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

The thread: Measured, not named — page 5

A projection is usually called conformal because that is its name. Here it is called conformal only after the angular deformation has been computed at several hundred points and found to be zero. Essays 97 to 120 of 292.
The world after a map of degree 2. The graticule of the sphere, sent through the square of the stereographic coordinate and then drawn in Mollweide. Every parallel is still a closed curve and every meridian still runs pole to pole, and the whole world has been wrapped round the sphere 2 times: the 2 meridians that used to be 180° apart now lie on top of one another. Nothing has been torn. The degree measured by integrating the area the map sweeps is 2.0004. The impossibility

How many times, not whether

Three rungs of this ladder answer yes or no and have no other kind of answer. The degree is the first quantity here that counts: a continuous map of the sphere to itself covers it a whole number of times, injectivity forces that number to ±1, and the number is recoverable three ways — by counting preimages, by integrating swept area, and from the rate at which the preimages coalesce.

 The circle a reprojection is about to turn into an ellipse. A circle drawn on the Mollweide plane, and its image under the map that carries that plane to Mercator — normalised so each pair has the same area, because the whole map can be rescaled and the shape is what is being shown. The dashed circle is what an undistorted reprojection would leave. This is Tissot's construction with a plane in place of the sphere, and it is the right picture for a reprojection because both ends are pictures. Worst angular deformation over the sampled points: 81.2°. Drawn on the source plane, in Mollweide. Measuring distortion

A projection between two projections

Every distortion measured on this site so far compares a map with the sphere. The operation a machine actually performs compares a map with another map — and that map has its own two principal scales, its own areal factor and its own angular deformation, none of which is the difference of the two it was built from.

 A reach set with a cost that depends on direction. Everywhere reachable in the time it takes to cover 3000 kilometres in still air, under a steady westerly, with the still-air set drawn inside it. The anisotropic set runs from 1653 kilometres against the flow to 4261 with it — a ratio of 2.58 — while the still-air set is round to 1.023, which is the lattice's own floor and not a shape. Drawn in an azimuthal equidistant centred on the source, so every radius on the page is a ground distance and none of the shape is the projection's. Paths and directions

A reach set with a cost that depends on direction

Three rungs build reach sets out of a distance, which is symmetric and isotropic by construction. Nothing anybody travels is: in a flow at 45 per cent of a vehicle's own speed the same vehicle gets 4,261 kilometres one way and 1,653 the other — a ratio of 2.58 — while the ground it covers grows by ten per cent.

 Everywhere Albers is exactly right, and the band round it. The set on which both principal scale factors are one — the only ground where a ruler on this map, at the map's own stated scale, measures the true distance in every direction. It is the parallels at 20.000° and 60.000°, drawn as a curve, with the band within 0.01 of true scale shaded round it. That band is 2.673 per cent of the sphere, and it narrows as ε as the tolerance tightens. The curve at its centre has no width at all, and no tolerance makes it have one. The impossibility

The places where a map is exactly right

Nine essays on this ladder say a map cannot be right everywhere. None asks where it IS right — and the answer is a curve, a pair of curves, or two isolated places, never a patch. Measured across fourteen projections the set's neighbourhood shrinks with an exponent of 0.48, 1.0 or 2.0, and the value the impossibility forbids is 0.

 The scale asked for and the scale the pyramid has, at 0°. A tiling scheme exists only at integer zoom levels, a factor of two apart in resolution, so a request for any scale between them is answered by the nearest rung. The ratio runs from 0.707 to 1.405 — 1/√2 to √2 — and repeats identically at every doubling, which is four times that in area. A request for 1:10,000 is served at zoom 16, which is 1:8,531; A request for 1:25,000 is served at zoom 14, which is 1:34,124; A request for 1:50,000 is served at zoom 13, which is 1:68,247. Nothing anywhere reports it, because the map that arrives is a perfectly good map of something. What a machine does with it

Zoom is a ladder

A tiling scheme exists only at integer zoom levels a factor of two apart, so a request for 1:25,000 is answered with 1:34,124 — 36 per cent coarser, and 86 per cent coarser in area. The mismatch runs from 1/√2 to √2 and repeats identically at every doubling, and nothing anywhere reports it, because the map that arrives is a perfectly good map of something.

 The boundary of the region the series is the map in. The curve where the transverse coordinate reaches 2.918, which is where the terms stop shrinking. It crosses the equator 83.81° from the central meridian and closes towards the poles, because the same longitude is a smaller transverse coordinate at a higher latitude — the boundary is a curve rather than a meridian. The narrow band beside it is a 3° zone, the width national grids actually use, drawn to the same scale: the practical world sits in about a fiftieth of what the series can reach. Drawn in Mollweide. The families

Where the series stops being the map

The transverse Mercator has no closed form on an ellipsoid, so every national grid computes a truncated series. Asking how many terms it needs has an answer everywhere; asking whether more terms always help has an answer only within 83.81° of the central meridian, and the boundary is computed rather than assumed.

