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

Every essay — page 2

Essays 25 to 48 of 339, in the same order.
One 20° × 10° cell at 60°, under four projections. The same patch of the sphere, drawn by four projections and scaled to fit. The shapes differ and so do the areas: the figure under each panel is the areal inflation relative to the same cell at the equator, so 1.00 means the projection treats the two fairly. What is taught wrongly

The projection that shows true size

There is no such thing, and the phrase hides a real question. Equal-area projections preserve area and destroy shape; nothing preserves size in the sense the phrase implies; and the tools that make the point best are not maps at all.

5 figures · Audit London to Tokyo, seen four ways. The same two routes on four projections. The gnomonic projection renders every great circle as an exactly straight line, which is what it is for; Mercator renders every rhumb line straight instead. Neither path changed — only the map did. Paths and directions

The gnomonic companion

One projection turns every great circle into a straight line, and it is the only one that does. It shows less than half the sphere, distorts enormously, and was indispensable for three centuries because of that single exact property.

5 figures · Paths Angular deformation against latitude, four projections. The same quantity for mercator, gallPeters, mollweide, winkelTripel, from the equator to 80°. A conformal projection sits flat on zero in the first of these plots and runs away in the second; an equal-area projection does the reverse. Nothing is flat in both. Measuring distortion

The two ways a map is wrong

Angles and areas fail independently. A projection can be perfect about one and catastrophic about the other, and a single number summarising "how distorted" a map is has already thrown away the distinction that matters.

7 figures · Tissot The scale factor was chosen, and the choice is a V. The worst departure of the grid scale factor from unity across a region 9° wide and 9° tall, for every scale factor between 0.998108 and 1.0002, on a central meridian of 2°W. A tangent projection touches at one meridian and is too big everywhere else, at 1987 ppm. Scaling the whole grid down slides that interval until it straddles unity, and the minimum sits at 1/√k_max = 0.999008102, where the worst departure is 993 ppm — a factor of 2.00, which is the most this construction can buy and is reached exactly. The published value marked beside it was chosen the same way, for this region, in the 1930s. Grids, and what a survey does

The scale factor was chosen

A grid's scale factor is not a constant of nature. Britain's 0.9996012717 is the reciprocal square root of the worst distortion a tangent projection would have had over the country, it halves the worst-case error exactly, and the halving is the most that construction can ever buy.

4 figures · Grid 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.

7 figures · Ellipsoid 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.

7 figures · Screen Transverse Mercator. The graticule of the Transverse Mercator projection at 30° of longitude and 15° of latitude. the only projection on which a constant compass bearing is a straight line. It is conformal. The families

Transverse Mercator and the series that computes it

The projection most of the world's survey data lives in has no closed form. What every national grid actually computes is a truncated power series in the flattening, and how far it can be trusted is an engineering parameter rather than a property of the projection.

8 figures · Ellipsoid What 4 parts per million of network strain leaves behind. Each arrow is where the best-fitting seven-parameter transformation leaves a marker, over 36 markers laid out as a grid across OSGB36's ground. The RMS residual is 1.64 metres and the worst is 3.00 metres. Seven parameters span the constant and linear parts of a displacement field; this one is quadratic, so no choice of the seven can reach it. The arrows are drawn 27× life size. What the numbers refer to

Where a fit leaves residuals

Seven parameters can carry a rigid motion and a size exactly. A triangulation network is neither, so the best possible transformation between two datums leaves metres on the table — in a pattern, not as noise — and which seven parameters come out depends on where the markers were.

9 figures · Datum London to Tokyo on Orthographic. 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 2.9e-1 of its own length. Paths and directions

The great-circle vertex

One quantity is constant along a shortest path on a sphere, and it fixes the highest latitude that path will reach before the journey starts. That number is why polar routes exist, and it can be read off the departure bearing without tracing the route at all.

8 figures · Paths Every projection in the library, measured against both properties. Maximum angular deformation across the bottom, maximum areal error up the side, both on logarithmic scales and both measured over several hundred sample points rather than taken from the projection's description. Conformal projections lie on the left edge, equal-area ones along the bottom, and the corner where both vanish is empty because a projection there would be an isometry of the sphere onto the plane. All 26 are plotted; the six named here are Mercator, Web Mercator, Gall–Peters, Mollweide, Equirectangular, Stereographic. Measuring distortion

Measuring instead of naming

A projection is called conformal because that is its name. Running the definition as a computation over several hundred points takes about twenty lines, catches a projection the whole internet uses, and is almost never done.

