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

Error budget — where it appears

The list of the separate error sources in a result, with each one's size, so that the largest can be found. A projection error quoted without one is unactionable: it may be the dominant term or a rounding error on the data, and the number alone does not say which.

Named by 13 essays across 6 fields — each of them below, with the objects they name alongside it.

What each configuration can see, and what it cannot. The smallest eigenvalue of the fit's own normal matrix — how much the residual changes for a unit move in the worst direction of parameter space — for three parameterised candidates against six configurations of the same size. A zero is not a hard fit: it is a direction the control points cannot see at all, so every value of the parameter along it gives an identical residual. 2 of 18 are at the floor of double precision, and they are not the ones a reader would guess. Where none is, the spread between the best and worst arrangement is still Infinity at a fixed point count.

Where the control points are

Five rungs fit a library to a map and ask how much to trust the winner. None asks whether the parameters are recoverable at all. On control points along one parallel an equirectangular's standard parallel is not merely hard to find — it is invisible, exactly, and five hundred and twelve points on the same parallel are as blind as eight.

wrong · Identify 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

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.

impossibility · Curvature 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.

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.

practice · Reduction 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.

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.

distortion · Gradient What a geoid model leaves out, against the degree it stops at. The RMS of everything above the model's highest degree, from Kaula's rule — the statement that the normalised coefficients at degree n are about 10⁻⁵/n². The line is R × 10⁻⁵ ÷ n, so a model to degree 360 omits 17.7 centimetres and one to 2190 omits 2.9. Every orthometric height derived from such a model carries that as an error, and it is not quoted with the height.

The geoid model stops at a degree

Eleven essays treat the geoid as a surface that exists. Every geoid anybody uses is a series truncated at a degree, so every orthometric height derived from one carries an omission error nobody quotes with the height — eighteen centimetres at degree 360 — and the same truncation removes two thirds of the slope, which does not converge at all.

datums · Height The solve's cost is a U in the shape; the curvature is not. Solid: how many conjugate-gradient steps the conformal solve needs, against the window's aspect ratio, at constant surface area. It has a minimum at the square window — 174 steps — and rises to 265 and 275 at the two extremes, which are elongated by the same factor in opposite directions. Dashed: the total Gaussian curvature the window encloses, on its own scale, which falls from 0.311 to -0.012 across the same sweep and is least at one end of it. The cost has its minimum where the window is square and the curvature has its minimum somewhere else, so whatever is making the solve expensive is not what is making the map spread.

A long window and a square one

Rung nine held the patch's area and found that where the cut goes still changes the map by a third, with the curvature it encloses predicting the change at r = 0.969. It recorded that it had held area and curvature and not shape. Sweeping the shape at constant area separates two things that had looked like one: the map is curvature and the cost is shape.

datums · Bodies What a seven-parameter fit's residual is made of. A published transformation accuracy is the root-mean-square residual at the common points, and here it is 1.51 metres. Almost all of it — 1.51 — is the network's own distortion, which a rigid motion and a scale cannot follow and which is present at every point of the country whether it was used in the fit or not. The transformation's own error, measured as the disagreement between the fitted parameters and the true ones over a clean grid, is 0.091 metres: 6 per cent of the quoted figure. The last bar is the control — the same fit with the distortion switched off, at 2.9e-4 metres.

What another common point buys

Rung three finds that a seven-parameter datum fit leaves a pattern rather than noise. Six per cent of the residual it reports is the transformation's own error and the other ninety-four is distortion no seven parameters can follow — so adding common points improves a term that was already small and cannot touch the one that is quoted.

datums · Datum What a rebinning loses depends on where the target's edges are. Two grids of fixed counts, fixed shapes and fixed resolution, with the target slid across the source from perfect alignment to a full cell. Nothing about either grid changes except where its boundaries fall. The loss runs from 27.5 per cent at zero to 56.3 at half a cell — a factor of 2.04 — and the longitude-only curve returns to its starting value at a full cell to six decimal places, which is the periodicity check. A cell boundary that coincides with a target boundary loses nothing, and a grid comparison that does not say where its boundaries are has left that out.

When the edges do not line up

Rung eight held the cell counts equal so that shape could be compared without the count ratio drowning it, and recorded a doubt: a longitude–latitude source shares its boundaries with a longitude–latitude target wherever their counts share a factor. The mechanism is real and worth a factor of two. It was not what the published number was made of.

applied · Cells The same 2-pixel road at three latitudes, zoom 5. The dark bar is the mark as drawn — 2 pixels, identical in all three panels, because that is what the stylesheet says. The pale band behind it is the ground that mark covers, drawn to one common ground scale: 9.78 kilometres at the equator, 6.92 at 45° and 1.70 at 80°. The reader sees the dark bar and is being told about the pale one.

The road is drawn two pixels wide

Seven rungs measure what a screen map does to position. Nothing on a map is a point: every mark has a width, the width is chosen in pixels, and a two-pixel road covers 9.78 kilometres of ground at the equator and 1.70 at 80° north. That is a generalisation applied at a strength varying by a factor of six across one sheet, by a stylesheet with no latitude in it.

applied · Screen 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 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 The bias is σ² over the length, over two decades. The amount by which a measured baseline is longer than the true one, against its length, for a twenty-millimetre error on each end. The line is the second-order prediction σ²/d. The measured bias times the length is constant to a factor of 1.0014 across the whole range, and it sits 8.2 per cent below the prediction — which is the fourth-order term the expansion drops. The estimate is antithetic, so the first-order scatter cancels exactly and a bias of thirty-seven microns is measured at a t-statistic of 349.

A length measured from noisy points is too long

A distance is a square root, a square root is concave, and the average of the distances is not the distance between the averages. The gap is a bias with one sign: 37 microns on a ten-metre baseline with twenty-millimetre marks, following σ² over the length across two decades, and it adds rather than cancelling — so the same boundary is 1.5 parts per million longer when it is measured in more pieces.

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.

VerificationConditioningNoisePrecisionLeast-squaresResolutionStandard errorAveragingResidualSurveyToleranceAggregation

All concepts