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

Every essay — page 10

Essays 217 to 240 of 339, in the same order.
The corner against where along the seam the feature crosses. A feature crossing the seam of a cube at right angles, drawn on each face's own map and unfolded, with the crossing moved along the edge. Every curve starts at zero, because the face's mirror through the edge's own midpoint reverses the along-edge direction and forces the shear term in the Jacobian to be odd. Away from it the gnomonic — the map with no corner at all in the rung below — reaches 44.4°, three times the equal-area map's and far past the conformal one's. The edge's half-length is 35.3°, so the right-hand end is still well inside it. The families

The corner is not at the midpoint

The rung below measured the corner a feature gets crossing a polyhedral seam, and found none at all under the gnomonic face map. It crossed at the edge's own midpoint every time — the one point on the edge where a face's own mirror symmetry forces the corner to vanish. Two fifths of the way to the vertex the gnomonic gives 20.15°, the equal-area map 7.28° and the conformal map under a degree, which reverses the ordering entirely.

6 figures · Polyhedral A parallel, and the places with the same northing as its middle. Both curves are drawn on a transverse Mercator zone 6° wide at 45° north. The upper one is the parallel — every place on it is at the same latitude — and it climbs 4379 metres of northing on the way to the zone edge. The lower one is the set of places whose northing equals the parallel's on the central meridian. Between them lies every pair the grid puts in the wrong order, and its widest point is 4.39 kilometres of latitude. Grids, and what a survey does

Further north on the grid is not further north

Thirteen rungs price a grid's origin, scale, units and zones, and every one treats a coordinate as a position. It is also an ordering, and the two disagree: a parallel climbs 4,379 metres of northing on its way to the edge of a UTM zone, so two places 4.4 kilometres apart in latitude can be listed in the wrong order — and the share of pairs it happens to is the convergence, in a different unit.

6 figures · Grid A plane fit swallows a datum shift, and keeps swallowing it. A map of OSGB36's ground drawn as though it were on WGS84, at seven sheet sizes. The upper curve is how far the drawing moves — a real 99-metre error on the ground — and the lower one is what survives the best similarity between the two, which is what a fit for the projection removes for free. The absorption runs from 99.68 per cent on a 1° sheet to 85.16 on a 60° one, and what is left is 3.9 parts per million of the map's own size even then — two microns on a sheet half a metre across. What is taught wrongly

The datum hides inside the projection's parameters

Six essays recover a projection from control points with the body it was drawn on taken as known. It is not known, and the fit cannot find it: a 99-metre datum shift is absorbed to 99.7 per cent on a two-degree sheet and to 85 per cent on a hemisphere, leaving eleven parts per million of the map behind.

6 figures · Identify A chain of stations, its positions' uncertainty and its baselines'. Nine stations held at the left-hand one, with every neighbour, second neighbour and third neighbour observed. The filled ellipses are each station's own uncertainty, which grows without limit as the chain runs away from the point it is held at — 7.5 mm at the near end and 574 mm at the far. The open ellipses above each leg are the uncertainty of the baseline, drawn at the same scale: they hardly grow at all, because almost everything that is wrong with one end is wrong with the other in the same direction. Measuring distortion

The difference of two coordinates

Three essays give a single coordinate a width. Every practical use of one is a difference of two — a distance, a bearing, a movement, an area — and the width of a difference is not the two widths combined, because the errors are not independent. Far from its datum a one-leg baseline is six times more certain than the positions it joins.

6 figures · Precision A conformal map of Vesta, in the coordinates that make one possible. The body's own isothermal coordinate, used as the map — which is Mercator's construction carried out on a body that has no axis to be Mercator about. The lines are Jacobi's ellipsoidal coordinates, which are the surface's lines of curvature, at 30° and 20° spacing; the two quadratures that turn them into an isothermal pair are one-dimensional. The measured angular deformation away from the marked points is 1.1e-6°, which is this site's noise floor, and the areal factor spans a factor of 162 — conformal, and emphatically not equal-area. The four marks are the umbilics, where the coordinate collapses and the map has nothing to say. What the numbers refer to

A conformal map of a body with three axes

This site built an equal-area map of a triaxial body and wrote down what it could not do: the conformal one, which needs an isothermal coordinate that a surface with no axis of revolution was said not to have. It has one, Jacobi found it in 1839 for a different reason, and it takes two one-dimensional integrals.

