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

The thread: Measured, not named — page 7

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 145 to 168 of 292.
Five stations, ten distances, three spare. A braced quadrilateral with a centre point. Every distance between the corners and every distance to the centre is observed, 10 in all, each with a standard deviation of 8 mm. Holding one station and one bearing leaves 8 unknown coordinates, so the network has three degrees of freedom: three independent statements the observations make that could be contradicted. Everything the adjustment can tell anybody about the quality of the work comes out of those three. Grids, and what a survey does

A coordinate is the output of a solve

Six essays measure a tape, close a traverse, spread a misclosure and reduce a chain. The coordinate that comes out of the far end is the solution of a least-squares problem, and the problem has a decision in it that is not a measurement: what to hold fixed. Change it and every coordinate moves by centimetres while not one residual moves at all.

 The widest zone a tolerance of 690 parts per million allows. At each latitude, the half-width at which a transverse Mercator grid with its scale factor rebalanced for that width reaches 690 ppm at its worst point. That tolerance is the one UTM actually meets at the equator, so the curve passes through UTM's own 3° there — and rises to 37.0° at 85° north, because a degree of longitude covers cos φ of the ground and the scale error goes as the square of the ground width. Six degrees is the answer at one latitude. Grids, and what a survey does

Sixty zones was a decision about one latitude

Twelve rungs price a grid, a zone, an origin and a reference, and every one of them works inside a single zone. The number of zones has never been asked about: six degrees meets its tolerance at the equator and is loose everywhere else, so a system spending the same tolerance evenly would use 51 zones at the equator and 8 at 82° — and UTM's worst error is 981 parts per million, not the 400 always quoted.

 Two unfoldings of the same 80-face solid. Both are edge unfoldings of the same solid along different spanning trees of its face graph, so both preserve every distance on the surface exactly. The left one is a net. The right one is not: 3 pairs of its faces occupy the same ground, so it cannot be cut out of paper and folded up. Nothing in the unfolding procedure prevents this, and past the regular solids most trees produce it. The families

A net can land on top of itself

Every one of the cube's 384 unfoldings is a net, and every one of the icosahedron's five million is too. Past the regular solids that stops being true: at 180 faces, 99 of every 100 randomly chosen unfoldings have faces sitting on top of each other, so choosing a net stops being a choice and becomes a search — except that the net anybody would actually draw works every time.

 Tissot's ellipse and the one that governs gradients, at 20°E 48°N. The solid ellipse is the image of a small circle — Tissot's indicatrix, semi-axes a and b. The dashed one is the image of a unit gradient, whose semi-axes are 1/b and 1/a because a gradient transforms by the inverse transpose of the Jacobian rather than by the Jacobian. Its long axis therefore lies where the indicatrix's short one does. On Mercator both are circles, so a gradient's direction survives; on Lambert cylindrical the two ellipses are the same shape turned through a right angle, so the worst direction for a gradient is the best direction for a shape. Measuring distortion

A slope is not a shape

Every map in this collection has carried geometry. An applied map far more often carries a field — elevation, pressure, a density — and the first thing anybody does with one is differentiate it. A gradient is a covector, it transforms by the inverse transpose of the Jacobian, and the ellipse that governs it is the indicatrix turned inside out.

 What a cut buys. The mean angular deformation of the interrupted sinusoidal against the total length of cut the interruption spends, for lobe counts from one to twenty-four. Goode's interruption — the one actually printed — is the marked point: it spends 100 thousand kilometres and returns 18.0°, where the even-lobed curve returns 9.2° for the same length. It is not on the frontier and it was never trying to be: its cuts are placed to keep continents whole. The impossibility

What a cut buys

Six rungs count cuts and none measures one. A cut is a curve on the sphere with a length in kilometres, the shape distortion it removes is a falling function of that length, and the interruption everybody prints spends a hundred thousand kilometres to reach a figure the even-lobed curve reaches with sixty.

 The set a reach map shows, drawn from eight bearings. A geodesic disc of 4,000 km and the polygon a fan of eight bearings draws round it, on an equal-area azimuthal page centred on the disc so that the shaded ground is proportional to the ground it stands for. Every vertex of the polygon is on the true boundary and every edge between two of them is a chord, so the drawn set is inside the true one — always, at every count, for any convex reach set. The area it misses is 7.53% of 48,635,855 km², and it is not an error that care removes. It is what a finite fan is. Paths and directions

Every reach set ever drawn is too small

An isochrone is drawn by walking out along a finite number of bearings and joining the points, so its vertices are on the true boundary and its edges are chords — which puts the drawn set inside the true one, always, at every count, for any convex reach set. The deficit falls as the square of the count, and a spherical cap loses less than a circle by exactly cos t (1 + cos t)/2.

 What a 3-figure grid reference names. The square is the 100-metre cell a 3-figure reference stands for, and the filled dot is the point the reference was made from. A reference is TRUNCATED rather than rounded, so the digits name the square's south-west corner — the open dot — which here is 9.1 metres west and 82.8 metres south of the point. Rounding would have landed on the nearest corner instead, 9.1 metres away east–west. The truncation is deliberate: it is what makes a shorter reference a larger square containing the same point. Grids, and what a survey does

A grid reference names a square

This ladder has priced everything about a grid except how a coordinate on it is written. A grid reference is truncated rather than rounded, so it names the south-west corner of a square rather than a point in it — and a population of references is displaced half a cell each way, which is a bias rather than scatter and does not average out.