5 figures · Tissot The least distortion possible over a 30° region. Scale factor along a radius of the cap, each projection normalised to unit scale at the centre. Chebyshev's criterion names the projection whose scale is constant on the boundary as the conformal map of least scale variation, and for a cap that is the stereographic projection centred on it — reaching exactly sec²(ρ/2) = 1.0718 at the rim, marked. Every other conformal projection drawn here rises past that line before it gets there. This is the only figure on the site showing an optimum rather than a comparison. The impossibility

Total curvature and the scale rule

The impossibility has a size. A region covering a fraction of the sphere carries a fixed amount of curvature that any flat map must absorb, and for a circular region the least distortion any conformal projection can achieve is a closed form nobody can beat.

4 figures · Curvature The least distortion possible over a 30° region. Scale factor along a radius of the cap, each projection normalised to unit scale at the centre. Chebyshev's criterion names the projection whose scale is constant on the boundary as the conformal map of least scale variation, and for a cap that is the stereographic projection centred on it — reaching exactly sec²(ρ/2) = 1.0718 at the rim, marked. Every other conformal projection drawn here rises past that line before it gets there. This is the only figure on the site showing an optimum rather than a comparison. What each projection optimises

Chebyshev's criterion

The only optimality theorem in the subject. Among all conformal projections of a region, the one with least scale variation is the one whose scale factor is constant on the boundary — a criterion with a proof, a unique answer, and a test the site can run.

5 figures · Choosing 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.

8 figures · Ellipsoid One pixel at zoom 11: 76.4 m projected, 47.6 m at 51.5°. The grid squares are pixels, at their own size. The open mark is the stored coordinate and the filled one is where it is drawn: rounding moves it 20.2 metres, against a worst case of 33.6 — half a pixel's diagonal, which is the whole of the bound. The second point sits 43 metres away, 0.58 of a pixel east and 0.70 north, so whether the two are drawn as one dot or two is decided by where the tile grid happens to fall: they merge for 12 per cent of the possible offsets, against the 12 per cent the two fractions predict. What a machine does with it

The pixel is a place with a size

Drawing a coordinate rounds it to a pixel, which moves it by up to half a diagonal — 16.8 metres at zoom 12. Whether two points 43 metres apart appear as two dots is not a property of the data at all: they merge for 12 per cent of the positions the tile grid could take, and the closed form predicts 12.4.

6 figures · Screen The angles of a triangle, and the curvature inside it. A triangle on the sphere with sides that are great-circle arcs. Its three angles sum to 212.26°, overshooting the flat 180° by 32.26°. Integrating the curvature over the interior gives 0.56306 against an excess of 0.56306 — the same number by two routes that share no arithmetic. That is Gauss–Bonnet, and it is how a flatlander measures the curvature of a world it cannot step outside. The impossibility

Measuring curvature from inside

A flatlander with a ruler can find out that its world is round, by drawing a triangle and adding the angles. Gauss–Bonnet turns that into an exact statement, and the total curvature of a closed surface turns out to be a number obtained by counting.

8 figures · Curvature Molodensky against the exact route, NAD27. The distance on the ground between where each shortcut puts the transformed point and where the exact Cartesian route puts it, at 0 metres of ellipsoidal height, on a logarithmic scale. The full formulae stay within 0.8 centimetres across every latitude drawn; the abridged form, which replaces two ellipsoid-difference coefficients with one combined term, is worst at 40 centimetres — a factor of 48. The abridged curve dips near 45°, where the combined coefficient happens to equal the pair it replaces. What the numbers refer to

Molodensky's shortcut

The exact way to shift a datum needs an iteration nobody wanted to run on a 1950s machine, so a direct formula was derived instead. It lands within centimetres — and the abridged version everybody quotes is a different formula, worse by a factor of fifty.

7 figures · Datum The least distortion possible over a 20° region. Scale factor along a radius of the cap, each projection normalised to unit scale at the centre. Chebyshev's criterion names the projection whose scale is constant on the boundary as the conformal map of least scale variation, and for a cap that is the stereographic projection centred on it — reaching exactly sec²(ρ/2) = 1.0311 at the rim, marked. Every other conformal projection drawn here rises past that line before it gets there. This is the only figure on the site showing an optimum rather than a comparison. Measuring distortion

Scale distortion is the third failure

A map's stated scale is its scale at one place. Everywhere else it differs, by a factor that no conformal projection can avoid and that neither of the two usual distortion measures reports. It is the failure everybody uses and nobody counts.

