Cartographic Projection
every map is wrong, and by exactly how much
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The thread: Computed, not quoted — page 3

Scale factors, areal ratios, angular deformations, geodesic distances and the areas of countries are all computed while the figure is drawn. None is a number recalled from a table. Essays 49 to 72 of 299.
The scale factor across Europe, from the middle outwards. Each band is the range the larger principal scale factor takes on the ring at that distance from the centre of Europe: 0 is the middle, 1 the frontier. two of the three projections drawn are conformal, and every one of those reaches its maximum at the right-hand edge, because the logarithm of a conformal map's scale factor is subharmonic. None of them reaches a maximum inside the region. Measuring distortion

Where the worst point is

The largest scale error on a conformal map of a country is always on the frontier, never inside it, whatever the country's shape and whichever conformal projection was chosen. It is a theorem rather than a tendency, and it is the reason Chebyshev's criterion works.

 London to Tokyo in three straight legs. The great circle, and the route a plan of three constant-heading legs actually follows between waypoints on it. The two touch at the waypoints and part between them by up to 456 km, and the flown route is 253 km longer than the direct one. The headings are 50°, 109°, 149°. Two routes. The great circle is 9559 km and is the shortest path on the sphere. The rhumb line holds a single compass bearing the whole way and is 11296 km — 1737 km further, or 18.2 per cent. On Orthographic the rhumb line departs from straight by 2.4e-1 of its own length. Paths and directions

Flying a curve in straight legs

Nobody steers a great circle, because a great circle requires the heading to change continuously. What is actually flown is a handful of constant-heading legs between waypoints on it, and the gap between plan and curve falls as the square of the number of legs.

 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. The impossibility

The curvature of the Earth is not one number

An ellipsoid's Gaussian curvature varies by 1.35% from equator to pole, so the sphere is not developable onto the ellipsoid any more than the plane is onto the sphere. The spherical approximation is a projection with an irreducible cost, and the cost is 6,739 parts per million.

 The scale along the Madrid–Tokyo corridor. The scale factor of four projections along the great circle from Madrid to Tokyo, each normalised to its own average over the route so the comparison is of variation rather than of size. The oblique Mercator whose own equator is laid along the corridor holds the scale to 0 parts per million; Mercator varies by 118.7% over the same line. A corridor is a curve, not a region, and the projection an area criterion picks is not the one a curve wants. What each projection optimises

Choosing for a line, not a region

Every criterion in this subject integrates over an area. A pipeline, a railway or a coastal survey is a curve, and the projection an area criterion picks for it is not the one it should have — measurably, by a factor of five thousand.

 Where the ground is going, six plates. The velocity of the ground under a coordinate, computed as the cross product of each plate's rotation vector with the position, and drawn along the band where that rotation is fastest rather than over an outline this site does not hold. The quickest arrow is 75 millimetres a year on the Pacific plate, which is 1.89 metres in twenty-five years. drawn in Robinson — the arrows' directions are the projection's as much as the plates'. What the numbers refer to

The epoch is part of the coordinate

A latitude and a longitude with no date on them are incomplete, because the ground they refer to is moving at between ten and seventy millimetres a year. Australia crosses an ordinary survey tolerance in under two years and moved 1.8 metres between two versions of its own datum.

 The equal-area conic, from cylinder to plane. The largest distance between the conic at cone constant n and each of its two limits, over a shared grid, with the free scale and offset removed. Both fall as the FIRST power of the distance from their end — the slope on these axes is one — so the conic is never nearly cylindrical: halving n only halves the difference. At n = 0.999999 the conic is the Lambert azimuthal to 1.3e-6, and at n = 0.000001 it is the Lambert cylindrical to 1.2e-6. The families

The conic is the whole family

Cylindrical and azimuthal are usually presented as two of three families beside the conic. They are the two ends of it. One parameter runs from the cylinder to the plane, and both limits are exact rather than suggestive — which is measurable, and measured here.