8 figures · Bodies 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.

9 figures · Screen Every projection against a picture that is not a map. The worst relative distance error over London, New York, Tokyo, Sydney, for each projection in the library and for the best flat arrangement of the same distances. Each azimuthal member is centred on the set's own centroid, which is the fair comparison. The free picture reaches 0.63% and the best projection, Azimuthal equidistant, reaches 8.28% — a ratio of 13.17. The free picture cannot lose, because every projection's own layout was handed to the search as a starting point; what the figure measures is how much the freedom is worth, and it is worth different amounts at different sizes. The impossibility

The best flat picture is not a map

A set of dots whose separations are as nearly right as separations can be made beats every projection in this collection — by a factor of thirteen on four world cities, and by eight per cent on sixteen. The collapse between those two numbers is not about cartography. It is that a picture of n places has 2n − 3 free numbers and n(n − 1)/2 distances to spend them on.

5 figures · Embedding Two members, their average, and the family's best answer to it. Two equal-area conics at cone constants 0.25 and 0.85, the average of the two, and the member of the family nearest that average — at 0.540, which is not the parameter midpoint. The average is not a conic at all: its parallels are still arcs but they are arcs of circles about different centres, so no single cone constant reproduces it and the residual is 0.1722. The families

A family is not closed under averaging

Nine rungs treat a family as a set of maps with a parameter running through it, and this collection's own compromise projections are averages of members. An average of two members is not a member: two conics far apart average to something 31 per cent of their own separation outside the family — and averaging within a family makes the map worse every time, while averaging across two makes it 16 per cent better.

6 figures · Families The convergence order across an edge, against the edge's own orientation. Each curve is one kernel, fitted the same way as every other convergence order on this site: the root-mean-square error against the grid spacing, in logs, over five refinements. At 27° they read 0.78, 0.58, 0.60, which is the measurement already published here — and 27° is one point. Turn the edge onto a parallel and the curves collapse, and the nearest-neighbour one goes negative, which is the fit's way of saying the error is not falling at all. A single number for "the order across an edge" is a number about the edge that was measured. What a machine does with it

One edge is not an edge

The three resampling kernels were measured across a discontinuity and came out at 0.78, 0.58 and 0.60 — one straight edge at 27° to the graticule. Across thirteen edges the same kernels span 0.19 to 0.87, the ranking between them reverses, and for an edge lying along a parallel the error does not fall with refinement at all.

7 figures · Dataset The same four requests, put to the two conditions. Each request is a stated field over a square region, and the bar is what is left over after the nearest map satisfying the condition has been found. The conformal condition refuses: its achievable set is decided by boundary values, and the residual is the part of the request no conformal map of any kind can supply. The equal-area condition never refuses — every one of these is met to 3.0e-5, which is the quadrature's own noise — because a positive areal request is granted by a construction with no iteration in it and no boundary data. What each projection optimises

The nearest equal-area map to an impossible request

Every number in the previous rung is inside the conformal achievable set, because Liouville's is the conformal condition. The equal-area set is one equation on two functions and never refuses: the same four requests are met to nine parts in a billion, and charged for in angle instead.

