therefore the LEAST curved — so K itself varies by 1.35%. A surface whose curvature varies cannot be laid on one whose curvature does not, so no map from the sphere to the ellipsoid is faithful either, and the least a conformal one can vary in scale is 6739 parts per million.

curvature-latitude

The radius of equal curvature, from equator to pole

Where it is called

Changing this changes every one of these figures.

Six surfaces and their Gaussian curvature. Curvature computed at the centre of each surface, by both available routes — from the way the surface sits in space, and from distances measured inside it alone. The two agree, which is Gauss's theorem. A surface with K = 0 can be unrolled flat without stretching: here the plane, the cylinder and the cone do, and the sphere, the torus and the pseudosphere do not. The impossibility

No map is faithful

Not "no map yet", and not "no map at page size". Gaussian curvature can be computed from inside a surface, a sphere has some and a plane has none, and that closes the question permanently.

 The same coordinate on four datums. One pair of numbers — 2.0° west, 54.5° north — read as a coordinate on OSGB36, ED50, NAD27 and on WGS84, and plotted where each reading puts the mark on the ground. The spread runs to 195 metres. The numbers are identical; only what they refer to differs. What the numbers refer to

What a coordinate refers to

A latitude and a longitude are not a place. They are a claim about a model, and the model has a shape, a position, a set of marked stones and a date — change any one of the four and the ground under the numbers moves by hundreds of metres.

 A false origin moves every number and no geometry. British National Grid, drawn twice at the same scale. On the left the coordinates are measured from the projection's own origin, where the central meridian meets the true origin's parallel, and 52% of the country takes a negative easting or northing — the worst reaching -323 km. On the right the authority's published false origin of 400 km east and -100 km north has been applied, and none of them does. Every distance computed from the two sets of numbers agrees to the last bit a double has left after carrying six figures, and every bearing agrees exactly; the only thing that changed is that no coordinate carries a sign. The margins say the origin was fitted to the land rather than placed arbitrarily below it: 77 km spare in the west and 50 m in the south, on a grid 988 km tall. Grids, and what a survey does

A grid has an origin that is not there

Every published national grid measures from a point in the sea. The British one sits 400 kilometres west and 100 kilometres south of where the projection's own zero is, and moving it changes every coordinate in the country and not one distance or bearing.

 OSGB36 to WGS84, one parameter at a time. Each bar is how far the mark at 2.0° west, 54.5° north moves under one of the seven parameters with the other six set to zero. seven of the seven are not zero for OSGB36. The units hide the comparison: one arcsecond of rotation moves this point 30.8 metres and one part per million of scale moves it 6.36 metres, so OSGB36's scale term contributes 130 metres — more than one of its three translations. What the numbers refer to

The seven parameters, and what each one does

A datum transformation is published as three metres, three arcseconds and one part per million, in three different units — which hides the fact that all seven are the same size of thing. Converted to ground displacement, the scale term beats two of the translations.

 Fitting a sphere to 10° of latitude at 54.5°. The Earth's Gaussian radius of curvature across the band, with two candidate spheres drawn against it. The global mean radius is 6371.0 kilometres and sits 2203 parts per million away from the ground here — 2.2 metres in every kilometre measured. The best local radius is 6385.0 kilometres and has no bias at all by construction, leaving 325 parts per million of spread that no sphere can remove, because the curvature varies across the band and a sphere's does not. The gain is a factor of 6.9. What the numbers refer to

A datum is fitted to a region

Forty countries adopted forty ellipsoids, and the usual explanation is that measurement was poor. It was not — a figure fitted to one country beats the global one over that country by a factor of seven, and what it removes is a systematic bias of two metres in every kilometre.

 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.

 Why there are two latitudes, at 45°. A meridian section of an ellipsoid with the flattening exaggerated 12× — at the true value of 1/298 this outline would be indistinguishable from a circle. Geodetic latitude is the angle the surface normal makes with the equatorial plane; geocentric latitude is the angle the radius makes with it. The two lines meet the plane at different points and the angles differ by 11.55 arcminutes on the real WGS84 ellipsoid, which is about 21 km of ground distance. What is taught wrongly

The Earth is a sphere, and when it is not

Every essay before this one treated the Earth as a ball, and said so. The flattening is one part in three hundred, which is nothing for a distance, everything for a latitude, and exactly enough to make the most-used projection in the world fail the property in its own name.

 The cut at 85.0511°, and the alternatives that are not square. Mercator's northing runs to infinity at the pole, so a tiling has to stop somewhere, and the latitude is not a rounding: 85.0511° is where the northing equals half the world's width, which is the only cut that makes the projected world a square. The bars are what other cuts would give — at 89° the world is 1.51 times as tall as it is wide, and a single square root tile cannot cover it. The price is 0.373 per cent of the Earth's surface, 1,901,487 square kilometres in two caps, computed from 2πR²(1 − sin φ) rather than estimated. What a machine does with it

The square costs the poles

The cut at 85.0511287798° is the most-quoted number in web mapping and is almost never derived. It is where Mercator's northing equals half the world's width — the condition for a square — and it drops 1,901,487 square kilometres. A two-tile root would have reached 89.786° and dropped 3,558.

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