Grids, and what a survey does
A grid has an origin that is not there
Every published national grid measures from a point in the sea. The British one sits 400 kilometres west and 100 kilometres south of where the projection's own zero is, and moving it changes every coordinate in the country and not one distance or bearing.
What a grid is made of
A projected coordinate declares five things — a datum, a projection, its parameters, a unit and an axis order — and four of them are invisible in the numbers. Every one has a documented case of being guessed wrong, and the cheapest of those cost a hundred metres.
The scale factor of a line
A scale factor is a property of a point and a distance is not measured at a point. The three rules in use for averaging one along a line differ by a hundred metres on a 362-kilometre baseline, and which of them is adequate is decided by the line's direction rather than its length.
The scale factor was chosen
A grid's scale factor is not a constant of nature. Britain's 0.9996012717 is the reciprocal square root of the worst distortion a tangent projection would have had over the country, it halves the worst-case error exactly, and the halving is the most that construction can ever buy.
The units are part of the coordinate
Two feet were in use in the United States until 2022, differing by exactly two parts per million. A plan drawn in the wrong one has every dimension right to a fifth of a millimetre and sits three-quarters of a metre from where it belongs, which is why it passes every check anybody runs.
Designing a grid for one region
A country with a grid of its own beats the international zone that would otherwise carry it by a factor of 1.51. That is worth having and it is much less than the argument is usually made to sound, and the difference between the two numbers is the whole case for and against national grids.
Where two zones meet
A twenty-kilometre baseline across a zone boundary is computed correctly in either zone — the two answers differ by nanometres — and taking one end from each returns 392 kilometres. Both zones are right and the seam between them is not a small error but a category one.
A grid scaled to the ground is not a map
Construction sites work in coordinates where a tape reading equals a computed distance, which is achieved by multiplying the national grid by a constant. The result is exact at one point, degrades east–west forty times faster than north–south, and is not a map projection at all.
What a tape measures
Four steps stand between an instrument reading and a coordinate, and their sizes are not in the order anybody expects. On a 77-kilometre line the slope reduction is 120 metres, the grid reduction 30, and a step nobody names — using the height above sea level where the height above the ellipsoid is wanted — is 583 millimetres.
A traverse must close
The check every survey specification leans on cannot see the error this whole field is about. Scale every leg of a closed figure by four hundred parts per million and it closes to the last bit of a double, while every dimension in it is wrong.
The two ways to spread a misclosure
Bowditch's rule and the Transit rule take the same closed figure and the same misclosure and disagree about which legs were wrong. Both close it exactly, neither puts the correction on the leg that actually carries the blunder, and no measurement can settle which is right.
Setting out runs the chain backwards
Turning a design coordinate into something to observe on the ground means undoing the reduction chain, and undoing a chain reverses the order as well as the operations. Two of the corrections do not commute, and getting them the wrong way round misses by a millimetre at fifty kilometres.
Two grids over the same ground
One point in the English Midlands is 406,788 east on the British grid and 167,478 east on UTM. Separating the difference properly leaves a surprise — only one of a grid's five parameters changes the map, and it is not any of the four that dominate the number.
The tolerance decides the model
Every correction inverts to a distance — the line length at which it alone exceeds a stated tolerance. The ranking those distances produce is the same at a millimetre and at a decimetre, and it is different for two jobs at the same tolerance on different ground.
A published coordinate is a result
Re-observe a control mark perfectly, with an instrument good to twenty millimetres, and the answer disagrees with the published value by a hundred metres. Neither number is wrong. The difference measures the interval between two definitions.
An area on the grid is not an area on the ground
A grid's scale factor is a property of lengths and what a surveyor sells is an area. Squaring a departure doubles it, so a hectare drawn on the British grid at its central meridian has 10,008 square metres of ground under it — and correcting an area with the line factor instead of its square leaves half the error behind.
What a closed figure cannot see
A closed traverse imposes exactly two conditions on its observations, so everything else is free — and the freedom is not spread evenly. At a twenty-millimetre tolerance the worst-placed station in a seven-station loop hides an angle blunder of 2.1 arcseconds and the station standing seventy-two metres from the close hides 57, because a rotation about a point near the finish moves the finish hardly at all.
The best grid a country could have had
A national grid is a conformal map chosen for one region, so its whole design problem is one number: the spread of its scale factor. That number has a theoretical floor, this site can now compute it, and the adopted grid turns out to be either exactly optimal or half as good again — depending entirely on which box the country is declared to be.
The chain the satellite does not have
The four-step reduction from a tape reading to a grid coordinate exists because a tape does not know where it is. A satellite observation does, so the same distance can be got by projecting both ends and subtracting — and the two answers agreeing to a fifth of a millimetre is the first check the chain has ever had.
