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

The thread: Measured, not named — page 6

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 121 to 144 of 292.
The shortest route, and the shortest route a vehicle can fly. A leg of 60 kilometres for a vehicle whose minimum turning radius is 5, arriving on a heading 120° off the line and required to leave on one -60° off it. The straight line is the geodesic; the curve is the shortest curvature-bounded path, which is Dubins's RSR — a turn, a straight, a turn — at 67.29 kilometres against 60. The second curve is the runner-up word, drawn to show that the choice between them is a real one rather than a formality. Paths and directions

The shortest route a vehicle can fly

Nine rungs find the shortest path under a metric and none asks whether the thing travelling can follow it. Bound the curvature and the route depends on two headings as well as two positions: the turning cost is a fixed 1.81 kilometres whatever the leg length, so it is 18 per cent of a short leg and 0.28 per cent of a long one, and the whole of it vanishes when the vehicle happens to be pointing the right way.

 Every library projection over Europe, on the two axes it can be wrong on. Each dot is one projection, scored over Europe on the two independent failures: how much it turns angles and how much it changes areas. The lower-left corner is the isometry that does not exist. The line joins the five projections nothing beats on both counts — the rest are inside it, and a reader who prefers either failure to the other should still not choose one of them, whatever weighting they hold. What is taught wrongly

The projections that are beaten on both counts

Two rungs of this ladder scored a rule of thumb over thirty regions and then forty-five. The same populations answer a harder question the ladder has never put: which library members are never the right answer at all. Two are beaten outright on both criteria everywhere, one is on no regional front in any population — and it is on the world's.

 Two nets of the same solid, cut to the same length. Every net of a Platonic solid severs exactly the same number of edges, all of the same length, so the total length of the cut is a constant and cannot choose between them. Each line joins the two places a severed edge ends up. Left: the net that keeps them closest, 11.30 edge lengths in total. Right: the net that puts them furthest apart, 17.97 — a factor of 1.59 for the same amount of cutting. The families

The net that loses the fewest neighbours

Every one of the cube's 384 nets cuts exactly seven edges of exactly the same length, so the quantity this collection has been pricing cutting by is a constant that cannot choose between them. Measured on the reader's side — how far apart a net puts two places that touch on the globe — the best net scores 11.30 and the worst 17.97, for identical cutting.

 Two latitudes for one point on Mars. A cross-section of Mars with its flattening exaggerated 6× so the two angles can be told apart. The line to the centre defines the planetocentric latitude and the outward normal defines the planetographic one; at 45° they differ by 0.338° on the real body, which is 20 kilometres along the surface. Both numbers are published for Mars, and a coordinate that does not say which it is is ambiguous by that much. What the numbers refer to

A coordinate on another body

Mars publishes two latitudes for every point and they differ by up to 0.338°, which is twenty kilometres of ground — almost exactly the same distance as the Earth's own 0.192°, because Mars is smaller by nearly the same factor. The libraries that compute either have known both numbers since this collection's early essays and had never been pointed anywhere but here.

 The signal, and four instruments' noise. The spherical excess of an equilateral triangle at 45° north against its side, with the standard deviation of a measured excess — σ√3 — ruled for four instrument accuracies. A fifty-kilometre triangle, which is about the largest anybody routinely observed, has an excess of 5.49 seconds of arc. A theodolite reading to one second gives that excess a standard deviation of 1.73 seconds, so the measurement carries about three significant bits. Everything in this rung follows from that ratio. The impossibility

How big a triangle it takes

Gauss proved that a surface-dweller can read the curvature off a triangle's angles. Doing it is another matter: a fifty-kilometre triangle has 5.49 seconds of excess, a one-second theodolite gives that excess a standard deviation of 1.73, and reading K to one per cent needs a side of 281 kilometres. Not one of the great surveys built a triangle within a factor of three of that.

