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

The thread: Every figure here is a projection — page 1

The page is flat, so there is no way to show the sphere except by projecting it. Even the globes are projections with their own distortion, and no picture here is the undistorted truth. Essays 1 to 24 of 27.
London to Tokyo on Mercator. 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 Mercator the rhumb line departs from straight by 5.0e-9 of its own length. Paths and directions

The shortest route is not straight

The shortest path between two points on a sphere is an arc of a great circle, and on almost every map it is a curve. The straight line on a Mercator chart is a different route entirely, and on some journeys it is twenty-eight per cent longer.

 The tile pyramid, four levels down to quadkey 120. The world as a square, quartered three times. Each level's tile is exactly half the width of its parent, so a tile is four tiles at the next level and never needs resampling to serve one — the property the whole scheme rests on, and one that holds only because the projected world is square. Level 3 has 64 tiles at 19567.9 metres per pixel, which at 51.5° north is 12181.3 metres of ground per pixel rather than the number the scheme publishes. What a machine does with it

A screen map is a pyramid of tiles

The scheme every slippy map runs on is a coordinate system with three integers and one projection, and almost all of it is forced. A square world is what makes the quadtree work, the levels are exact powers of two, and the published resolution — 156,543 metres per pixel at zoom zero — is a distance on the ground at exactly one latitude.

 Six surfaces and their Gaussian curvature. Curvature computed at the centre of each surface, by both available routes — from the way the surface sits in space, and from distances measured inside it alone. The two agree, which is Gauss's theorem. A surface with K = 0 can be unrolled flat without stretching: here the plane, the cylinder and the cone do, and the sphere, the torus and the pseudosphere do not. The impossibility

No map is faithful

Not "no map yet", and not "no map at page size". Gaussian curvature can be computed from inside a surface, a sphere has some and a plane has none, and that closes the question permanently.

 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.

 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.

 The icosahedron's faces, drawn on the sphere. The edges of the icosahedron projected radially onto the sphere, which is the partition of the world a polyhedral map uses: everything inside one spherical polygon is drawn on one flat face. Each face spans 25.8° from its centre to its own boundary, and the gnomonic map onto it reaches 6.6° of angular deformation at the corners. Drawn in an orthographic projection of the embedding, which is a map with its own distortion. The families

The globe on a solid

Cylinder, cone and plane are not the only surfaces a sphere can be laid on. Project it onto a polyhedron and the curvature goes entirely to the corners — π at each of the tetrahedron's four, π/5 at each of the dodecahedron's twenty, and always 4π in total, which is exactly the curvature of the sphere it replaced.

 A straight line stored in Web Mercator, and where it really goes. Two points 5570 kilometres apart, joined by a straight segment in the plane the file's coordinates are in. Drawn on that plane it is a straight line and looks like the route; on the ground it is the curve marked measured, and the geodesic between the same two endpoints is the other one. The profile below is the distance between them along the line, reaching 718 kilometres at 49 per cent of the way across. Nothing here is an error in the data: both endpoints are exact, and the whole of the discrepancy is the word straight. What a machine does with it

A straight segment is a claim about a plane

Two exact endpoints, joined by a straight line in the plane the file is stored in. On the ground the line is 718 kilometres from the route it claims between New York and London, and 2,961 between London and Tokyo. The departure grows as the square of the length — fitted exponent 2.001 — so a stated tolerance costs vertices as a square root.

 How each projection escapes being one to one. Eighteen projections, and the theorem allows no fourth column. A map of the whole sphere either leaves ground undrawn, or draws one place as a curve, or is cut so that one ground curve appears twice — and 11 of the eighteen do more than one of those. The three columns are a solid angle, a count of places and a page length, which is why they are three columns rather than one score: there is no rate at which a hemisphere converts into a pole. The impossibility

No map of the whole sphere is one to one

Eleven essays establish that no map preserves distance, and every step of that argument needs a distance. There is a second impossibility underneath it that needs nothing at all: a sphere is compact and has no boundary, so a continuous map of it into the page cannot also be one to one. Eighteen library projections, eighteen escapes, and not one of them free.

