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

Every essay — page 3

Essays 49 to 72 of 339, in the same order.
A feature across nine tiles at zoom 4, and the nine labels it gets. A rectangle from -10° to 30° east and 36° to 62° north, drawn in the projected plane with the tile boundaries over it. A renderer that draws each tile from the geometry inside that tile puts the label at the centroid of the piece, marked hollow; the whole feature's centroid is the filled mark. The furthest piece's label is 1863 kilometres from it. The pieces' areas sum to the whole to 2.2e-16 of a relative part, so the clipping is exact and the displacement is the operation rather than an error in it. What a machine does with it

A tile is drawn without its neighbours

Independence is what makes the scheme scale: one request touches one square of the world. It also means a feature crossing nine tiles is nine features, each labelled at its own centroid — up to 1,863 kilometres from the whole one — and the displacement is bounded by the size of the feature rather than the size of a tile, so it grows as the reader zooms out.

6 figures · Screen The same six projections, ranked over Europe and Chile. Each column orders the projections by Kavrayskiy's regional criterion — the root-mean-square departure of the two principal scales from unity, integrated over the region with the area element. The lines cross, which is the point: Lambert conformal conic leads over Europe and comes fifth over Chile. A table of projections ordered by distortion is a table about somebody's region. Measuring distortion

Distortion over a region

An indicatrix describes a point and every practical question is about a country. Going from one to the other means integrating, integrating means choosing a weighting, and the weighting is the step that turns a measurement into somebody's opinion.

7 figures · Tissot Where the inverse problem stops converging, 3° around the antipode of London. Every target in a 6° square centred on the point diametrically opposite London, shaded by how many iterations Vincenty's inverse formula needed to find the geodesic to it. 3 of 625 — 0% — never converged at all, and the worst success took 113 steps against a handful anywhere else on Earth. The failure is not a defect in the formula: near the antipode the shortest path is nearly ambiguous, and an iteration looking for one answer is being asked which of many. Paths and directions

The route with no shortest path

Between a point and the point diametrically opposite there are infinitely many shortest routes and no shortest route, and the standard formula for the distance between two places stops converging in a neighbourhood of it. The failure is a property of the question rather than a defect in the answer.

6 figures · Paths 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. What is taught wrongly

Datum shifts dwarf projection errors

The projection argument is conducted in parts per million and the datum question is answered in hundreds of metres. A coordinate whose datum is unstated is out by more than any projection choice could ever put it, and almost nobody checks.

8 figures · Ellipsoid What a grid of one's own is worth. For every candidate central meridian, the best worst-case distortion achievable over the region once the scale factor has also been optimised — so each point on the curve is already the bottom of its own V. The minimum is at 2.9°W with k₀ = 0.9993009, giving 700 ppm. The dashed line is UTM zone 30, whose meridian and scale factor were chosen for no region in particular, at 1059 ppm. The national grid is better by a factor of 1.51 — which is the whole of the answer to why a country publishes a grid rather than using the zones, and it is a smaller factor than the argument is usually made to sound. Grids, and what a survey does

Designing a grid for one region

A country with a grid of its own beats the international zone that would otherwise carry it by a factor of 1.51. That is worth having and it is much less than the argument is usually made to sound, and the difference between the two numbers is the whole case for and against national grids.

7 figures · Grid How large a patch can be treated as flat, at 10 parts per million. The smallest scale distortion any map of a circular patch can have, against the radius of the patch, on logarithmic axes. The line is straight with a slope of 2.00: the error grows as the SQUARE of the size, so a patch ten times wider is a hundred times worse. A tolerance of 10 ppm is reached at a radius of 40.3 km — 81 km across — and that is the number behind the boundary between plane surveying and geodesy. The impossibility

How small is flat enough

A builder works in plane coordinates and a national mapping agency does not, and the line between them is not a convention. The unavoidable error of treating a patch of the Earth as flat grows as the square of its size, and the size at any stated tolerance is a number.

