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

Averaging — where it appears

Taking a mean of repeated observations, which reduces the standard error as the inverse square root of their number only when they are independent. Under a correlated spectrum it falls far more slowly, and the hours that buy nothing are most of them.

Named by 10 essays across 5 fields — each of them below, with the objects they name alongside it.

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.

How many triangles it takes

Pricing one triangle left the fact that a survey observes hundreds unaccounted for. 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.

impossibility · Curvature 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.

A family is not closed under averaging

A family is a set of maps with a parameter running through it, and the compromise projections here 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.

families · Families How aligned a region's ellipses are, for every projection and every region. The resultant length of the doubled indicatrix orientations, over 12 projections and 9 regions. One means every ellipse in the region points the same way; zero means they are spread evenly and cancel. Three cylindrical rows are 1.00 throughout, and everything else varies down the row and across it — which is the answer to the question the number was first stated without: alignment is a property of the pair. Mercator's row is the exception that is not a measurement, because a conformal projection's indicatrix is a circle and a circle has no orientation.

Whether the ellipses point the same way

Alignment was found to decide whether the average of a region's deformations is above or below the deformation of its average, and stated it at ten projections over one region. Swept over a hundred and eight pairs, the answer is that alignment belongs to the projection 28 per cent, to the region 38, and to neither 33 — and two rows of the table turn out not to be measurements at all.

distortion · Tissot A season of crossings pins the stretch to a tenth only if it learns from where the streak was. The share of 400 seasons at each length whose learned stretch lies within a tenth of 1.25. From the floors' rank histogram: 25 crossings 19 per cent, 50 crossings 27 per cent, 100 crossings 32 per cent, 200 crossings 43 per cent, 400 crossings 55 per cent, 800 crossings 67 per cent. From where the streak turned out to be: 25 crossings 56, 50 crossings 72, 100 crossings 85, 200 crossings 98, 400 crossings 99, 800 crossings 100. At a hundred crossings, 85 against 32 per cent.

A season learns the stretch from where the streak was, not from the floors

An ensemble whose members spread four fifths as widely as the forecast's real error covers the diversion floor in 94 crossings of a hundred instead of 95, and the repair is to stretch every member away from the centre by a factor learned from past crossings. Learned the way the rank histogram teaches it — from where the truth's floor fell among the members' floors — a season of a hundred crossings lands within a tenth of the right stretch one time in three, and a season of eight hundred two times in three. Learned from where the streak turned out to be, which the analysis after each crossing supplies whether the streak reached the gap or not, a hundred crossings land within a tenth 85 times in a hundred and two hundred almost always. The floor is the quantity the plan is about; it is the wrong quantity to learn the spread from.

paths · Paths 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 After a revision narrows the ensemble, a learner that never forgets is still well short of the new stretch three hundred crossings later. The stretch each learner holds before each crossing, the median over 400 winters, from 100 crossings before a forecast revision to 300 after it. Before it the members spread 0.8 of the truth's spread and the right stretch is 1.25; after it they spread 0.6 and the right stretch is 1.667 (dashed). never forgets: 1.28 at 25 crossings, 1.34 at 100, 1.44 at 300; memory 200: 1.31 at 25 crossings, 1.44 at 100, 1.59 at 300; memory 50: 1.42 at 25 crossings, 1.62 at 100, 1.68 at 300; restarts on the date: 1.63 at 25 crossings, 1.67 at 100, 1.67 at 300. Shaded: the middle eight winters in ten for the learner with a memory of 50 crossings, 1.55–1.78 at the end.

A revised forecast is relearned in as many crossings as the learner remembers

An ensemble's stretch is a property of the forecast system, and forecast systems are revised. When a revision narrows the members from four fifths of the real spread to three fifths, a learner that weights every crossing since the start of the winter equally still holds 1.44 for a new stretch of 1.67 three hundred crossings later, and leaves the plan under the floor in 93.8 crossings of a hundred instead of 95. A learner that forgets regains ninety-five per cent coverage after about as many crossings as its memory — 31, 57, 113 and 206 for memories of 25, 50, 100 and 200 — and pays for forgetting only when nothing has changed. A memory of fifty costs nothing measurable over a winter. And a bias the whole ensemble shares, which the stretch alone reads as extra spread, is found to a kilometre by a learner that tracks the mean departure as well.

paths · Paths 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 An office that overshoots headlands draws every coast wrong by that coast's own curvature. Four coasts at one map size: the compiled coast whose seams are counted, and three control coasts an office might trace with the truth known — an ellipse as thin as the counted coast (axis ratio 2.88), a nearly round one (1.12) and a coast of five headlands. Solid: each coast. Dashed: the same coast as an office would draw it whose every hand pushes a headland out and pulls a bay in by a fixed amount per unit of curvature, exaggerated a hundred and twenty times so that it shows. The curvature's spread round each coast, its variance in units of the coast's size: the sheet being counted 16.9, control, as thin as the target 15.6, control, nearly round 0.1, control, five headlands 29.8. The habit's size on a coast follows that spread, and the habit's slope does not change from coast to coast.

A known coast charges an office's habit, and a bending one takes it out

An office whose hands all share a habit leaves it in every tracing of every sheet, where the spread between hands cannot see it. One control sheet — a coast known independently, traced the office's usual way — reads the habit's size and restores the charge, if the habit is a drift. If the habit answers to the coast's own curvature, a size read on a rounder coast charges a thinner one at little more than half, but the same control read differently does better than charge the habit: its departures, set against its curvature, give a slope that is the office's own on every coast, and subtracting it leaves about one per cent of the habit in the mean. The slope also tells the two kinds of office apart. A coast that hardly bends teaches neither, and corrects worse than nothing.

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 Two places, one written coordinate. The lattice a coordinate written to 5 decimal places lives on, at 45° of latitude, where a cell is 1.11 metres north to south. Two real places a fifth of a cell apart are drawn as open marks and the single value they both round to as a filled one. The distance between them, computed from what was written, is exactly zero — an error a noise model cannot produce, since independent noise never puts two different points in the same place.

Rounding is not noise

Eight rungs treat a coordinate's error as noise that averages down. A published coordinate has a second error that does not: it is deterministic, it is shared between every point in the same cell, and at five centimetres apart ninety-three per cent of pairs come out as the same place — which no amount of independent noise can produce.

distortion · Precision

Named alongside it

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

BiasNoiseEstimatorVerificationCompilationCorrelationError budgetResidualSeamStandard errorClosed formCoverage

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