 Everywhere within 3,000 km of London, drawn in Lambert cylindrical. The set of places exactly 3,000 kilometres from London by the shortest route, projected point by point. On the ground it is a circle — every point of it is the same distance from the centre, in every direction. On this page the longest radius from the drawn centre is 4.17 times the shortest, so a reader with a ruler measures two different distances for one ground distance depending on which way the ruler points. The two extreme radii are drawn. Paths and directions

A circle of a distance is not a circle

Eleven essays in this field have followed a route across a map. A range ring is not a route — it is the edge of a set — and drawing one exposes a failure the route essays cannot: the same ground distance comes out 4.17 times longer in one direction than another on a common projection, and 13.03 times at 70° north.

 A map for pointing, not for locating. Craig's retroazimuthal projection, centred on Mecca. The straight line from any point to the centre makes an angle with the map's vertical equal to the true initial bearing from that place to the centre — checked here at 169 points, worst disagreement 5.7e-14 degrees. The bearings printed beside each line are computed on the sphere with no projection in them. What each projection optimises

A map that cannot be read backwards

Craig's projection answers one question exactly — lay a straight edge from any place to the centre and read the compass course, right to 6 × 10⁻¹⁴ of a degree. It pays by folding: 78°S and 48°S on the same meridian are drawn at the same point, so no inverse exists and nothing else can be read off it at all.

 Everywhere within 200 km of the route from London to Tokyo. The shortest route between the two places, and the set of places within 200 kilometres of it, with both edges computed on the sphere and then projected. On the ground the set holds 3.949 million square kilometres — the band 3.823 and the two end caps 0.126, which between them make one disc of the corridor's own width. Drawn in Mollweide the corridor is visibly wider at one end than the other, and the ground it stands for is not. Paths and directions

A corridor has a width the page cannot keep

Buffering a line is the most-run operation in spatial analysis and the corridor it produces is a reach set with two failures nobody separates: stroked on the page, one width covers 190 to 471 kilometres of ground along a single route; computed in closed form, the formula stops being the area at a width the route's own length fixes, and eventually claims more ground than the sphere has.

 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.

 How far each page reorders the shapes. The number of pairs of shapes whose order on the page differs from their order on the ground, out of 36, for ten projections. Four of them put a shape other than the geodesic disc at the top — Equirectangular, Lambert azimuthal equal-area, Robinson, Miller cylindrical — which means the shape that attains the isoperimetric bound on the sphere is not the most compact thing on those sheets. The projection with none is not the equal-area one; it is whichever one's stretching happens to leave this particular set of shapes alone. Paths and directions

The most compact shape depends on the paper

A geodesic disc attains the isoperimetric bound on a sphere — its score is one, exactly, at any radius. Score the same nine regions from their images on ten projections and four of them put something else on top, an oval and its own 45° rotation come out 4.7 per cent apart, and the projection that preserves the order is not the equal-area one.

 The score does not settle at any resolution. The compactness of one stated boundary — a circle with cosine ripples at eight geometrically spaced wavenumbers, so it has structure at every scale — read at sixteen vertices up to two thousand and forty-eight. The ground score falls from 0.980 to 0.834, and it keeps falling: the boundary's length grows without bound as it is resolved while the area it encloses converges, so the quotient has no limit. The four page curves sit within a fraction of a per cent of the ground curve and of each other, which is the comparison this rung exists to make. Paths and directions

The score is not stable at any scale

One boundary, read at eight resolutions from sixteen points to two thousand and forty-eight: the compactness score falls from 0.980 to 0.834 and is still falling. Changing the projection instead moves it by 0.69 per cent. The two decisions are made by the same person on the same afternoon and only one of them is ever reported.

