Cartographic Projection
every map is wrong, and by exactly how much
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The thread: The trade-off is forced — page 2

Conformal means the principal scales are equal; equal-area means their product is one. Both at once forces them both to one, which is an isometry, which cannot exist. The trade-off is two lines of algebra. Essays 25 to 48 of 97.
A direction carried once round the sphere. A vector transported round a closed loop on the sphere, kept as parallel to itself as the surface allows at every step — the component that leaves the tangent plane is removed and nothing else is done to it. The heavy arrows are its direction at the start and at the finish, drawn from the same point; the light ones are its direction along the way. It comes back turned through 282.0 degrees, which is 0.783 of a revolution, and nothing in the transport turned it. Drawn in an orthographic projection of the embedding, which is itself a map and has its own distortion. The impossibility

A direction carried round a loop

Carry a bearing round a circuit, keeping it as parallel to itself as the surface allows, and it comes back turned. Round a parallel at 45° the turn is 4.443 radians and the cap enclosed is 1.840, and they sum to exactly one revolution — so the turning is not the curvature, and on a cone it is all of one and none of the other.

 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.

 "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.

 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.

 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. What is taught wrongly

The normal section is not the geodesic

A theodolite at A sighted on B swings in one plane and the instrument at B sighted back swings in another, so the two observations trace different curves on the ground — 79 metres apart over 3,026 kilometres — and the shortest path is neither of them. The lengths differ by 2.4 millimetres, so the wrong curve measures the right distance along the wrong ground.

 The azimuthal family, as one function of one variable. Every azimuthal projection is a rule for how far from the centre to draw a point at angular distance ρ, and nothing else. five named projections are plotted as their radial functions — tan ρ, 2 tan(ρ/2), ρ, 2 sin(ρ/2), sin ρ — and the two dashed curves are the solutions of the two conditions, integrated from the centre with no more than "true scale at the centre" to start from. They land on the stereographic and the Lambert azimuthal to 8e-10. The names are the answers to the equations, not descriptions of viewpoints. The families

The azimuthal family is one function

Five azimuthal projections are taught as five viewpoints — from the centre, from the far pole, from infinity. They are one projection with five choices of how far out to draw a point at angular distance ρ, and each named property is a differential equation in that one function: f′ = f/sin ρ integrates to the stereographic projection and f f′ = sin ρ to Lambert's, from nothing but true scale at the centre.

 One length, three corrugations. A straight segment shortened by a factor of 0.9, then wiggled across its own direction until its length is back to what it was. The wiggle's amplitude is what buys the length and the number of wiggles is free, so all three curves have exactly the target length while the third stays 16 times closer to the shortened segment than the first. That is the mechanism: the family converges to a map that is not isometric, while every member of it is. The impossibility

Impossible in two derivatives, possible in one

The impossibility this whole collection rests on computes a second derivative, so it is a statement about maps that have two. Take one away and it is false: a corrugation restores an exact length while converging to the map that does not, and iterating it gives a flattening whose derivative converges and whose curvature runs to half a million.

 Six projections, at 40° north. Flexion (solid) and skewness (light) against direction of travel, drawn about a zero circle at one point of each projection. The three conformal ones have curves of exactly equal size, offset by exactly 90°, because on a conformal map both quantities are components of one vector — the gradient of the log of the scale. The others have neither property: ratios run from 0.72 to 2.72. Measuring distortion

Bending and stretching are one failure

The ladder was written expecting flexion and skewness to be two independent ways for a map to be wrong. On a conformal projection they are not independent at all: the two extremes are the same size to four figures and sit exactly ninety degrees apart, because both are components of a single vector.

 A 20° shape across the antimeridian, in the space where the numbers live. Longitude runs across the page from −180° to 180°, which is where the failure is: the shape is one rectangle on the ground and two pieces in the numbering, and every operation that treats longitude as a real number sees the two. The bounding box comes out 359° wide instead of 20°, the planar area comes out 17 times too large because the shoelace encloses the complement, and the midpoint of a segment from one edge to the other lands 20015 kilometres away — the antipode of where it belongs. The true area, from the closed form, is 4,920,667 square kilometres. What a machine does with it

The antimeridian is a cut in the numbers

A twenty-degree box across 180° has a bounding box of 359.4°, a planar area seventeen times too large, and a midpoint 20,015 kilometres from where it belongs — which is the antipode, exactly. Moving the cut moves the failure and never removes it, because a circle cannot be numbered by an interval.