 The same map on four differently prepared pages. Mollweide at 30°E 20°N, then the same projection with a rotation and a magnification applied to the page, then with one axis stretched by 1.6, then with a shear of 0.5. The similarity changes nothing: flexion, skewness, ω and the anisotropy are identical to every printed figure, and only the last column — the same turning measured per unit of page arc rather than per unit of ground arc — moves, by exactly the magnification. The stretch and the shear change all of them, and flexion by 35 per cent. Measuring distortion

The second derivative is not an invariant

The first-order ladder established which quantities survive a change of coordinates and which are artefacts of the parameterisation. Asked of the second order, the answer is that flexion survives a rotation and a magnification of the page exactly, and survives nothing else: a stretch of 1.6 in one axis moves it by eleven per cent, a shear of 0.5 by thirty-five, and a shear of 3 leaves a ranking of eight world maps with a rank correlation of −0.07 to the one it started with.

 How thinly a sphere can be covered by a few equal caps. The covering density of the best arrangement of n equal caps found for each n — the total area of the caps divided by the sphere's, so a value of one would be a perfect tiling with no overlap. The horizontal line is 2π/√27 = 1.2092, the thinnest covering density of the PLANE by equal discs, which this site has used for the sphere since its first atlas essay. It is wrong in both directions: at 2 caps the sphere is covered more thinly than any plane can be, because a cap may be a hemisphere, and at every count from 3 upwards more thickly — 1.5092 at 3, and 1.3377 at 14. The ringed points are the four counts whose optimum is proved: 2 at 90.00°, 4 at 70.53°, 6 at 54.74°, 12 at 37.38°. Everything else is an upper bound from a search, drawn as one, and the bound loosens as the count rises — the search reaches the proved optimum to 3.4 per cent at these counts and has no such check anywhere else. What each projection optimises

The sphere is not the plane at small counts

The site's atlas arithmetic multiplies an ideal sheet count by 2π/√27, the thinnest covering density of the plane. At the four counts whose optimal covering of the sphere is a theorem the plane's number is 21 per cent high at two caps and 9, 5 and 2 per cent low at four, six and twelve — wrong in both directions, and the direction changes with the count.

 Two parameter sets 100 metres apart, over the region they were fitted to. Each marker is drawn at a size proportional to how far the two transformations put it apart. The second set differs from the first by 100 metres of translation along the direction this network can least see, with the rotations and the scale re-fitted to absorb it — which is what a second agency's adjustment does when it chooses a different constraint. The worst disagreement anywhere in the region is 5.64 metres and the mean is 3.61. Applied at south-eastern Australia the same two sets differ by 193 metres, because the rotation that absorbed the translation here is a rotation of the whole Earth. What the numbers refer to

Two parameter sets, one transformation

Agencies publish seven-parameter datum transformations that differ by hundreds of metres in translation, and the usual reading is that one of them is better. Over the region either was fitted to they are the same transformation: a hundred metres of translation, re-absorbed by the rotations and the scale, moves a British coordinate by 5.6 metres and an Australian one by 193.

 The same ranking, with the right answer removed from the library. A map drawn in Mercator, fitted by every candidate except Mercator. Something still wins: Conformal conic, by a factor of 1.64 over the runner-up, leaving 0.4 per cent of the map's width unexplained. With Mercator in the library the winner's margin is 1.2e+13. The ranking always produces a name; what tells the two situations apart is how far ahead the name is. What is taught wrongly

When the answer is not in the library

Fitting twenty candidates to a map and ranking the residuals always produces a winner, which makes it a ceremony unless it can also produce a refusal. Held out of its own library, a Mercator map is named as a conformal conic, leaving 0.4 per cent of the map's width unexplained — and the quantity that tells the two situations apart is not the residual but the margin, which is 1.6 when the truth is absent and 10¹³ when it is present.

 A 1-hectare parcel across British National Grid, at 52°N. The departure of a plan's dimensions from the ground's, across the width of the zone, as a length and as an area. The area curve is the length curve doubled — an areal scale factor is the square of a linear one, and squaring a small departure doubles it — so a parcel whose sides are each 399 parts per million short on the central meridian is 797 short in area. On a 1-hectare parcel that is 8.0 square metres at the central meridian and 8.0 at 0.00° out. Grids, and what a survey does

An area on the grid is not an area on the ground

A grid's scale factor is a property of lengths and what a surveyor sells is an area. Squaring a departure doubles it, so a hectare drawn on the British grid at its central meridian has 10,008 square metres of ground under it — and correcting an area with the line factor instead of its square leaves half the error behind.

