Residual — where it appears
Named by 24 essays across 5 fields — each of them below, with the objects they name alongside it.
Where a fit leaves residuals
Seven parameters can carry a rigid motion and a size exactly. A triangulation network is neither, so the best possible transformation between two datums leaves metres on the table — in a pattern, not as noise — and which seven parameters come out depends on where the markers were.
When a formula is not enough
National mapping agencies distribute datum shifts as tables of numbers on a grid rather than as parameters. What decides the spacing that table needs is not the size of the shift — a hundred-metre shift tabulates coarsely, and a thirty-centimetre ripple riding on it does not.
Three conditions are one too many
Two distances fix a point in a plane and a third has no freedom left to be satisfied with. Chamberlin's trimetric construction averages the three positions that satisfy two conditions each, and the spread between them — never zero anywhere, 22 km over North America, growing as the cube of the region — is the price of the extra clause.
A map does not say what it is
Every map on this site is one the site drew, from a projection it chose. Every map a reader has ever used is the other kind — a picture whose projection is a sentence in a corner, a legend, or nothing at all. The graticule is enough to recover it: fit every candidate to the crossings and rank what is left over.
Two projections that cannot be told apart
Over a small enough region every projection is the same picture, so the question is how small. The answer is not one number: two conformal projections need twelve degrees of extent before their graticules can be separated, and two projections with different anisotropy separate below one.
A local model has an order
Every georeferencing tool fits a polynomial between two coordinate systems and the choice of degree is usually made by counting control points. What it buys is an order of convergence — 2, 3 and 4, measured — and the first term an affine model cannot hold is the second derivative this ladder has spent five essays on.
What a careless copy hides
A photocopier that stretches one axis is a nuisance to remove before a projection can be identified. Removing it costs more than it looks: an affine fit absorbs the entire difference between the cylindrical equal-area projections, so Gall-Peters and Behrmann become the same picture at any size, to sixteen decimal places.
Two parameter sets, one transformation
Agencies publish seven-parameter datum transformations that differ by hundreds of metres in translation, and the usual reading is that one of them is better. Over the region either was fitted to they are the same transformation: a hundred metres of translation, re-absorbed by the rotations and the scale, moves a British coordinate by 5.6 metres and an Australian one by 193.
When the answer is not in the library
Fitting twenty candidates to a map and ranking the residuals always produces a winner, which makes it a ceremony unless it can also produce a refusal. Held out of its own library, a Mercator map is named as a conformal conic, leaving 0.4 per cent of the map's width unexplained — and the quantity that tells the two situations apart is not the residual but the margin, which is 1.6 when the truth is absent and 10¹³ when it is present.
A residual has more than one explanation
The method names a projection by fitting every candidate to a set of control points and taking the smallest residual. It has never been asked what else a small residual could be. A map drawn in the right projection from coordinates on the wrong datum leaves a residual of one part in a million — indistinguishable from noise, at every region size, because a similarity fit absorbs a datum shift almost exactly.
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.
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 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.
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 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.
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.
The nearest equal-area map to an impossible request
Every number in the previous rung is inside the conformal achievable set, because Liouville's is the conformal condition. The equal-area set is one equation on two functions and never refuses: the same four requests are met to nine parts in a billion, and charged for in angle instead.
A rotation is not absorbed the way a shift is
The previous rung expected a datum's rotations to be the part a plane fit could not swallow. They are the part it swallows best — 93 times better than the translations over a hemisphere, and 24 times better per metre moved — and the reason is that the rotation which matters is a change of longitude, which is a symmetry of the map.
A condition imposed at points is not a condition
Nine rungs state a condition and solve it, and every solve imposes the condition at a finite set of samples because that is what a linear system is. With barely more equations than unknowns the residual the solver reports is 8.3 times too good — and refining the collocation twentyfold does not improve the map at all, it only makes the report honest.
The answer is a set
Eight rungs have produced a best fit — one projection, ranked first, with a margin. A best fit without a spread is not a measurement, and the spread is free: a control point has a width, and every candidate whose residual is inside that width has not been ruled out. At one per cent noise a four-degree region admits ten of twenty candidates and a forty-degree one admits exactly one.
The nodes were evenly spaced
The previous rung showed that refining an evenly collocated fit improves the solver's report and leaves the map alone. Moving the same number of nodes to the Chebyshev positions — crowded toward the corners, where a conformal map of a polygon is singular — makes the fit settle: 7.030 × 10⁻⁵ at forty-eight nodes and exactly that at every count above it, against an even fit that wanders by 43 per cent and never converges at all.
The sheet moved before it was measured
Nine rungs take control points off a map and assume the sheet they came from is the sheet the cartographer drew. Paper shrinks across its grain three times as fast as along it, and on a map whose grain runs along its own axis that shrinkage is EXACTLY a change of standard parallel — one per cent moves the recovered parallel by 0.57 degrees with the residual sitting at the solver's floor.
A map with no graticule
Ten rungs are handed control points, and a great many maps have none. Handed an outline with no labels on it at all, the method still works — and works better: the correspondence between ink and ground is recoverable exactly, because a similarity preserves ratios of arc length, and the margin on clean observations is 1.6 × 10¹⁰ against a graticule's 9.9 × 10⁶. What breaks it is noise, at three parts in a thousand.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Least-squaresVerificationToleranceSimilarity transformationDatumIdentifiabilityConformalityDegeneracyControl pointsIdentificationProjection identificationConditioning