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

The thread: Computed, not quoted — page 8

Scale factors, areal ratios, angular deformations, geodesic distances and the areas of countries are all computed while the figure is drawn. None is a number recalled from a table. Essays 169 to 192 of 299.
The ground two divisions disagree about. seven places across the world dividing the sphere twice under a steady westerly. Outbound, a place belongs to the site that can reach it soonest; inbound, to the site it can reach soonest. The pale tints are the outbound division where the two agree; the dark ground is where they do not, and it is 50.7% of the sphere — 258,530,666 km². Every site's own share of the world barely moves between the two divisions, by at most 3.8%. The two divisions assign nearly the same AMOUNT of ground to each site and assign completely different ground. Paths and directions

A partition under a directed cost has two versions

Dividing a surface among several sites is one question when the cost is a distance and two questions when it is not. Under a steady flow at 0.45 of a vehicle's own speed, the division by who can reach a place soonest and the division by which place can be reached soonest disagree about 50.7 per cent of the sphere — while no site's own share of the world moves by more than 3.87 points.

 Believing a group takes away the redundancy that would test it. The thirteen degrees of freedom in this network, divided between the two groups, against the weight ratio the adjustment was told. At a ratio of a quarter the edges carry 12.1 of them and at sixteen they carry 0.07. A group's own variance can only be estimated from its own share, so a surveyor who is confident about an instrument has taken away the arithmetic that would have caught the confidence. The two curves sum to thirteen at every ratio, which is the identity that makes this a redistribution rather than a loss. Grids, and what a survey does

The weights are a guess the solve believes

Rung seven finds a decision inside the least-squares problem no residual can see: what to hold fixed. There is a second, made more often and thought about less. Every observation enters with a weight nobody measured, the weights move the coordinates by a factor of 1.8, and the standard check on them can be made to pass by a scaling that moves nothing at all.

 A mass, the geoid it raises, and the plumb lines that lean towards it. A sphere of 5 km radius buried 8 km down, denser than its surroundings by 500 kg per cubic metre — a salt dome, or an ore body. Its mass is 261.8 × 10¹² kg, and outside itself its field is a point mass's exactly, so everything above is a closed form. The geoid rises 22 centimetres over it, drawn at 20,000× the true slope; the plumb line leans by at most 2.21 arcseconds, and it does so 5.7 km to the side rather than above the body, because the deflection is the geoid's SLOPE and a slope is zero at a summit. That is the number this site has been able to relate and unable to compute since its practice phase. What the numbers refer to

A deflection is the slope of a mass

The site has computed the exact relation between a geoid slope and a deflection of the vertical — one arcsecond is 4.85 millimetres in a kilometre — and computed no deflection anywhere, because a deflection is the gradient of a geoid and the geoid is a citation. State a mass instead, and every quantity is a closed form: a salt dome five kilometres across bends the plumb line by 2.21 arcseconds.

 Steepest descent on Lambert cylindrical, computed on the ground and on the page. Fifteen routes, each started at the same place twice. The solid line follows the true direction of steepest descent on the sphere; the dashed line follows the direction read off the page at every step, which is what an analysis of a projected grid does. Both take the same length of step on the ground, so the only difference between them is the direction. They part by up to 1517.8 kilometres, against a bearing error of 45.6°. The faint lines are contours of the field, which is seven caps at stated centres and widths. Measuring distortion

Water runs downhill on the ground, not on the page

A drainage network is the set of steepest-descent trajectories of a field, so it is built entirely out of directions. A conformal map preserves those directions exactly and therefore preserves the whole network; an equal-area map does not, and sends a route up to 892 kilometres away from where the water actually goes.

 Where Robinson's valley breaks, against how large the region is. The threshold at which the set of near-optimal aspects stops being one connected piece, for square regions of growing size at 38° north. It falls from 2.13 at 6° to 0.27 at 30°, a factor of 8.0. The previous rung measured this at one region and quoted "about twice the optimum"; that value belongs to a small region, and the prediction that it should fall as the region grows is what this tests. What each projection optimises

Where the valley breaks in two

The previous rung found a near-optimal set that is one connected sheet at a loose threshold and fourteen basins at a tight one, and explained the transition without testing it. The explanation is a prediction about region size and projection sharpness: swept over both, the threshold falls from 2.13 to 0.27 as a region grows from 6° to 30°, three projections lie on nearly one curve, and a fourth declines to join for a reason worth having.

 Three implementations of one projection, and three different distortions. The spread in angular deformation between linear, Catmull–Rom and Aitken interpolations of Robinson's own table, at 60° of longitude. It is exactly zero at every fifth degree, because all three pass through the tabulated rows, and it is not zero anywhere else: 37 of 87 latitudes differ by more than half a degree and the worst is 8.42°. The graticules the three draw differ by 4.2 parts per thousand of the map's own span, which is under the width of a printed line. The families

A projection defined by a table has an interpolation in it

One member of this library has no formula: Robinson set nineteen pairs of numbers by eye and the table is the definition. Three published interpolations of it draw graticules that differ by four parts in a thousand of the map's span and report angular deformations that differ by 8.4 degrees.