6 figures · Condition The height of a 4000-metre summit, against the density assumed beneath it. The geopotential number is 39204 m² s⁻² and is not in doubt. Turning it into a length divides it by the mean gravity along the plumb line, which is inside the mountain — and reconstructing that from the gravity measured at the surface needs a density. Taking the rock to be 2400 rather than the 2670 it actually is puts the summit 185 mm low; taking it to be 2900 puts it 157 mm high. Skipping the reduction entirely puts it 691 mm high, which is why the reduction exists. What the numbers refer to

The line a height is measured along

Five rungs have argued about the surface a height is measured *from* and every one of them took the line it is measured *along* to be straight and known. It is neither: through a stated buried mass a plumb line arrives 47 millimetres from the point below the summit, and the height it gives depends on the density of rock nobody has seen — 342 millimetres of spread at 4,000 metres and 1.37 metres at 8,000.

6 figures · Height Which rotation a projection cannot see is decided by the projection. The same rotation — 2.455 arcseconds, DHDN's polar one — applied about each of the three axes in turn, with the residual after the best plane fit drawn on a logarithmic scale. A cylindrical and a conic in their normal aspects hide the polar rotation to arithmetic noise, 3e+5 times better than either equatorial one, because a change of longitude is a symmetry of both. A pseudocylindrical hides none of them — its horizontal coordinate carries a factor in latitude, so a longitude shift is a shear rather than a translation. And an azimuthal centred on the equator hides the equatorial rotation instead. What is taught wrongly

A rotation is not absorbed the way a shift is

The previous rung expected a datum's rotations to be the part a plane fit could not swallow. They are the part it swallows best — 93 times better than the translations over a hemisphere, and 24 times better per metre moved — and the reason is that the rotation which matters is a change of longitude, which is a symmetry of the map.

7 figures · Identify Four averages of one region's indicatrices — Robinson over the whole sphere. The faint ellipses are the indicatrix at ninety sampled places; the four drawn over them are four candidate averages of exactly that set. The arithmetic mean of the tensors gives semi-axes 1.1621 and 1.0182; the log-Euclidean mean gives 1.0053 and 0.9608; the Karcher mean gives 1.0042 and 0.9619; and taking the mean of a and the mean of b separately — which is what a published average distortion usually is — gives 1.2335 and 0.8200. Their maximum angular deformations are 7.57°, 2.60°, 2.47°, 23.23°, against a mean of the pointwise deformations of 20.67°. Five numbers, one set of ellipses. Measuring distortion

An average of ellipses is not an ellipse

Rung three integrates distortion over a region and finds the weighting is somebody's opinion. It never asked what was being averaged. An indicatrix is a positive-definite matrix, matrices form a cone rather than a vector space, and the arithmetic average of an equal-area map's indicatrices comes back inflating area by up to 11 per cent.

8 figures · Tissot Four layouts, the same five stations, the same ten distances. Every panel has five stations, all ten distances between them, the same instrument precision and the same three degrees of freedom. The ellipses are the error ellipses of the adjusted coordinates, drawn at one common exaggeration, and they are computed from the geometry and the weights alone — no observation value enters any of them. The worst semi-axis runs from 11.1 millimetres to 232, a factor of 20.9, and the difference is entirely where the marks were put. Grids, and what a survey does

The network's answer is decided before it is measured

Nine rungs measure what an adjustment does with observations. Every quantity a specification is written about — the error ellipses, the redundancy numbers, the smallest detectable blunder — is a function of the geometry and the weights alone, and does not contain an observed value anywhere. Four layouts of five stations with the same ten distances differ by a factor of 20.9 in their worst coordinate.

6 figures · Reduction The share of its distances a flat picture can hold. A picture of n places on a plane has 2n coordinates and is unchanged by two translations and a rotation, so 2n − 3 numbers in it are genuinely free; each distance held exactly is one equation. So at most 2n − 3 of the n(n−1)/2 distances can be right, whatever the places are and however hard the map maker tries. The share falls as 4/n: five of six for four places, thirteen of twenty-eight for eight, and 197 of 4,950 — 4.0% — for a hundred. The dashed curve is 4/n, which the count approaches from below and never crosses. The impossibility

Five distances of six, and never more

A flat picture of n places has 2n − 3 free numbers and n(n−1)/2 distances, so it can be exactly right about 2n − 3 of them and no more — five of six for four places, twenty-nine of a hundred and twenty for sixteen, and 3.98 per cent for a hundred. The bound is reached by a construction with no fitting in it, and an azimuthal equidistant map spends only n − 1.

