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

Correlation — where it appears

How far two orderings of the same objects agree, used here to compare rankings by different criteria rather than to relate measurements. Spearman's rank version is used here because the criteria produce scores on incomparable scales and only the ordering they imply is worth comparing.

Named by 11 essays across 6 fields — each of them below, with the objects they name alongside it.

Six projections, at 40° north. Flexion (solid) and skewness (light) against direction of travel, drawn about a zero circle at one point of each projection. The three conformal ones have curves of exactly equal size, offset by exactly 90°, because on a conformal map both quantities are components of one vector — the gradient of the log of the scale. The others have neither property: ratios run from 0.72 to 2.72.

Bending and stretching are one failure

The ladder was written expecting flexion and skewness to be two independent ways for a map to be wrong. On a conformal projection they are not independent at all: the two extremes are the same size to four figures and sit exactly ninety degrees apart, because both are components of a single vector.

distortion · Flexion Which projection wins, by the first derivative and by the second. Each column is a region, with the projections listed in the order Kavrayskiy's first-order criterion puts them in and the figure at the right of each row giving that projection's rank under the second-order criterion — flexion and skewness aggregated the same way. The rank correlations are Europe 0.83, the conterminous United States 0.81, the tropics 0.90, a cap of 30° radius 0.88, so the two orders agree broadly and disagree in detail. Where it matters is the winner: over a cap of 30° radius the choice moves from Albers equal-area conic to Lambert azimuthal equal-area, while Europe and the conterminous United States and the tropics keep theirs. A criterion that changed every answer would be suspect and one that changed none would be decoration.

The second derivative over a region

Flexion has been measured at points and over the whole sphere, and never over a region — which is the only unit anybody chooses a projection for. Doing it finds that the second-order criterion moves the winner in one region of four, and that one projection in the library has no second derivative at all.

distortion · Flexion 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.

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.

distortion · Precision The same resolution, the grid moved, and a different answer. The same field at a fixed 18 × 9 division, with the grid slid by fractions of a cell. Nothing is lost — every cell is the same size as before and there are exactly as many of them — and the largest reported value moves over a range of 12.59 per cent. That is more than a whole halving of the resolution costs, which is 10.15 per cent on the same field. The scale effect has an excuse and this one has none.

The answer depends on the cells it was counted in

Nine essays price the cell as a shape. The number reported out of it is priced nowhere: sliding a grid without changing its resolution moves the largest reported value by 12.6 per cent, which is more than halving the resolution costs.

applied · Cells The mountain's gravity follows the ground and the root's does not. A profile 260 km long across the stated mountain patch. Top: the height of the ground. Bottom: the gravity at the ground from the terrain's own rock, a sheet at 2,670 kg/m³ (solid), and from a root at 30 km carrying the same mass with the opposite sign (dashed). Over the whole patch the rock's gravity correlates with the height at 0.99, so of its 54.0 mGal rms only 7.8 is not a function of the height under it. The root's, smoothed over thirty kilometres, correlates at −0.85, and of its 10.0 mGal 5.3 is free of the local height. A levelling loop misses by exactly that free part, which crosses the ground's contours.

A levelling loop weighs the root, and a triangle weighs the mountain

A mountain range is carried partly by a root of light crust beneath it, and a survey that knows the ground can ask how much. Put a root thirty kilometres down under a stated range and it moves a levelling loop's misclosure by 85 per cent as much as the mountains' own rock does, and a triangle's excess by 16 per cent: the root's gravity is a smoothed copy of the ground, and a smoothed copy crosses the contours. Read against their noise, pinning the root's share to a tenth takes about 2,700 fifty-kilometre loops, a million triangles or 550 astronomic stations — and if the rock's density is unknown too, a triangle cannot tell a compensated mountain from a lighter one.

impossibility · Curvature A triangle never sees the root against its own noise; a levelling loop sees it until it is deep. For a root at each depth carrying all of the terrain's mass, the rms of each reading's response to it over 160 placements, divided by that reading's noise — the background the root sits in plus the observing error. A 25 km loop: 34.2 per cent at 10 km, 15.6 at 30, 3.9 at 60. A 50 km loop: 37.6 per cent at 10 km, 20.1 at 30, 5.5 at 60. A 100 km loop: 34.9 per cent at 10 km, 20.2 at 30, 6.1 at 60. A 50 km triangle: 3.1 per cent at 10 km, 1.0 at 30, 0.2 at 60. The triangle's response falls with depth more steeply than the loop's, which is what made their ratio look like a gauge of depth, but at every depth it is a few per cent of its noise or less.

A loop reads a root's depth from its own pattern, and a triangle reads none of it

A levelling loop and a geodetic triangle respond to a buried root in proportions that change with its depth, so their ratio looked like a gauge of how deep a range is compensated. It is not, because a triangle's response to a root at 30 km is a hundredth of its own noise. The depth is there in the loops alone: a root changes the pattern of misclosures from loop to loop as it deepens, and 1,600 loops of 50 km place a 10 km root within a factor of 1.5 three times in four and a 30 km root about half the time. A 60 km root is not placed even by 6,400, and when the depth is fitted rather than stated, such a root is read as a light shallow one carrying 25 per cent of the mass where it carries 60.

impossibility · Curvature Seven published numbers, twenty-eight actual ones. The correlation matrix of a seven-parameter fit to 64 common points over a region 9° across. Only the diagonal is ever published — seven standard deviations — and the twenty-one off-diagonal entries are not small: the strongest is ty against rx at 0.940. A translation and the rotation that mimics it over a small patch are very nearly the same parameter, so the fit cannot tell them apart and its errors in the two are locked together.