 Hilbert order on one face, as a curve. The order in which Hilbert numbering visits the 64 cells of one cube face at level 3. The line never leaves a cell without entering one that shares an edge with it — that is what makes it a space-filling curve, and it is why two cells with nearby identifiers are usually near each other on the ground. What a machine does with it

The address is a curve through the sphere

A database does not fetch a set of cells, it reads ranges of identifiers — so the cost of a query is how many runs its cells form, not how many cells it needs. Hilbert order wins that measurement and loses the one usually quoted for it: its neighbouring cells are further apart in identifier than row-major's, on average and at worst.

 The four places, and every distance between them. London, New York, Tokyo, Sydney on a Mollweide projection, with every great circle between a pair drawn. The six ground distances run from 5,570 km to 16,994 km, and they are the whole of the input to every figure in this ladder — no coordinate, no projection and no coastline enters any of them. The arcs are drawn only to say which pair each number belongs to; on this page they are curves, and on the ground they are the shortest routes. The impossibility

Four cities that cannot be drawn to scale

Every impossibility in this field so far has been about a surface. This one is about four numbers: London, New York, Tokyo and Sydney have six distances between them, and no four dots on any sheet of paper have those six separations. The test is a determinant Cayley wrote down in 1841, and it comes out −3.34 × 10²³ where zero is required.

 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

Ten rungs find the shortest path 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.

 OSGB36 then DHDN, done four ways. How far each route lands from the truth — applying the two transformations one after the other — at 54.5° north, 2.0° west. Composing them into one affine map is exact, because two affine maps compose into an affine map and nothing is dropped. Describing that composition with seven parameters again leaves 0.19 millimetres, because two linearised rotations compose into something with a symmetric part that seven numbers cannot hold. Adding the fourteen published numbers in pairs — which is what a chain is usually done with — leaves 7.4 millimetres. Logarithmic from a micrometre. What the numbers refer to

A chain of transformations does not close

Two datum transformations applied one after the other are not the sum of their fourteen published numbers. The gap is seven millimetres in Britain, doing the two in the other order moves the answer twenty, and the rotation matrix everybody prints is not a rotation.

 The part of a request no conformal map can supply — scale falling with distance from the centre. The difference between the requested scale field and the nearest achievable one, over the patch, with the sign shown by colour. The RMS is 1.81e-1 in the logarithm of the scale and the worst single point is 4.91e-1. This is not an error: it is the part of the request that no conformal map of any kind can grant, and its SHAPE is the answer — it says where the request was impossible, which a single number cannot. What each projection optimises

The nearest map to an impossible request

Rung seven found that not every distortion can be asked for. It never asked what happens when one is asked for anyway — and the answer has a shape: the achievable fields are the solutions of an elliptic equation, a request is a point off that set, and the nearest point leaves a residual whose floor is the curvature rather than the size of the ask.

 The same kernels, across an edge. Each kernel measured over the same five rasters as the smooth measurement, on a field that carries a step across a tilted line instead of a smooth function. The solid lines are the edge and the faint ones the smooth field. The orders on the smooth field are nearest 1.00, bilinear 1.98, cubic 2.93; across the edge they are nearest 0.78, bilinear 0.58, cubic 0.60 — within a factor of 1.33 of one another, and the fastest of them belongs to nearest-neighbour, which does no interpolating at all. The ranking a smooth field establishes does not survive a discontinuity, and a real raster is mostly edges. What a machine does with it

An edge has no order of convergence

On a smooth field the three resampling kernels converge at orders 1, 2 and 3 and the choice is obvious. Across a discontinuity they converge at 0.78, 0.58 and 0.60 — within a factor of 1.4 of each other, in an order that puts nearest-neighbour first, and a real raster is mostly edges.

 A geodesic circle on Mollweide, and the two models of where it goes. A circle of geodesic radius ρ about 20°E 45°N, projected. The dashed outline is where the indicatrix says it goes — the ellipse a Tissot figure draws — and the thin solid one adds the quadratic term, which is the flexion and skewness this ladder measures. At 16° the indicatrix is out by 13.5 per cent of the figure's own size and the second-order model by 2.20 — a factor of 6.16, which is what one more derivative buys. Measuring distortion

The size at which the second derivative arrives

Six essays have measured a projection's second derivative at points, over regions and under transformations, and none of them says at what size it stops being a curiosity. The answer needs a figure with an extent rather than a point, and it is smaller than anybody drawing a national map would guess: the indicatrix alone places a shape to one part in a thousand out to thirteen kilometres.