 The cut makes 20 faces out of 14 regions and needs no more colours. The same partition on a sheet cut at the antimeridian. six of the 14 regions are drawn in two pieces, one against each edge, and they are shown darker. The sheet has 20 faces where the globe had 14 regions, and it needs exactly the same four colours — because the two pieces of a split region between them touch exactly what the region touched, so identifying them gives back the sphere's own graph, edge for edge. The impossibility

Four colours, and what a cut cannot do to them

Every projection removes a set, and a map cut at the antimeridian draws six of its fourteen countries in two pieces — twenty faces where the globe had fourteen regions. The obvious guess is that a map with split countries is the exclave problem and needs a fifth colour. It needs exactly four, and the reason is that the cut adds faces and adds no edges.

 The set that can be reached is not the set that can reach. Two sets under a steady westerly: everywhere reachable from the marked place in a stated time, and everywhere from which the marked place can be reached in the same time. They have the same area to a fraction of a per cent — reversing a uniform flow is a reflection — and they overlap on only 32.3 per cent of their union. 67.7 per cent of the ground in one of them is not in the other. Paths and directions

The set that can be reached is not the set that can reach

The moment a cost stops being symmetric, two questions that read alike stop having the same answer. Under a flow at 45 per cent of a vehicle's own speed the set reachable from a place and the set from which the place is reachable have the same area to five significant figures and share 32 per cent of their union — so 68 per cent of the ground in one of them is not in the other.

 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.

 Two explanations for one residual, against the size of the region. A map in the right projection whose control coordinates are on the wrong datum, and a map fitted with the wrong projection and the right datum, both measured as a residual after the reproduction's scale, rotation and offset have been removed. The datum shift's residual is flat: it is the same 1.3e-6 at every size, because a similarity fit is very nearly what a datum shift is and it absorbs the rest. The wrong projection's grows by three orders of magnitude with the region. They are the same number below about a degree, which is where a residual stops saying anything about either of them. What is taught wrongly

A residual has more than one explanation

The method names a projection by fitting every candidate to a set of control points and taking the smallest residual. It has never been asked what else a small residual could be. A map drawn in the right projection from coordinates on the wrong datum leaves a residual of one part in a million — indistinguishable from noise, at every region size, because a similarity fit absorbs a datum shift almost exactly.

 Where the middle of a 20° × 20° region is, in five planes. The region is drawn in longitude and latitude — which is itself a projection, and one of the ones being compared. Each filled mark is the shoelace centroid computed in one projected plane and inverted back to the ground; the hollow mark is the centre of area on the sphere, by integration. They spread over 273 kilometres. The equal-area member is 66 kilometres out, because a centroid is a first moment and preserving area says nothing about where the area sits. What a machine does with it

A centroid belongs to a plane

Every renderer labels a region at its centroid, and every centroid is a shoelace over coordinates as stored — which is a statement about the plane they are in. Six planes put the middle of one 20° × 20° region up to 273 kilometres apart, the equal-area member is 66 kilometres out, and the disagreement falls as the square of the region's size.

 One level surface, 1000 m up at the equator, in two height systems. A single equipotential surface — the shape a body of water takes — with the number each height system gives it, all the way from the equator to the pole. The orthometric height, which is the distance up the plumb line and therefore a length, falls by 5.28 metres along it, because level surfaces converge polewards. The dynamic height, which is the geopotential number divided by one constant gravity value, is flat to 0.000 millimetres — it is the same number everywhere on the surface, and it is not a distance from anything. What the numbers refer to

A height that is not a length

Level surfaces converge polewards, so the surface a lake sits on is 5.28 metres lower at the pole than at the equator and a height system that reports lengths says a lake runs downhill. The fix reports a number that is constant on the surface and is not a distance from anything: a hundred-metre climb raises it by 99.73 metres at the equator and 100.26 at the pole.

 What the 5th decimal place of a coordinate is worth on the ground. A latitude written to 5 decimal places steps 1.112 m north for one unit in its last digit, at every latitude, because the meridian does not care where it is measured. A longitude written to the same 5 places steps 1.112 m east on the equator and 0.097 m at 85°, because a degree of longitude is a degree of a circle whose radius is R cos φ. The same written precision means two different distances at the same point, and a different pair at every other. Measuring distortion

A coordinate is a number with a width

Every number on this site so far has been exact. A written coordinate is not: five decimal places of a degree is 1.112 metres of latitude everywhere and 1.112 metres of longitude only on the equator, falling to 0.097 at 85°. The same written precision carves the ground into a cell that is square in one place and eleven times longer than it is wide in another.