 Vesta, with its own coordinate lines. Vesta drawn in an orthographic projection — every figure here is a projection, including this one — with the parametric coordinate lines its published coordinates are written in. Its three semi-axes are 286.3, 278.6 and 223.2 km, so the equator is an ellipse rather than a circle and the body has no axis of revolution to define a latitude against. The surface's own radius runs from 223.2 to 286.2 km, a ratio of 1.282, and the outward normal departs from the direction out of the centre by up to 14.10°. What the numbers refer to

A map of a body with three axes

Every projection on this site is written in a latitude, and a body with three unequal axes has none: the coordinate lines are not perpendicular, Lambert's equal-area formula spreads areas by 28 per cent on Vesta, and the equal-area map that does work has a different one on every meridian.

 The shortest route between two regions, and the pair it runs between. Europe and the conterminous United States, with the shortest route between them: 3,581 km, attained at one point on each boundary. The pair is not a corner, not a centre and not anything a reader could name — it is wherever two edges happen to come closest, which depends on the whole shape of both. The dashed route is the pair Gall–Peters makes look nearest: 4,085 km, which is 505 kilometres long, and its two ends sit 2135 kilometres from the true pair between them. Paths and directions

The shortest route between two coasts

A shortest path is usually asked for between two points. Between two regions the answer is attained at a pair of boundary points, and which pair is not the pair any page makes look nearest — a cylindrical map picks a pair 2,135 kilometres from the right one across the Atlantic, and 1,716 kilometres wrong between Chile and New Zealand, where its route is 22.5 per cent long.

 Equal bands of screen, unequal bands of ground. A section through a camera pitched 60° from straight down, at the height that makes an untilted view exactly the flat map — 1152 pixels above the plane for a 768-pixel screen and a 36.87° field of view. The rays are drawn at equal spacings up the screen and the blocks below them are the ground each of those equal bands covers: 1.75 plane units per pixel at the bottom of the frame and 16.88 at the top, a factor of 9.7. Nothing about this is a projection of the sphere; it is a photograph of a map, and the map underneath it is still Web Mercator. What a machine does with it

A tilted view has no zoom level

Every essay about the screen so far assumes the map lies flat on it, which was true until about 2015. Pitch the camera sixty degrees and one frame asks for 3.6 zoom levels at once, a square tile covers ground four and a half times deeper than it is wide, and the pyramid has one integer per tile to answer with.

 The quickest way out of the ice bends at the edge. A ship twelve kilometres inside a straight ice edge, making 6 km/h in the ice and 25 km/h in open water, bound for a point ten kilometres out into the water and twenty along. The dashed line is the straight route, 3.244 hours. The solid one is the quickest, 2.849 hours — 24 min sooner — and it reaches the edge only 2.55 km along, leaving the slow ice as directly as it can. At the edge it meets the perpendicular at 12.0° in the ice and leaves at 60.2° in the water, and the sines of those angles stand in the ratio of the speeds to within 4e-17. Paths and directions

A crossing bends by a law only a conformal chart can show

A ship leaving pack ice for open water takes its quickest route by bending at the ice edge, so that the sines of its angles to the perpendicular stand in the ratio of its two speeds. The law has nothing in it but angles, so on Mercator it can be checked with a protractor — and on a plate carrée at sixty degrees north the same crossing reads as a ship making ten kilometres an hour in the ice rather than six.

 One line cannot point both ways between New York and Tokyo. The great-circle bearing from New York to Tokyo is 333.0°, and from Tokyo back to New York 25.1°. A flat picture draws one straight line between them, whose two directions differ by exactly 180°; the two bearings differ by 180° and another 127.9°. The dashed direction is the best a straight line can do for both ends at once, and it is 63.9° from the truth at each — which is a floor on the bearing error of every flat picture that contains these two cities. The impossibility

A flat picture has one direction between two places, and the Earth has two

The bearing from New York to Tokyo and the bearing back are not reverses of each other: they miss by 128 degrees. A flat picture draws one line between the two cities, so no picture containing both can have a bearing error below 64 degrees. On any set of places up to about fifty degrees across, the best flat picture sits exactly on that floor — and away from the poles, so does Mercator.