4 figures · Curvature Two square worlds: Mercator's areas against an equal-area scheme's angles. Both projections give a world exactly as wide as it is tall, so either could carry a quadtree of square tiles. Mercator keeps every angle and inflates area by sec²φ — 132-fold at 85°. The cylindrical equal-area scheme whose world is square has standard parallels at ±55.65° — the solution of π cos²φ₀ = 1 — and keeps every area exactly, at a cost of nothing there and 145° of angular deformation at the edges. At the standard parallel itself Mercator's areal factor is 3.142, which is π, because the square-world condition and sec²φ are the same equation. What a machine does with it

The pyramid did not have to be Mercator

The usual defence is that a quadtree needs a square world and Mercator supplies one. So does the cylindrical equal-area with standard parallels at ±55.654° — the solution of π cos²φ₀ = 1 — and it needs no polar cut at all. What Mercator actually buys is conformality, and the price of giving it up is 13.8° of shear at 60° north.

7 figures · Screen Both zones are right, and one of each is not. A 20 km baseline straddling the boundary between UTM zones 31 and 32 at 52°N, computed from grid coordinates three ways, on logarithmic bars. Taking both ends in zone 31 and taking both in zone 32 give answers 0 nanometres apart — the resolution a double has left after carrying a six-figure easting rather than a disagreement — so a job may use either and the overlap belt every zone publishes exists to let it. Taking each end from the zone it nominally belongs to gives 392 km, because the two eastings are measured from meridians six degrees apart and subtracting them measures nothing at all. The same ground point is 706 km east in one zone and 294 km east in the other, and both coordinates are correct. Grids, and what a survey does

Where two zones meet

A twenty-kilometre baseline across a zone boundary is computed correctly in either zone — the two answers differ by nanometres — and taking one end from each returns 392 kilometres. Both zones are right and the seam between them is not a small error but a category one.

7 figures · Grid How Equirectangular distorts, by latitude. Angular deformation and areal scale plotted against latitude along the meridian at 0°. On Equirectangular the angular deformation reaches 114.2° and the areal factor reaches 11.5. What is taught wrongly

The plate carrée, the projection nobody chooses

Plotting latitude against longitude on ordinary axes is a projection. It preserves nothing, its angular deformation reaches 108° and its areal error eightfold, and it is probably the most widely produced map in the world because it is what happens when nobody decides anything.

4 figures · Audit 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.

7 figures · Tissot 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.

8 figures · Paths 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.

7 figures · Curvature 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.

5 figures · Choosing 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.

8 figures · 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.

5 figures · Families 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.

7 figures · Paths What a pole line buys, and what it costs. four pseudocylindrical projections placed by the two numbers the decision moves. Across the bottom, the angular deformation averaged over the whole sphere, where the pole-line projections — Eckert IV and Robinson — are the better maps. Up the side, on a log axis, the factor by which the last parallel is stretched, where they are worse by more than tenfold: 49× at best against 4.6× at worst for a pole drawn as a point. A pole line represents one point of the globe by a line of the map, and that is the price. The families

Where a pseudocylindrical puts its error

Every projection in this family has to decide what to do at the pole, which on the globe is a point where every meridian meets. Drawing it as a point and drawing it as a line are the two answers, and the trade between them is measurable in both directions.

5 figures · Families A grid scaled to the ground, and where it stops being one. A surface coordinate system on the British National Grid: the grid multiplied by 1.0004044, the reciprocal of the combined factor at a project origin 120 m above the ellipsoid, so that a grid distance equals a ground distance there. It is exact at the origin by construction and wrong everywhere else, and the two directions are not comparable — the axis is logarithmic across five decades. 60 km due east the set-out distance is out by 5559 mm, and the same distance due north by 20.5 mm. The scale factor of a transverse Mercator depends on the easting and almost not at all on the northing, so a system sold as good "within a few kilometres" has a useful area that is a strip rather than a circle. Grids, and what a survey does

A grid scaled to the ground is not a map

Construction sites work in coordinates where a tape reading equals a computed distance, which is achieved by multiplying the national grid by a constant. The result is exact at one point, degrades east–west forty times faster than north–south, and is not a map projection at all.

9 figures · Grid 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.

7 figures · Curvature 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.

4 figures · Dataset 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.

7 figures · Choosing 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.

6 figures · Audit 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.

8 figures · Height 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.

7 figures · Tissot