5 figures · Embedding A finer grid makes a measured slope worse. The error in the direction of steepest ascent, against the spacing the field was sampled at, for four noise levels. With exact values the curve falls at a fitted slope of 2.00 — second order, which is what a central difference is. Add noise and the same curve turns over: a finite difference divides the noise by the spacing, so halving the grid doubles the noise in the slope while quartering an error that was already negligible. The minimum is where the two meet, and it is not at the fine end. Measuring distortion

The slope of a field that was measured

Three rungs differentiate a formula, which is what makes the projection the only thing under test. A real field is a grid of numbers with an error on each of them, and differencing such a thing divides the noise by the spacing — so a finer grid gives a worse slope, there is a best spacing, and it is the cube root of the noise.

8 figures · Gradient Give the edge a width and the kernels get their orders back. Every edge this collection has resampled across has been exactly discontinuous, which is not what a sensor produces: a footprint, an atmosphere and a lens all smooth a boundary over a cell or two before anything is resampled. Convolving the edge with a Gaussian of stated width and refitting gives 1.23, 1.97 and 3.60 at one degree of blur, against 0.78, 0.58 and 0.60 with no blur at all. The blur is held fixed in degrees while the grid refines, which is what happens to a real sensor's data as its resolution improves. What a machine does with it

A real edge has a width

Thirteen edges were measured and every one of them was exactly discontinuous, which no sensor has ever produced. Convolving them with a point-spread function of one degree — a cell or two — takes the three kernels from 0.78, 0.58 and 0.60 back to 1.23, 1.97 and 3.60, and takes the edge along a parallel, which converged at −1.49, up to 1.92 for bilinear and 3.73 for cubic.

6 figures · Dataset The basin has three widths, and they differ by a factor of 4.6. Two sections through the near-optimal basin of the Robinson aspect over Japan, drawn at one scale. Each ellipse is the set of aspects whose score is twice the optimum's, from the objective's own second derivative at the optimum: 32.1°, 16.0°, 6.9° along the three principal directions. A single number for "the width of the basin" is the cube root of their product, 15.3°, and it is not any of them. What each projection optimises

The basins have widths as well as depths

The previous rung measured the height of the pass and recorded a shortfall: shape means widths too. Measured, the basin has three of them — 32°, 16° and 7° at Japan — it gets wider rather than narrower as the region grows, and the exponent it predicts overshoots the measured one by half again.

7 figures · Choosing A conformal map solved rather than written down. The same patch of parameters on two bodies, mapped to the plane by solving the discrete Cauchy–Riemann equations — one complex equation per triangle, 1568 triangles, least squares, conjugate gradients, and no formula for either surface. Left: a sphere, where the answer is known in closed form and is not used. Right: a body with a bump on it, which has no isothermal coordinate and therefore no closed form at all. The parameter lines cross at right angles in both, to a median of 0.60° and 1.25° of angular deformation. What the numbers refer to

A conformal map of a body that is not a quadric

Jacobi's ellipsoidal coordinates give a triaxial body a conformal map by two quadratures, and this collection wrote down what that argument uses: the surface has to be a quadric. A real body is not. Solving the discrete Cauchy–Riemann equations instead — one complex equation per triangle, two thousand triangles, conjugate gradients — gives a conformal map of a bumped body to a median of 1.10°, converging at first order in the mesh, with the areal factor spreading by 3.07 and refusing to converge at all.