 London to Tokyo, round a 15° exclusion. The direct great circle, dashed, runs through the disc. The admissible shortest route leaves it along a great circle tangent to the rim, follows the rim, and leaves along another tangent — which is the closed-form answer and is checked against a shortest-path search over the rim that knows nothing about tangents. It costs 39 kilometres on 9559, which is 0.41 per cent. The rim stretch is 293 kilometres of it, and it is the only part of the route that is not a geodesic anywhere along its length. Drawn in Orthographic. Paths and directions

A route that must go round

Every route on this site so far has been free to go anywhere, and no real route is. The shortest path past a circular exclusion is two tangent great circles and an arc of the rim — a closed form that agrees with a shortest-path search to three metres in 9,598 kilometres — and it costs not the obstacle's size but the square of how far the obstacle reaches past the route.

 53° north, 2° west: four places one pair of numbers could be. The same two numbers, read as three datums and as the pair in the other order. The centre is the WGS84 reading; the marks are where the ground is if the numbers were measured against OSGB36, ED50, NAD27 instead, at 103 m, 138 m, 195 m. Every one of those is a plausible position — near enough to look like a survey difference and far enough to be a different field. Reversing the two numbers instead moves the point 7942 kilometres, and is the mistake nobody worries about because it is usually obvious. What a machine does with it

A coordinate without its system is not a location

The same two numbers name ground 103 metres apart on one datum, 195 on another, and 7,942 kilometres away if the pair is read in the other order. The datum errors are the dangerous ones because they are plausible — and the axis swap, which everyone calls obvious, does nothing at all along a line that crosses three continents.

 Every mixture of Equirectangular at 50.46° and Aitoff. Airy's criterion over the whole sphere, for every weighted average of the Winkel tripel's own pair. Each mixture is normalised to its own best constant scale first, so the comparison is about shape rather than size. The curve has its minimum at w = 0.45, where the criterion is 0.3037 against 0.4114 and 0.4290 at the two ends. That is a projection nobody constructed beating both projections somebody did, by 26 per cent. What each projection optimises

The average of two projections

The Winkel tripel is literally the arithmetic mean of two other projections, and this site's implementation of it agrees with that mean to the last bit. Averaging beats both ingredients by 26 per cent — and it preserves conformality exactly, destroys equal-area completely, and can turn eighteen per cent of the world inside out without either distortion measure saying so.

 A triangle of 25.0° excess, drawn on Mercator. The three geodesic sides as curves and the three straight sides a ruler draws, with each vertex labelled by how far the ruler's angle is from the true one. The straight-sided triangle's angles sum to exactly 180° because it lies in a plane, and the real one's sum to 205.02°, so the three errors have to account for the whole 25.02° of spherical excess between them — and they do so on this conformal projection exactly as they do on any other. What is taught wrongly

Conformal does not mean the angles are right

A conformal projection preserves angles between curves at a point. Draw a triangle on one with a ruler and its angles are wrong by degrees — and the total error is fixed before the projection is chosen, because a plane triangle sums to 180° and the real one does not.

 Three surfaces, and the two heights between them. The ellipsoid, the geoid and the ground, with the two heights a coordinate can carry. Ellipsoidal height h is what a satellite fix returns and is measured from a surface defined by four constants. Orthometric height H is what a level and a staff measure and is referred to the geoid — the equipotential surface that best fits mean sea level. They differ by the separation N, drawn here as 45 metres because that is a stated input rather than a computed one: a geoid model is a data product with a truncation degree in it, and this site computes rather than downloads. The arithmetic h = H + N is exact whatever N is. What the numbers refer to

Height above what?

A satellite reports a height above a mathematical surface. A level and a staff report a height above the surface water settles on. The two disagree by tens of metres, both are correct, and only one of them decides which way a pipe drains.