 Two lines of position replace two circles thousands of kilometres across. A ship at 50° N 20° W observes two bodies at altitudes of 45.60° and 42.69°, which place it on two circles 4,938 and 5,261 km in radius, drawn faint. From an assumed position 150 km to the north-east, each body's azimuth is laid off (dashed) and its intercept marked — 12.2 km and 111 km — and a line of position drawn perpendicular to the azimuth through that point. On Mercator the two lines cross 3.28 km from the ship: each line is only its circle's tangent at the intercept, and by the time the two lines meet they have left their circles. Paths and directions

A line of position is Newton's method, but only on a conformal chart

An altitude of a star puts a ship on a circle five thousand kilometres across, and the intercept method replaces that circle with a straight line. On Mercator the fix those lines give is one step of Newton's method: its error falls as the square of the assumed position's, and re-running it turns 300 kilometres into 24 metres in two steps. On a sheet that does not correct longitude for latitude it keeps a fixed share of every east–west error, one less the cosine of the latitude — 36 per cent at 50° north.

 A ground that is not deforming, read off a Mercator sheet. Every place on this map is carried by a rigid rotation of the whole Earth at forty millimetres a year — the motion that deforms nothing, and that this collection has already shown deforms nothing. The circles are the strain rate a geodesist would report from the grid coordinates alone, up to 17.4 nanostrain/yr. None of it is on the ground. It is the projection's own scale factor changing along the displacement, which is a second derivative of the map arriving in a first-order measurement. What the numbers refer to

The strain a map adds to the ground's

A ground carried rigidly at forty millimetres a year deforms nothing, and read off a Mercator sheet at sixty degrees north it reports 10.9 nanostrain a year. Along a profile through a real boundary the invented part is 0.0 per cent of the answer where the zone is loud and 884 per cent where it is quiet.

 A legend saying "illuminated from 315°" is true at one longitude. A hillshade's azimuth is measured from the top of the sheet, because the shading is computed on the projected raster. The top of the sheet is grid north, so the compass bearing the light comes from is the declared azimuth plus the meridian convergence, and that varies across the sheet. On a conic it swings by 55.3 degrees over eighty degrees of longitude. On a cylindrical projection in its normal aspect it does not swing at all, which is the flat line — the only case the legend is right everywhere. Measuring distortion

The light comes from a page direction

A hillshade's illumination azimuth is declared from the top of the sheet, and the top of the sheet is grid north. On a conic the light therefore swings 52.6° across eighty degrees of longitude, and 72.8 per cent of the sheet is shaded differently from what the legend claims.

 One rule, applied once, to one dataset. Two features 3 metres apart on the ground, and a snapping tolerance of 5 map units on Web Mercator. The tolerance's reach is the curve; the pair's separation is the line. They merge everywhere below about 52.8° and stay separate above it — the same features, the same rule, the same run. A pipeline that cleans a global dataset in one pass produces a topology that changes at a latitude nobody chose and nothing records. What a machine does with it

A tolerance in map units is not a tolerance

A snapping tolerance is a number, and the number is in whatever units the file is in. Five map units on Web Mercator is 4.97 metres of ground at the equator and 0.87 at eighty degrees — so a rule that merges two features three metres apart merges them everywhere below 52.8° north and refuses everywhere above it, in one pass, over one dataset, with nothing recording where the boundary is.

 Two conventions for one indicatrix field on Mercator. The left panel is the convention nearly every published field uses: every ellipse drawn at the same area on the paper, so the picture carries the shape and throws the size away. The right panel draws the image of the same ground circle at every place, so the drawn area IS the areal scale factor. On Mercator the axis ratio spans 1.0000 and the areal factor spans 14.93, so the left panel shows nothing the other one shows. Neither caption states which is which, on any published map this collection has found. Measuring distortion

The ellipses are a sample, drawn at a size somebody chose

Thirteen essays measure with the indicatrix and none audits it as an instrument. A published field has a gauge nobody states and a placement nobody states: on Mercator the standard convention draws twenty-five identical circles while the areal factor runs over a factor of 14.9, and the average a reader takes off any of these fields is between 22 and 64 per cent too high.

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