 Where the projection's pole should go, and the answer the third rotation moves it to. Every dot is a pole position the search tried, sized by the best score it can reach there when the third rotation is also free — the score is the distortion of Robinson over Europe by Kavrayskiy's criterion, so smaller is better and the large green dots are the good regions. The circled mark is the two-parameter optimum and the square is the three-parameter one: they are 74 pixels apart on this map, which is a different aspect rather than a refinement of the same one. The search costs 8 times the evaluations of the two-parameter one. Drawn in Mollweide. What each projection optimises

The third parameter, run

An aspect has three numbers and this site has been searching two of them, with a note admitting it. Searching all three is worth up to 2.1 times — and the obvious way to do it, starting from the two-parameter answer and letting the third move, finds a fraction of that or nothing at all.

 Two readings of the same triangle, and the band between them. The same three vertices, joined two ways: with straight lines in the plane the coordinates are stored in, and along the ground. Every dot is a point the two readings disagree about — inside on one and outside on the other. The band covers 29.5 per cent of the polygon, which is 4241 thousand square kilometres, and it is not an error in either reading: the file does not say which one it means. Densifying the stored boundary removes it, which is the only fix there is. Drawn in Equirectangular. What a machine does with it

Inside is a claim about the edges

Whether a point is inside a polygon is not a property of the point and the polygon. It is a property of the plane the edges were understood to be straight in, and between two readings of the same file there is a band of disagreement — 29 per cent of one triangle's area, 4.2 million square kilometres, and the file does not say which reading it means.

 Where a ridge, a valley and the pass between them curves the other way. The Gaussian curvature of a stated terrain over a 80-kilometre window, with the sign shown by the colour and the size by the ink. Positive on the summits and in the hollows, negative everywhere between — which is most of it: 79 per cent of the curved area. The extremes are 2.2e+4 times the Earth's own curvature, which is the quantity twelve rungs of this ladder have taken to be the curvature of the thing being mapped. The impossibility

Where the surface curves the other way

Twelve rungs argue the impossibility on surfaces whose curvature is positive everywhere. The surface a map of the ground actually depicts is not one of them: a stated terrain is saddle-shaped over 79 per cent of its curved area, its curvature runs to twenty-two thousand times the Earth's own, and even a single smooth hill is concave over 89 per cent of itself.

 One instrument, one covariance, drawn where Mollweide puts it. The same measurement is made at every point: 5 m east by 5 m north, uncorrelated, which on the ground is a circle. Each ellipse is that covariance pushed through the projection's own Jacobian and drawn 26,000 times life size. On Mollweide the axis ratio reaches 4.15, so an instrument that is equally good in every direction is drawn as though it were not. Measuring distortion

An error ellipse is an indicatrix

A positional covariance pushed through a projection is the same matrix sandwich that produces Tissot's indicatrix, so the error ellipse drawn on a map and the distortion ellipse drawn beside it are the same ellipse. A five-metre circular accuracy is drawn at an axis ratio of 3.04 on one common projection — and the same projection draws a genuinely lopsided 304 by 100 metre error as a perfect circle.

 An equal-area map nobody would publish. Mollweide, with a row-dependent horizontal displacement applied to the page afterwards. That plane map has Jacobian determinant 1 everywhere, so every areal scale factor is untouched: the worst departure from 1 anywhere sampled here is 6.0e-12, which is the arithmetic's own floor. It satisfies the equal-area condition exactly and completely, and it is a ruin. The angular deformation at 30°E 20°N has gone from 11.0° to 60.5°. What each projection optimises

Every equal-area map is every other one

Take Mollweide and slide every row of the page sideways by an amount that depends on the row. The result satisfies the equal-area condition to 6 × 10⁻¹², exactly as well as Mollweide does, and it is a ruin — the angular deformation at one ordinary point has gone from 11° to 60°. Equal-area is one equation, and one equation leaves a whole function free.

 The scale spread of every grid Britain could have adopted. A grid is conformal by requirement, so its angular deformation is zero everywhere and the whole design problem is the spread of its one remaining number: the largest scale factor over the region divided by the smallest, in parts per million. The grid's own scale factor does not enter — multiplying every scale by a constant leaves the ratio alone, which is why it is chosen last. The bottom bar is Chebyshev's optimum, the conformal map of this region whose scale is constant on its boundary, which no map of any family can beat; the adopted grid sits 1.98 times above it. Measured over a stated box rather than a coastline, because a coastline would put the vendor's generalisation into the answer. Grids, and what a survey does

The best grid a country could have had

A national grid is a conformal map chosen for one region, so its whole design problem is one number: the spread of its scale factor. That number has a theoretical floor, this site can now compute it, and the adopted grid turns out to be either exactly optimal or half as good again — depending entirely on which box the country is declared to be.

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