 Which way Sinusoidal stretches the ground. The major axis of Tissot's indicatrix at 180 points, drawn as a stroke rather than an ellipse because the argument is the orientation. Longer strokes mark more elongated indicatrices. The direction departs from the graticule by up to 44.4°, so the scale "along the meridian" is not the scale along the direction that is actually stretched most. Measuring distortion

Distortion has a direction

Tissot's indicatrix is an ellipse, an ellipse has an orientation, and the orientation is never reported. On most projections it is not along the meridian, on a conformal one it does not exist at all, and both facts are computable from the same four derivatives as everything else.

 The image of a circle on Mercator, against its own indicatrix. A circle of three radii on the ground at 30°, 40°, projected exactly — the solid curve — against the ellipse the indicatrix predicts for it, dashed. The filled dot is the image of the circle's centre and the hollow one is the centre of area of what was actually drawn, which is not the same point. The departure runs from 3.14 per cent at 4° to 15.80 per cent at 16°, so it grows in proportion to the radius rather than to its square: halving the circle halves the relative error and does not quarter it. Measuring distortion

The indicatrix is a limit

Tissot's ellipse describes an infinitesimal circle, and every published one is drawn finite. The error of the description falls as the radius rather than as its square — halving the circle halves the lie — and on the Robinson projection at a table entry it does not fall at all: sixty metres and six kilometres are both 1.17 per cent wrong.

 "Within 0.01°" on the ground, at five latitudes. The same condition drawn at five latitudes, all at one scale. A degree of latitude is a fixed distance — 1106 metres here, varying by less than one per cent from equator to pole — while a degree of longitude collapses as cos φ, from 1113 metres to 289 at 75°. So the "circle" is an ellipse of 3.86:1 there, and it encloses 26 per cent of the ground the same condition covers on the equator. Even on the equator it is not round: M is smaller than N by the flattening, so the shape is 6694 parts per million shorter north–south than east–west. What a machine does with it

A degree is not a unit of length

"Within 0.01 degrees" is a condition anybody can write and no instrument can measure. On the ground it is an ellipse — 1,106 metres north–south and 558 east–west at 60° — and even on the equator it is not a circle, because the meridian's radius of curvature is smaller than the parallel's by the flattening.

 Sheets for a tolerance, from Chebyshev's bound and a covering. For each stated tolerance on the scale error, the cap radius at which the best possible conformal projection just meets it — sec²(ρ/2) − 1 = tolerance, which is Chebyshev's bound and has no fitting in it — and then the number of such caps needed to cover the sphere at the packing density a real arrangement achieves. One part in a thousand costs 1210 sheets of 403 kilometres radius. The slope is -0.989: a factor of ten in what the job will accept is a factor of ten in the atlas. What each projection optimises

How many sheets an atlas needs

A tolerance on the scale error inverts, through Chebyshev's bound, into a sheet radius — and a covering problem turns the radius into a count. One part in a thousand costs 1,210 sheets of 403 kilometres radius, the count goes as the reciprocal of the tolerance exactly, and the projection multiplies it by anything from one to fifty-six.

 How much curvature varies from place to place, body by body. The Gaussian curvature along a meridian, each body against its own smallest value, so the curves are comparable. Written from the bodies actually drawn: Earth varies by 1.0135, Mars varies by 1.0239, Vesta varies by 2.71, Phobos varies by 4.16, from the flattest place on each to the sharpest. A surface of constant curvature is a sphere and nothing else is, so the impossibility argument this field is built on has a local version on every real body: how flat a patch is depends on where the patch is. The impossibility

Curvature that varies from place to place

The impossibility this site is built on was argued on a sphere, where the curvature is one number. On the Earth it varies by 1.35 per cent, on Jupiter by 31, on Vesta by a factor of 2.7 — and on a body with three axes it varies along a parallel, which no formula in latitude can express.

 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. What is taught wrongly

Equal-area on the wrong body

The site's headline is that Web Mercator puts geodetic latitudes into a spherical conformal formula and stops being conformal. The same sentence is true with "equal-area" in it and nobody says it: the areal factor is 1.00674 at the equator, 0.99332 at 88°, and averages to almost exactly one — so every check that adds up areas passes while every cell is wrong.