 How large a blunder has to be before the test notices. A blunder of increasing size put into the least-checked observation of a braced quadrilateral, with the standardised residual it produces. The horizontal line is the critical value the test uses, and the vertical one is the minimal detectable bias — δ₀σ/√r, which is 87 millimetres for this observation and is computed from the network's DESIGN, before any observation is made. Below it nothing is flagged; above it everything is. The observation's redundancy number is 0.144, so it is checked by a seventh of an observation and hides six-sevenths of whatever is wrong with it. Grids, and what a survey does

The blunder the network cannot see

A least-squares adjustment has no concept of a mistake. The smallest blunder its test will find in the least-checked leg of a braced quadrilateral is 87 millimetres, and by the time it fires a station has moved by nearly ten times the accuracy the same adjustment reports for it.

 Every projection's angular deformation, and the tolerance that judges it. The whole library on one logarithmic axis, with the tolerance drawn as a line. The population is bimodal: the projections that satisfy the condition sit at 2.09e-6 and below, the ones that do not at 3.85e-1 and above, and there is nothing between. The tolerance could be moved anywhere in that gap — a factor of 1.84e+5 — without changing one verdict. Larger marks are projections that claim the property. What is taught wrongly

The tolerance that decides the verdict

Eight rungs of this ladder hand out verdicts, and every one rests on a tolerance chosen once, in the site's second phase, at sixty times a measured noise floor. Swept, it decides nothing: the population is bimodal, the tolerance sits in a gap 238,000 times wide for conformality and 646 million times wide for equal area, and the verdict with the least room is a passing one whose margin is the arithmetic's rather than the map's.

 The corner an equal-area face map puts in a feature, and the one the gnomonic does not. A great circle crossing the seam between two faces, drawn on each face's own map and unfolded flat, with the angle between the incoming and outgoing tangents plotted against how obliquely it crosses. Under the equal-area face map every solid gives a corner: zero for a perpendicular crossing, where the two faces are symmetric about the edge, peaking near thirty degrees of obliquity and falling again as the crossing lies down along the edge. Under the gnomonic it is zero at every angle on every solid, which is the flat line on the axis. The families

The gnomonic crosses a seam without a corner

Eight rungs choose a solid, a face map and where to cut. This one is about a seam the net does not cut, where two faces stay joined and carry two different maps — and the expectation was a corner in every feature crossing one. At the midpoint of the edge — which is where every crossing here is taken, and which turns out to be the one place on it a face's own mirror forces the corner to vanish — the gnomonic gives none at any obliquity on any Platonic solid, and the equal-area face map gives one of up to 0.39 degrees.

 The least error a flat picture can have, against how much sphere it spans. The same configuration of places, shrunk about its own centroid so that every bearing is kept and only the span changes, with the least worst-case relative error of the best flat picture at each size. Both axes are logarithmic. The fitted slope over the rows below ninety degrees is 2.0246: the error falls as the SQUARE of the diameter. That is Gauss's theorem arriving as a number for a finite set — curvature is a second derivative, so its first effect on a distance is quadratic in the separation — and it is why a county fits on a sheet and a hemisphere does not. The impossibility

How wrong a flat picture has to be

The rung below proves no flat picture of four places is exact and leaves the size of the failure to a determinant nobody can read. Measured directly, the least error falls as the square of how much sphere the places span — fitted exponent 2.0088 — and the same exponent comes back from five different arrangements while the constant in front of it moves by a factor of seventeen.

 The plan, the crossing that was flown, and the one a right forecast would have given. A crossing of NaN km planned against a jet forecast to sit on the fortieth parallel, flown through a jet that is actually 6 degrees further south, and re-planned six times on the way. Committing to the plan costs 167.81 hours. Re-planning costs 165.84. A vehicle that had known where the jet was would have taken 163.21. So the wrong forecast is worth 4.60 hours and re-planning recovers 1.97 of them. Paths and directions

A crossing is a chain of decisions

Eleven rungs hand back a curve and stop, and nothing anybody flies is decided once. Re-planning is worth exactly nothing when the forecast turns out right — a sub-path of an optimal path is optimal, and the chain is the plan. When it turns out wrong it recovers 19 per cent of the cost at two decisions and 83 at eighteen, and never all of it.

 The pass between the basins, measured rather than bracketed. The height of the lowest path from one near-optimal basin to the other, for regions of growing size. It is found by sorting the score surface and joining cells in order, so it is the minimax path's own height rather than the level at which a bisection stops finding two pieces. The fitted exponent is -1.60 at a coefficient of determination of 0.913. The previous rung's level-set measurement gave −1.33 and the argument predicts −1, so measuring the height directly moves the answer FURTHER from the prediction rather than towards it. What each projection optimises

The height of the pass between two basins

The previous rung recorded a shortfall: the fracture threshold falls as the −1.33 power where the argument predicts −1, and the difference was supposed to be the height of the pass. It is not. Measuring the pass directly gives −1.60, which is further from the prediction, and the two basins turn out to be exact symmetric copies with nothing to decompose.

 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.

 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.

 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.

 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.

 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.

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