5 figures · Tissot Two feet, and the one that is not a rounding. The US survey foot is 1200/3937 m and the international foot is exactly 0.3048 m, so the first is longer by 2.0000 parts per million. Read a coordinate in the wrong one and the error is proportional to the coordinate, not to any distance in the job: at 2.5 million feet from the false origin it reaches 1.52 m, and it crosses a survey's own closing tolerance of twenty millimetres at 33 thousand feet. Across a 300-foot site the same mistake changes every dimension by 0.18 mm — below every tolerance there is, which is why a plan in the wrong foot passes every check and is in the wrong place. Grids, and what a survey does

The units are part of the coordinate

Two feet were in use in the United States until 2022, differing by exactly two parts per million. A plan drawn in the wrong one has every dimension right to a fifth of a millimetre and sits three-quarters of a metre from where it belongs, which is why it passes every check anybody runs.

8 figures · Grid What treating the Earth as a sphere costs, per journey. The ellipsoidal geodesic minus the spherical great circle, in kilometres, for five journeys. The correction is a few tenths of a per cent and it changes sign: a route running east–west at mid latitude is longer on the ellipsoid, and one running along a meridian is shorter, because an oblate body is fatter round the equator and flatter pole to pole. Both distances are computed — Vincenty's iteration against the haversine formula — and the ellipsoidal one is checked against a geodesic obtained by integrating its own differential equation. Paths and directions

Geodesics on the ellipsoid, and why they are hard

The shortest path on a flattened Earth is not a plane curve, has no closed form, and can be longer or shorter than the spherical answer depending on which way it runs. Every practical method is a series or an iteration, and the correction changes sign.

7 figures · Paths Sinusoidal, cut into 6 lobes. six lobes, cut through the oceans so each continent stays whole. Each lobe is the same projection about its own central meridian, so each point is near a line where the shape distortion vanishes: the mean angular deformation over the mapped world falls from 38.6° uninterrupted to 17.8° here. What is given up is that the map is no longer one surface — the graticule stops at every cut because there is nothing on the other side of it. What each projection optimises

Giving up continuity

Cutting a map into lobes really does reduce the distortion, by a factor that can be measured. What is paid is that the map stops being one surface — and the size of the tear is the number that pictures of interrupted maps never carry.

8 figures · Choosing 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. What is taught wrongly

Geodetic against geocentric latitude

Two angles, both called latitude, differing by eleven and a half arcminutes at their worst. One is what every coordinate means and the other is what every spherical formula assumes, and the gap between them is twenty-one kilometres on the ground.

7 figures · Ellipsoid What grid spacing a tolerance buys. The worst bilinear interpolation error across OSGB36's ground, against the spacing of the table it is interpolated from, both axes logarithmic. The smooth shift falls by exactly four for every halving, which is what second-order interpolation does. Adding a 30 centimetre ripple 2 degrees across — three parts in a thousand of the 99 metre shift it rides on — moves the spacing needed for 20 millimetres from 1° to 0.125°, which is 64 times as many nodes. What the numbers refer to

When a formula is not enough

National mapping agencies distribute datum shifts as tables of numbers on a grid rather than as parameters. What decides the spacing that table needs is not the size of the shift — a hundred-metre shift tabulates coarsely, and a thirty-centimetre ripple riding on it does not.

8 figures · Datum Mercator swung through every tilt, over Chile and the tropics. The regional distortion of one projection as its axis is tilted from the normal aspect at 0° to the transverse at 90°, each curve divided by its own value in the normal aspect so the two regions can share an axis. Chile is best at a tilt of 90°, a factor of 183.5 better than north-up; the tropics is best at a tilt of 0°, which is north-up. Rotating the sphere costs nothing and changes none of the projection's own properties, which makes this the cheapest improvement available and the one most often left unmade. What each projection optimises

Fitting the aspect to the region

Choosing a projection is a choice among a few dozen named things. Choosing its aspect is a choice among a continuum, it costs nothing, it changes none of the projection's own properties, and for a long thin country it is worth a factor of 183.

6 figures · Choosing American polyconic. The graticule of the American polyconic projection at 30° of longitude and 15° of latitude. a different tangent cone for every parallel — true to scale along all of them, and along the central meridian. It is neither conformal nor equal-area. The families

The projections that gave up being one thing

The polyconic is built from a different cone for every parallel, which means it is built from no cone at all. It preserves nothing the usual tests look for, it has an exact property neither of them measures, and its sheets do not fit together — a defect discovered in the field rather than at the drawing board.

4 figures · Families