6 figures · Bodies One of these is a corner. The corner reported at the exact conformal seam, and the corner reported at a cube's gnomonic seam fifteen degrees along its edge, each measured over a shrinking arc. The gnomonic's is 21.4572° at every span from twenty-four degrees down to three — the same number to four decimals — because it is a corner. The conformal one halves whenever the span does, fitted exponent 0.990, because a tangent read from a finite chord of a curved image departs from the true tangent in proportion to the chord. It is not a corner; it is the instrument. The families

The exact map says the seam is smooth

The previous rung could only bound the conformal seam's corner at about two degrees, because the series it was fitted with holds its boundary condition to three parts in a thousand. An exact map exists — the stereographic projection composed with ∫dt/√(1 − t⁴) — and it settles it: the corner falls in exact proportion to the arc the tangent is read over, fitted exponent 0.99, while the gnomonic's 21.4572° is the same to four decimals at every span.

5 figures · Polyhedral One pair of numbers, a hundred and twenty places. The easting 412,000 and northing 5,678,000, interpreted in each of the sixty zones and in both hemispheres. The northern candidates are a ring of sixty at 51.247° north, spaced exactly six degrees apart; the southern ones are a second ring at -39.043°, which is not the mirror of the first because the southern convention subtracts the northing from ten million. Every one of the hundred and twenty is a perfectly valid reading of the same two numbers. Grids, and what a survey does

One pair of numbers, a hundred and twenty places

A UTM coordinate is two numbers and a zone. Drop the zone and the numbers are still valid in each of the sixty; drop the hemisphere too and the pair names a hundred and twenty places. They form two rings at one latitude each, spaced exactly six degrees apart, and every one of them has the same grid convergence and the same scale factor — so no further geometric measurement can choose between them.

5 figures · Grid Four of the seven on the front can never be first. Every library projection over the whole sphere, with both errors normalised to the table's own range. The seven filled circles are the Pareto front — nothing beats them on both counts. The line through three of them is the lower convex hull, and a weighted sum of the two errors is a straight line in this space, so only a hull vertex can ever come first. Four projections — Web Mercator, Miller cylindrical, Equirectangular, Winkel tripel — sit in the dents, undominated and unchoosable. What is taught wrongly

Which projection a weighting can make best

Rung eight finds the seven world projections nothing beats on both counts and tells a reader with a preference that one of them is their answer. Four of the seven are not: they sit in dents of the front, undominated and unreachable, and no weighting of angle against area can ever put them first. Robinson can be first, on 2.8 per cent of the weight range.

6 figures · Audit The page's curvature is not a free parameter. The same trilateration, carried out on a sphere of stated radius instead of on a plane, with the error of the distances the construction decides rather than holds. At the radius the distances were measured on it is 5.1e-15 — exact, by construction, which is the refusal this figure exists to make. Two per cent either side of it the worst decided distance is already out by around 10%. Below 90 per cent of the Earth's radius the construction cannot be completed at all: two circles that must cross do not, and the page is simply too small to hold the places. The flat sheet is the right-hand limit, at 152%. The impossibility

The escape is not a dimension

Four places that will not lie in a plane surely lie in a space — and they do not. The double-centred matrix of their great-circle distances has a negative eigenvalue, so no Euclidean space of any dimension holds them, and buying a third dimension improves the picture by nothing whatever, to fifteen decimal places. What does work is a page with curvature, and the curvature is pinned to within two per cent.

6 figures · Embedding Two expansions of one integral. The worst error along the whole meridian, against the number of sine terms kept, for the series in the third flattening and the classical series in e². Both are checked against a Simpson's rule on the defining integral, which shares no algebra with either. At one and two terms they are the same number to four digits and the e² series is fractionally ahead; from the third term the n series pulls away, and at four it is 1175 times more accurate — 7.6e-8 metres against 9.0e-5. What is taught wrongly

Which small quantity the series is in

Every ellipsoidal formula in this collection is a truncated series in the third flattening, inherited from Krüger in 1912 and justified nowhere. Measured against the classical expansion in e², at four sine terms it is 1,175 times more accurate — and at one and two terms it is fractionally worse, which is not what the folklore implies.

6 figures · Ellipsoid