The parameters are not independent

Rung seven gives the seven parameters their own uncertainty and stops at seven numbers. There are twenty-eight, and the twenty-one nobody publishes are not small: a translation and the rotation that mimics it correlate at 0.94, the normal matrix has a condition number of 4 × 10¹⁶, and propagating from the diagonal alone overstates the transformation's uncertainty by up to a factor of thirty-six.

datums · Datum Beside a bending river, every reachable point is reached by a route that meets the bank at most three times. Everywhere within a 10 km walk of the centre, beside the stated river meandering ±2 km on a 4 km wavelength with bridges every 2.5 km, shaded by the fewest times a shortest route to it has to meet the bank — a contact being one run along the bank, however many corners it passes. 70.3% of the ground is seen from the centre across dry land. One contact reaches 28.0% more, 85% of it beyond the river and the rest in bays of the near bank hidden from the centre by a spur; two reach 1.18% and three the last 0.47%, 70% of those two beyond the river, in the lee of the far bank's bends beside the outer bridges. Nothing needs a fourth.

A layer is a meeting with the bank, not a corner turned

A reach set drawn in layers — rays from the centre, then from every place a route can leave the bank — ends in the exact answer, and how deep it must go depends on what a layer is. Counted by corners turned, the depth belongs to the polygon the bank is drawn as: five turns on a bank drawn every 400 metres, eighteen on one drawn every 50. Counted by meetings with the bank, it is three at every resolution. One meeting anywhere leaves 1.6 per cent of a ten-kilometre walk unreached, where rays from the bridges alone leave 6.4; two leave 0.47; three leave nothing. A longer walk needs more — up to six meetings at twenty kilometres across thirty-six rivers — and the bend's steepness, which prices the first layer, ranks the depth at 0.24.

paths · Reach Averaging divides the hand's wander and leaves its habit: a shared part sets a floor no stack goes below. The drift left in the mean of N tracings, as a share of one tracing's, when a stated part of the hand's variance is a habit shared by every tracing: none (thick), a quarter, and half. With none it falls as one over the square root of N, to 25 per cent at sixteen; a quarter shared stops it at 50 per cent, half at 71. Dotted: the drift at which an independent hand's stacks read the strong seam's easing (two tracings, 71 per cent) and the weak seam's (sixteen, 25).

Averaging tracings reads the easing, until the hand repeats itself

One tracing by a drifting hand cannot show how far a copy was eased, because an easing as long as the drift is a stretch of drift. Average several tracings and the easing stays while the drift shrinks: two tracings read the easing's width at the seam where the copy departs strongly, sixteen read it at the seam where it barely departs, and as the drift falls the rigid count starts buying pieces for the easing again while the eased count stays at two. All of that assumes each tracing drifts afresh. A hand that repeats part of its drift in every tracing leaves that part in the mean, where it sets a floor no number of tracings goes below; and a digitiser who learns the hand from the spread between tracings measures only the part that averages away, and charges the mean too little, so that both counts buy every piece they are offered.

wrong · Identify Sixteen tracings by four hands leave half the drift that sixteen by one hand do, unless the hands share a habit too. The drift left in the mean of sixteen tracings of one sheet, as a share of one tracing's, measured over 120 sheets, for sixteen tracings by one hand, eight each by two, four each by four, two each by eight and one each by sixteen. When half of each hand's drift is its own habit (darker): 1 × 16 74, 2 × 8 54, 4 × 4 40, 8 × 2 31, 16 × 1 25 per cent. When a quarter is the hand's habit and a quarter a habit every hand in the office shares (paler): 1 × 16 73, 2 × 8 63, 4 × 4 58, 8 × 2 56, 16 × 1 54. Dotted: 71 and 25 per cent, the drift at which an independent hand's means read the strong seam's easing and the weak seam's.

The spread between hands charges the mean, and an office's habit hides in the sheet

One hand's tracings cannot see the part of its drift it repeats, so a mean of sixteen by one hand is charged for a third of the error it carries. Sixteen tracings by four hands carry half the error of sixteen by one, and the spread between the hands' means, divided by the number of hands, charges that mean correctly to within a fifth on every sheet from three hands on — without ever estimating how much of a hand is habit. The same spread reads a hand's share of habit from two hands of two. What it cannot read is a habit every hand in the office shares: tracings of a sheet with one are tracings of a different sheet without it, and the charge they give falls further short the more hands and tracings are added.

wrong · Identify Waiting longer buys almost nothing under the spectrum that is actually there. The scatter of block means against the block length, for the three spectra. White noise falls with a slope of -0.484 — the inverse square root everybody assumes. Flicker falls with a slope of -0.111, so a hundredfold longer average buys a factor of 1.67 rather than ten. A random walk is very nearly flat.

The error that does not average down

Five rungs give a coordinate a width, and all five assume the observations it was averaged from are independent. They are not. Under the noise spectrum every published analysis of a position time series reports, the standard error falls as the eleventh root rather than the square root — so a factor of ten costs a hundred observations if the noise is white and a thousand million if it is not.

distortion · Precision

Named alongside it

The objects these essays reach for when they reach for this one.

Least-squaresNoiseAveragingEstimatorIdentifiabilityPrecisionVerificationBiasCompilationCovarianceDatumError budget

All concepts