 The two reductions, at a grid factor of 1.0004. A measurement made on the ground has to be brought to the ellipsoid and then to the grid, and the two corrections have opposite signs. The elevation factor is R/(R+h) and always shrinks; the grid factor here is 1.0004, which stretches. Their product is the only number a surveyor can use. They cancel exactly at 2551 metres. At 1500 metres the combined factor is 165 parts per million, which is 1.65 metres on a 10 kilometre baseline. What the numbers refer to

The ground is not the grid

A tape measure on a hillside has to be brought down to the ellipsoid and then out onto the map, and the two corrections have opposite signs. On a grid whose scale factor exceeds one there is exactly one elevation where they cancel — 2,551 metres, for a factor of 1.0004.

 How large a map has to be before a wrong projection stops fitting it. Three rivals fitted to a Mercator graticule centred at 45° north, over regions from 1° to 70° of half-extent. The residual is the shape difference alone, with the best scale, rotation and offset removed, and both axes are logarithmic. The horizontal rule is a fifth of a per cent of the map's width — about the width of a drawn line on a printed sheet — and where a curve is below it, no measurement of that map can tell the two projections apart, however carefully it is made. What is taught wrongly

Two projections that cannot be told apart

Over a small enough region every projection is the same picture, so the question is how small. The answer is not one number: two conformal projections need twelve degrees of extent before their graticules can be separated, and two projections with different anisotropy separate below one.

 One face of an icosahedron, two ways. The same spherical face mapped onto the same flat triangle by two different rules, with a grid of marks whose size is the local areal factor. The gnomonic map draws every great circle straight and stretches the corners by a factor of 1.50; the area-preserving map holds the areal factor at one to a part in a million and pays in shape, reaching 11.9° of angular deformation against the gnomonic's 8.0°. Both take the face's boundary to the face's boundary, which is what lets the pieces still fit together. The families

What a face can preserve

The obvious map onto a polyhedron's face is the gnomonic, and it draws every great circle straight while stretching the corners by a factor of 1.50. Replace it with a construction that holds the areal factor at one to a part in a million and the shape error rises from 8.0° to 11.9° — the same trade the whole sphere forces, arriving on a piece of it a twentieth the size.

 The half-extent at which each rival stops fitting a Gall–Peters graticule. For each candidate, the size of region at which its best fit to a Gall–Peters map first leaves a residual of a fifth of a per cent of the map's width. Below that size the two are the same picture. The numbers are half-extents in degrees of latitude, at 20° north, with the plane affine transformation removed; a bar at 90° is a rival that never separates at all within the range searched. What is taught wrongly

What a careless copy hides

A photocopier that stretches one axis is a nuisance to remove before a projection can be identified. Removing it costs more than it looks: an affine fit absorbs the entire difference between the cylindrical equal-area projections, so Gall-Peters and Behrmann become the same picture at any size, to sixteen decimal places.

 Which corrections a 10 mm job may leave out. Each correction inverted: the line length — or, for the last row, the patch radius — at which that correction alone reaches 10 mm, for a job 150 m above the ellipsoid on 2° of ground, 0.0° from the British National Grid's central meridian. The tightest is the slope reduction at 16 m. Three of the four rows are linear in the tolerance, so this ranking is the same at a millimetre and at a decimetre; what does change it is the site — move the job up a mountain or onto steeper ground and the order is different, which is why a specification's list is not transferable. Grids, and what a survey does

The tolerance decides the model

Every correction inverts to a distance — the line length at which it alone exceeds a stated tolerance. The ranking those distances produce is the same at a millimetre and at a decimetre, and it is different for two jobs at the same tolerance on different ground.