 Every one of the cube's 384 nets, scored. All 384 spanning trees of the cube's face graph, unfolded and scored on the total separation their cuts leave: how far apart, in edge lengths, the two copies of each cut edge end up on the page. Every one of them is a valid net — no Platonic unfolding overlaps — and they range from 11.30 to 17.97, a factor of 1.59. The heuristic of unfolding outwards from a chosen face lands on the first of them, exactly. The families

What the net heuristic cannot find

Rung seven chose a net by unfolding outwards from a face and showed the choice beats guessing, and recorded that its own limit was unknown. Enumerated in full, the heuristic turns out to be exactly optimal on every solid small enough to check — rank one of 384 — and above that ceiling no sample can tell whether it still is, because eight hundred random nets never reach it.

 What is left after the best rigid motion of the page. Every projection in the library, rotated about its own symmetry axis by five degrees, with the best rigid motion of the page fitted and the leftover measured against the picture's own size — on a logarithmic scale, because the answers span four orders of magnitude. Fourteen sit at 2e-6 or below, which is the axis search's own floor. Seven sit between 9e-3 and 3e-2. There is nothing in between, so the split is a fact rather than a threshold — and the fourteen are exactly the cylinders, the cones and the planes. The families

The family is a symmetry, not a shape

Rung seven dissolves the taxonomy: eleven named maps fitted to one perspective formula, six reproduced exactly and five refused, so the developable surface describes how the projections were discovered rather than what they are. What is left is the question it does not ask. If the shapes are not the classification, what is?

 A 6° query against a cube scheme, and the cells it fetches. The cells of a tangent-warped cube scheme at level 5, with the 29 cells a query of 6° radius touches shaded. The disc's own area is 16.84 cells; the count is 29, because every cell the disc's boundary crosses is fetched as well as every cell inside it. In Hilbert order those cells form six contiguous ranges of identifiers, which is six range scans, and the span from the lowest to the highest covers 91 cells against the 29 wanted. Drawn in Mollweide, with the mesh shown only near the query. What a machine does with it

A query is a disc, and a disc is not a cell

Everything a cell system does is an address lookup except the one question anybody actually asks it: find everything within five kilometres of here. That is a disc, and the number of cells it fetches is not its area divided by a cell's — at the radii a query is really made at, it is three to seventeen times that.

 The same six contours on Mercator and Lambert cylindrical. The value of a harmonic sum with a summit and a basin in the northern mid-latitudes travels with the point, so the set of points at a stated level is the same set on every map and each contour is exactly right on both panels. Everything a reader measures from them is not: the spacing between neighbouring contours, their lengths, and the area between two of them all change from one panel to the other, and the two panels are the same field. Measuring distortion

The contour is right and the reading is wrong

There is exactly one thing about a field that no projection can get wrong: which points share a value. The contour lines on any two maps of the same field are the same set of points. Every quantity a reader takes off them — the spacing, the length, the area between two of them, the hypsometric curve — is not, and the two kinds of map get different ones wrong.

 Shape predicted from field, against shape as published. Clairaut's theorem gives a body's flattening from two numbers of its gravity field: J₂, which is how its mass is arranged, and m = ω²a³/GM, which is how fast it spins. For a body in hydrostatic equilibrium the prediction is the shape, and the diagonal is where such a body sits. Earth is on it to 0.05 per cent — 12 metres at the pole, out of twenty-one kilometres of flattening. Mars is 12.5 per cent off it, which is 2.23 kilometres, and the excess is Tharsis: a body carrying a continent-sized volcanic load is not a fluid figure, so its ellipsoid is not one of its own level surfaces, and its zero of height has to be chosen rather than found. What the numbers refer to

A body with no sea level

On Earth the zero of height is found rather than chosen — water settles onto the equipotential surface by itself. Nowhere else has one, and the difference is measurable: Clairaut's theorem predicts the Earth's flattening from its own gravity field to twelve metres at the pole and misses Mars's by 2.2 kilometres.

 The set of aspects within a stated distance of the best, for Robinson over Japan. Each row takes every point of a 36 × 19 × 24 grid in the three aspect parameters that scores within (1 + t) of the best, joins neighbouring points, and identifies the pieces the exact degeneracy relates. At t = 3 it is one connected piece spanning 170° of pole; by t = 1 it has broken into 14 pieces; and by t = 0.3 the largest of them spans 11°. So it is not one valley and it is not one basin — it is a sheet that fractures. What each projection optimises

The shape of the valley

An aspect search returns three numbers, two searches return triples that differ by a hemisphere, and the maps they produce agree. One cause is an exact degeneracy and the rest was called a valley and left unmeasured. Sampled densely, it is neither a valley nor a basin: a connected sheet spanning 170° of pole that fractures into fourteen pieces once the threshold tightens.

 Bigger triangles, and how much bigger depends on what is fixed. How well a survey can resolve Gaussian curvature, against the side of its triangles, under three things being held fixed. One triangle: the accuracy improves as the inverse SQUARE of the side, fitted exponent -2.000. A chain of fixed length, which is what every great arc was: bigger triangles mean fewer of them, and the exponent is -1.503 — exactly three halves. A network covering a fixed area: -0.999, exactly one. The trade depends on what a survey is short of. The impossibility

How many triangles it takes

Rung eleven priced one triangle and recorded that a survey observes hundreds. Averaging n of them divides the noise by √n, and n is not free: a chain of fixed length holds fewer big triangles than small ones, so the accuracy improves as the side to the power three halves rather than two. Struve's 141 triangles of forty kilometres are worth exactly Gauss's seven of eighty-five.

 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.

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