 The equator, with its ellipticity exaggerated forty thousand times. The dashed circle is the equator every projection formula on this site assumes. The solid curve is the equator satellite geodesy reports, drawn with its departure multiplied by 40,000 so that seventy metres on a six-thousand-kilometre radius can be seen at all. The long axis is at 14.9° west and the short one ninety degrees from it, and the difference between them is 70.0 metres — a real quantity, about the height of a twenty-storey building, on a body every geodetic computation treats as a surface of revolution. What is taught wrongly

The equator is not a circle either

Eleven rungs price what pretending the Earth is a sphere costs, and every one of them replaces the sphere with a surface of revolution — a body whose equator is a circle. It is not. The two equatorial radii differ by seventy metres, the two surfaces part by thirty-five, and the auxiliary latitudes every ellipsoidal formula is written in stop existing.

 Two routes to one grid distance, 76.7 kilometres apart. The first four bars are the tape's chain: the chord between two marks, the height difference taken off it, the reduction to the ellipsoid, and the projection's own scale applied. The last is the satellite route — project both marks and subtract — which needs none of those steps because the observation already contains the positions the chain exists to supply. They agree to 0.17 millimetres, and that agreement is what makes the chain checkable at all: two routes to one number, sharing no arithmetic. Grids, and what a survey does

The chain the satellite does not have

The four-step reduction from a tape reading to a grid coordinate exists because a tape does not know where it is. A satellite observation does, so the same distance can be got by projecting both ends and subtracting — and the two answers agreeing to a fifth of a millimetre is the first check the chain has ever had.

 Going out and coming back, at three orders. A point three degrees from the central meridian, taken forward into the projection and back again with the series truncated at the same order both ways, and the ground distance between where it started and where it returned. The second-order pair is out by a tenth of a metre at some latitudes; the third by half a millimetre; the fourth — the order every national grid formula in ordinary use is written to — by 0.39 mm. The dips are where one of the two series passes through a node, not where the map is better. The families

The inverse of the series is not the series of the inverse

Transverse Mercator on an ellipsoid has no closed form, so every national grid computes it as a truncated series. There are two of them — one out and one back — and they are separate approximations. Composing them does not give the identity: at the order every grid formula in ordinary use is written to, a point three degrees from the central meridian comes back 0.39 mm north of where it started.

 The same address length, a tenth of the area. Every cell of a lon/lat quadtree at level 4 carries an identifier of the same length. The heavy curve is each cell's area as a fraction of the largest, against its latitude: a polar cell is 10.2 times smaller than an equatorial one. The light curve is the inverse of the cell's aspect ratio, which falls from 1.00 near the equator to 0.10 at the top — the cells stop being anything like square long before they stop being usable. What a machine does with it

An address is an area

A cell identifier does not name a place, it names a region — so its precision is an area rather than a length. On the obvious lon/lat scheme that area varies by a factor of 10 at level 4 and 163 at level 8, and the factor doubles with every level: the same identifier length means less ground the further north it is used.

 One slice of the aspect objective, at the best γ. The Kavrayskiy score for Robinson over Japan, as the pole is moved over the whole sphere with the third rotation held at the value the search settled on. Dark is good. The marks are local minima of the full three-dimensional grid that happen to lie in this slice: there are 6 of them here and 58 in the cube, and a search that walks downhill from a random start reaches the best of them 7 per cent of the time. What each projection optimises

The landscape the search walks on

The three-parameter aspect search was run and its answer recorded with a note admitting nothing proved it global. Mapping the objective finds 26 to 34 local minima for every projection and region tried, a downhill walk from a random start reaching the best of them 6 to 35 per cent of the time — and one seed from the coarse grid the search already uses reaching it in all four cases. The score is reproducible to two per cent across a sevenfold refinement; the pole it names moves 60 degrees.

 The error of a spherical formula is twice the flattening. The worst angular deformation of the spherical Mercator formulae applied to each body's own latitudes, against that body's flattening, on logarithmic axes. The points are measured by differencing the projection; the line is 2f radians, which is not fitted. The two agree to a third of a per cent for Mercury, the Earth and Mars, and depart by 3.3 and 5.1 per cent for Jupiter and Saturn, whose flattenings are too large for the first term to be the whole story. The site's headline number — 0.3848° on Earth — is an instance of this law rather than a fact about Web Mercator. What the numbers refer to

The same projection on a different body

A projection's distortion does not depend on the body's size at all — only on its flattening — so Web Mercator's 0.3848° of angular deformation is not a fact about Web Mercator. It is 2f radians, and the same mistake made on Mars measures 0.6765°, on Jupiter 7.68°, and on a body scaled ten times larger than Mars the identical number to twelve figures.