 Normal gravity, derived from four constants. Gravity on the surface of the level ellipsoid, by Somigliana's closed form, for WGS84. Nothing here is measured: a, f, GM and ω go in and the whole curve comes out, rising 5186 milligal — 0.53 per cent — from equator to pole. The two open marks are the published values of equatorial and polar gravity for WGS84, which the derivation reproduces to ten significant figures rather than borrowing. What the numbers refer to

The ellipsoid is a level surface

WGS84 publishes two dozen constants and defines four of them. The other twenty are consequences — polar gravity, the potential of the ellipsoid, the coefficient that dominates the Earth's gravity field — and every one comes back here from a, f, GM and ω to the last digit published.

 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 shortest route and the quickest one, in a zonal jet. A craft making 20 km/h through the medium, from 40° north, 34° west to 52° north, 6° east. The great circle is 3313 km and takes 111.6 hours in this flow; the quickest track is 169 km longer — 5.1 per cent further — and takes 99.1 hours, saving 11.2 per cent of the time. The strokes are the flow at its own scale, and the whole difference is that the track bends into the helping part of it. Drawn in Lambert conformal conic. Paths and directions

The quickest route is not the shortest

Every route on this site so far is in a medium that does nothing, so length and time are the same question divided by a constant. Once the water moves, they are different questions with different answers: the quickest track sails five per cent further and arrives eleven per cent sooner, and the journey back takes four times as long as the journey out.

 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.

 Where two equal geodesics meet, from 15° north. A geodesic leaving at azimuth α and its mirror image at −α have the same length wherever they meet, and by symmetry they meet on the antipodal meridian. On a sphere they all meet at one point, the antipode, to 1.4e-7° — the flat line. On the ellipsoid the meeting latitude moves with the azimuth, from -15.563° to -15.009°, so the set of points with two shortest routes is an arc 61 km long, and the distance to it varies by 31 km along its own length. Paths and directions

Where the shortest route stops being the only one

On a sphere there is exactly one point with no shortest route from a given place: the antipode. On the ellipsoid the Earth actually is, that point is an arc — sixty-six kilometres of the antipodal meridian for a point on the equator, half a kilometre for one at 85°, and every point of it reachable by two different geodesics of exactly equal length.

 The area of one 20° × 10° cell at 50–60° north, seven ways. The cell has an exact area — R²Δλ(sin φ₂ − sin φ₁), 1,416,580 square kilometres — so every other row is a measurement of the method rather than of the ground. The equal-area projection returns it to 1.000000 and the spherical polygon formula to 1.000000, which is three routes agreeing — and the same cell integrated on the ELLIPSOID comes out 0.45 per cent away from all three, because the sphere is a model. Taking the shoelace in Mercator gives 3.06 times too much, and treating degrees as a length gives 1.75 times — about sec φ at the cell's middle, which is where that error comes from. What a machine does with it

Computing an area needs a surface

A shoelace over a ring of coordinates returns a number whatever the coordinates are. For one twenty-by-ten-degree cell it returns 1.75 times the true area in degrees, 3.06 in a conformal plane, and exactly the closed form in an equal-area one — and the closed form itself is 0.45 per cent out, because the sphere is a model too.

 A closed traverse cannot see a scale error. The same closed figure with two different errors in it. On the left every leg is 400 parts per million too long, which is roughly what forgetting the grid scale factor costs — and the figure closes to 2.3e-13 m, which is the last bit of a double rather than a measurement. Scaling every leg of a closed figure by the same factor produces a similar figure and a similar figure is still closed, so the check every specification leans on is blind to it. Every dimension in that figure is wrong: the perimeter is out by 1.17 m. On the right one angle is 20 seconds out — a far smaller disturbance in its own units — and the figure fails to close by 0.062 m. Closure tests the shape and says nothing about the size. Grids, and what a survey does

A traverse must close

The check every survey specification leans on cannot see the error this whole field is about. Scale every leg of a closed figure by four hundred parts per million and it closes to the last bit of a double, while every dimension in it is wrong.

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