 A raster warped to Lambert azimuthal equal-area and back, nearest against bilinear. The left panel is the field the raster carries — a smooth analytic function, so that the error of an interpolation is the interpolation's error and not a photograph's history. The other panels are what is left after warping into Lambert azimuthal equal-area and back to Equirectangular, shown as the difference from the original at six times the contrast. Nothing moved: the coordinates go through the maps exactly. What is lost is that a target pixel's centre does not fall on a source pixel's centre, so a value has to be invented for it. Bilinear is closer to the field — RMS 0.0022 against 0.0212 — and has given up 0.63 per cent of its variance to get there. The panels are drawn at 48 by 32 cells; the measurement is made at the same resolution. What a machine does with it

Reprojecting a raster invents values

Moving a picture from one projection to another moves no coordinate — the maps are exact both ways. What is lost is that a target cell's centre does not land on a source cell's centre, so a value has to be made up for it, and the making-up has an order of convergence: 1.00 for nearest, 1.98 for bilinear, 2.93 for a cubic, measured by refining the grid.

 The conformal map onto a square face of a cube. The spherical face of a cube carried onto its flat face by a map that is conformal everywhere — the measured angular deformation over the drawn interior is 1.63e-6°, which is the arithmetic's own floor. The rings and spokes are circles and radii on the sphere, and they cross at right angles here because that is what conformal means. The map was solved for as 16 terms of a series rather than written down: the face's edge comes out straight to 0.33 per cent of its own half-width, and that residual — not the conformality — is what more terms buy. At each corner the map behaves like ζ^0.75, so the scale factor there is infinite. The families

A conformal map onto a face

The polyhedral ladder ended owing a conformal face map, on the grounds that it needs elliptic functions. It does not: a conformal map of the sphere is an analytic function of one conformal coordinate, so the map is a power series, choosing it is a least-squares fit — and its scale factor is infinite at the corners, which is the angle deficit arriving as a singularity.

 The map that is continuous, and the pair it pays with. The orthographic is defined and continuous at every place on the Earth — it is written in the components of the place itself, with no longitude in it to jump. What it gives up is being one to one, and it gives it up almost everywhere: 47 per cent of the sphere shares its page point with the place directly behind it. Borsuk–Ulam guarantees at least one ANTIPODAL pair among those, and here it is exactly one — the centre and the place on the far side of the world, both at the middle of the picture, found to a residual of 1.5e-14. The impossibility

Two opposite places on the same spot

A projection may be continuous everywhere or one to one everywhere, and the first two rungs price both. What neither says is that the choice is not symmetric: a map that keeps continuity does not lose injectivity somewhere arbitrary. It loses it, always and at minimum, on a pair of places directly opposite each other on the Earth.

 Past the five solids: what more faces buy, and what they cost. The icosahedron subdivided 2, 3, 4, 6, 8 ways, giving 20, 80, 180, 320, 720, 1280 faces. The worst angular deformation inside a face falls from 13.1° to 0.26°, with a fitted exponent of -0.947 against the face count — the reciprocal, as it must be, because a face's angular size goes as the inverse square root of the count and the gnomonic's deformation goes as the square of that. The total cut length rises from 11.6 to 96 sphere radii, fitted at 0.509. Both exponents together say the whole economics of the family in one line: halving the distortion costs √2 times the cutting, for ever. The families

More faces, less distortion, more cutting

The five regular solids are where polyhedral mapping stops, and they stop because there are only five rather than for any reason about maps. Subdivide instead, and the family runs to any number of faces with two fitted exponents: distortion falls as the reciprocal of the count and cutting rises as its square root.

 five planes a dataset might be stored in, scored on three operations. Each candidate measured over -10° to 30° east and 35° to 60° north: the worst areal error, the worst angular deformation, and the spread of the scale factor, which are what an area query, a shape and a distance respectively depend on. The best plane for areas is Gall–Peters, for shapes Lambert conformal conic, and for distances Lambert conformal conic — three different answers, and no fourth candidate would collapse them, because a projection exact in two of these columns has a = b = 1 everywhere and is the isometry Gauss's theorem forbids. area of a polygon costs 3.06× too large in the wrong plane; drawing a line between two points costs 194 km from the ground it claims. What a machine does with it

The operation decides the coordinate system

Five candidate planes over one region, scored on the three things a spatial operation depends on. The conformal conic wins shape and distance and is 11.7 per cent out on area; the equal-area member is exact on area and 38.9° out on shape. No candidate is exact in two columns, and no candidate ever will be, because one that was would be an isometry.

 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.

 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.

 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.

 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.

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