 10,000 observations of one place, averaged 120 times. Left: 400 single observations of the same place, with a 60-kilometre standard deviation east and north, drawn across ±400 km. They are scattered about the truth and their average is unbiased on the ground. Right: 120 independent averages of 10,000 such observations each, computed on the page and taken back to the ground, drawn across ±1.8 km. The cloud is tight, as averaging ten thousand things should make it, and it is not centred on the cross: it sits 489 m away, against 403 m predicted by the projection's second derivative alone. Measuring distortion

The average of noisy positions moves

Average sixty thousand scattered observations of one place on a Mercator map and the answer is 404 metres too far north — at every sample size, because it is a bias and not noise. The same average on the Lambert cylindrical equal-area is 404 metres too far south, the two being ½ (σ²/R) tan φ and its exact negative, and on the plate carrée it is not displaced at all.

 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.

 One construction, one number. A ray from a point on the equatorial plane, through the sphere, onto a tangent cylinder. Where the centre sits is the whole of the family: on the axis it gives tan φ, at the far side of the sphere 2 tan(φ/2), and at infinity sin φ. The same expression with angular distance in place of latitude, and a tangent plane in place of a cylinder, gives the gnomonic, the stereographic and the orthographic. The surface decides whether the answer is read as a height or as a radius and nothing else. The families

The developable surface was never necessary

Eight essays on this ladder start from cylinders, cones and planes, and none of them has asked which named projection is actually a projection onto one. Fitting a one-parameter perspective construction to eleven of them: six come back exact, and the three landmarks are the gnomonic at h = 0, the stereographic at h = 1 and the orthographic at h = ∞.

 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.

 On a triaxial body, latitude depends on longitude. Walk round each body at a constant planetocentric latitude of 45° and watch the direction of the surface normal, which is what the planetographic latitude is. On Mars it does not move at all — that is what having an axis of revolution means. On Vesta it swings by 1.40° and on Phobos it swings by 6.40°. A body without an axis has no latitude that is a function of position alone, and every coordinate on it is a convention with a body-fixed frame attached. What the numbers refer to

A body that is not an ellipsoid

Vesta's three axes are 286.3, 278.6 and 223.2 kilometres, all different, so it has no axis of revolution — and on such a body the planetographic latitude of a point at 45° planetocentric swings by 1.4° as one walks round it in longitude, and by 6.4° on Phobos. Latitude stops being a function of position.

 The indicatrix is a limit, and here it is being taken. The axis ratio of the image of a circle of radius ρ, measured from the picture rather than computed from the derivatives, against the ratio the derivatives give. It converges, and it converges as ρ² — each halving of the radius quarters the departure. At the smallest radius drawn the two agree to 7.6e-4 relative. This is what "the indicatrix is a limit" means when it is working: there is a number for the ratio to settle on, and it settles. Measuring distortion

The indicatrix at a point that has none

Ten essays on this ladder take the derivative for granted. Rung three proved the indicatrix is a limit and every figure since has assumed the limit exists. There are points on the maps this collection draws where it does not, and they are not exotic: the horizon of a gnomonic map, and every corner of every polyhedral net.

 A hexagonal tiling of the sphere, and its pentagons. 362 cells — 350 hexagons and 12 pentagons, the pentagons marked — drawn on Orthographic. The twelve are not a defect of the construction and cannot be removed by subdividing further: Euler's formula requires exactly twelve however many hexagons there are. Each pentagon here has 0.52 times the area of an average hexagon, so a count aggregated over these cells has twelve entries that mean something different from all the others. What a machine does with it

Hexagons cannot tile the sphere

Hexagons are the best cell shape a plane offers and the sphere will not take them. Euler's formula forces exactly twelve pentagons into any such tiling — twelve at 42 cells and twelve at 642, while the hexagon count rises twenty-one-fold — and each of the twelve is measurably smaller than the hexagons around it.

 What each configuration can see, and what it cannot. The smallest eigenvalue of the fit's own normal matrix — how much the residual changes for a unit move in the worst direction of parameter space — for three parameterised candidates against six configurations of the same size. A zero is not a hard fit: it is a direction the control points cannot see at all, so every value of the parameter along it gives an identical residual. 2 of 18 are at the floor of double precision, and they are not the ones a reader would guess. Where none is, the spread between the best and worst arrangement is still Infinity at a fixed point count. What is taught wrongly

Where the control points are

Five rungs fit a library to a map and ask how much to trust the winner. None asks whether the parameters are recoverable at all. On control points along one parallel an equirectangular's standard parallel is not merely hard to find — it is invisible, exactly, and five hundred and twelve points on the same parallel are as blind as eight.

 The bound was spherical, and the country is not. Three numbers per region, all in parts per million of scale spread. The first is the Chebyshev bound computed on the sphere. The second is that same optimal map used on the ellipsoid, which is what adopting it would actually deliver. The third is the bound with the ellipsoid-to-sphere factor put into the boundary condition, which is the real optimum. The penalty for using the spherical answer reaches 1.22 times — while a named candidate barely moves, because an optimal map has cancelled its own variation and has nothing left to hide a new one in. Grids, and what a survey does

The bound on the body the country is on

The best conformal grid a country could have had was computed against a bound solved on a sphere, with a note saying the flattening was second order and unquantified. It is second order for a named projection — every family's best candidate moves by under 0.7 per cent — and it is 22 per cent for the bound, because an optimal map has already cancelled its own variation and has nothing left to hide a new one in.

 The parameters are not reproducible and the map is. The three-parameter aspect search run at three grid resolutions and compared with the finest, twice over. Compared on the numbers it returns, the answers are 65° of pole apart. Compared on what they do to the region — the root-mean-square difference in angular deformation at every sample — they are 0.29° apart, against a map whose own deformation over that region averages about a degree. The disagreement recorded as a shortfall is a disagreement about coordinates for one map. What each projection optimises

Report the map, not the parameters

The previous rung found the aspect search returning the same score to 2.3 per cent from poles sixty degrees of latitude apart, and recorded that as a shortfall: the answer was not reproducible. The shortfall assumed the disagreeing triples make disagreeing maps. They do not — the three answers agree on the distortion field to a quarter of the deformation the map already has.

 Which scale fields a map could have, and which are only wishes. Liouville's equation — the Laplacian of log k equals 1/k² on the page — is the whole condition for a conformal map of a unit sphere to have a stated scale factor. The first two rows are the scale fields of real projections and they satisfy it to the differencing step. The rest are requests a designer might write, and every one of them fails — except one, which turns out to be a projection somebody already found. Asking for no distortion anywhere fails by exactly one, which is the curvature of the sphere. What each projection optimises

Not every distortion can be asked for

Six essays have written projections as conditions and asked how much freedom a condition leaves. The reverse question has never been put: a cartographer knows what distortion they want, so can they ask for it? For a conformal map the answer is a single equation, it is the Theorema Egregium in disguise, and asking for no distortion anywhere fails it by exactly the curvature of the sphere.

 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.

 Equal area or steady shape, and not both. Four cell schemes plotted by how much their cells vary in area and how far from square the worst of them is. The bottom-left corner is the scheme that has both, and it is empty: the equal-area cube holds area to 1.003 and has the most elongated cells, the tangent-warped cube has the tightest shapes and lets area vary by 1.20, and the lon/lat scheme is off the scale on both. Neither axis can be driven to one while the other stays there. What a machine does with it

A cell system trades area for shape

A grid can hold every cell to exactly the same area or hold every cell nearly square, and the measurement says it cannot do both: the equal-area cube's areas agree to a part in a thousand and its worst cell is 1.29 times as long as it is wide, while the tangent-warped cube holds shape to 1.19 and lets area vary by 20 per cent.

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