The bound on the body the country is on
The best conformal grid a country could have had was computed against a bound solved on a sphere, with a note saying the flattening was second order and unquantified. It is second order for a named projection — every family's best candidate moves by under 0.7 per cent — and it is 22 per cent for the bound, because an optimal map has already cancelled its own variation and has nothing left to hide a new one in.
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.
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.
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.
The weights are a guess the solve believes
Choosing what to hold fixed is a decision inside the least-squares problem that no residual can see. 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 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.
Further north on the grid is not further north
Thirteen rungs price a grid's origin, scale, units and zones, and every one treats a coordinate as a position. It is also an ordering, and the two disagree: a parallel climbs 4,379 metres of northing on its way to the edge of a UTM zone, so two places 4.4 kilometres apart in latitude can be listed in the wrong order — and the share of pairs it happens to is the convergence, in a different unit.
The network's answer is decided before it is measured
An adjustment is usually judged by what it does with observations. Every quantity a specification is written about — the error ellipses, the redundancy numbers, the smallest detectable blunder — is a function of the geometry and the weights alone, and does not contain an observed value anywhere. Four layouts of five stations with the same ten distances differ by a factor of 20.9 in their worst coordinate.
One pair of numbers, a hundred and twenty places
A UTM coordinate is two numbers and a zone. Drop the zone and the numbers are still valid in each of the sixty; drop the hemisphere too and the pair names a hundred and twenty places. They form two rings at one latitude each, spaced exactly six degrees apart, and every one of them has the same grid convergence and the same scale factor — so no further geometric measurement can choose between them.
One sentence, and the ground between its readings
Every land boundary in the world is defined by a sentence, and a sentence naming two monuments does not name a curve. "A straight line" between the Rio Grande and the Colorado admits at least three answers 7.53 kilometres apart at their worst; "the forty-ninth parallel" admits four, 96 kilometres apart, with 130,972 square kilometres between the extremes.
The line a commission can actually run
A boundary commission cannot run a parallel of latitude. It can sight a straight line between monuments, and a straight line between two points of equal latitude passes poleward of the parallel — by s² tan φ / 8R, which at a mile of spacing is fifty-eight millimetres and at a hundred kilometres is two hundred and twenty-six metres. The described line and the marked line are different curves, and the marked one governs.
A meridian boundary moves when its datum does
The 141st meridian is the one boundary description in this collection with no geometric ambiguity in it: every reading of it is the same curve, exactly. It has a different one. A longitude refers to a datum, the 1825 convention named none, and the line of longitude exactly 141° west sits 129.0 metres apart on NAD27 and WGS84 — a ribbon 1,039 kilometres long and 134 square kilometres in area.
An equidistance line belongs to a surface
A maritime boundary is very often defined as the line equidistant from two coasts — a description with no coordinate in it and no curve to choose between. It has a third ambiguity: equidistant measured how. On the ellipsoid, on the sphere, and with a ruler on three different charts, the same four basepoints give lines up to 39.6 kilometres apart and 2,816 square kilometres of seabed between them.
A tripoint defined three times
A tripoint is very often not a coordinate in any treaty. It is a description — the point where the boundary between A and B meets the boundary between B and C — and each of those boundaries is itself a description. So the point is defined three times, once by each pair, and under one convention throughout the three definitions agree to half a micrometre. Under three they enclose 6.69 square kilometres.
A river boundary goes where the river goes, or stays where it was
A boundary that follows a river moves when the river moves, and the law of every such boundary turns on one word: whether the river moved gradually or suddenly. Over one meander's life, gradual movement changes the country of about twenty-eight square kilometres of land a loop. A single flood that cuts the loop off strands twenty-six — nearly the same land, arriving as one parcel on the wrong side of the river.
The highest summits stay where they are, and the watershed does not
A treaty that puts a boundary on the highest crests which divide the waters names one line where the water divides on the summits and two where a river has cut back through the range. On a stated range the two part by up to eighteen kilometres. They also differ in kind: an error of ten metres in every height moves the summits three metres and the watershed on the plateau more than a kilometre.
The sea is measured from a tide, and the headlands decide how much
A territorial sea is measured from the low-water line, and the low-water line is where a chart's tidal datum meets the shore — so a datum a metre lower moves it twenty metres at a headland and a kilometre across a tidal flat. On a stated coast the twelve-mile limit barely notices the flats: fifty-six square kilometres of shore are uncovered and eleven of sea are gained. A bank that dries at the lower datum adds three hundred and thirty-nine in one step.
A straight baseline moves more water inward than sea outward
The seventh article of the law of the sea lets a coast fringed with islands measure its territorial sea from straight lines joining chosen points, and gives no length, no angle and no scale for the choice. On a stated coast the lines a common reading permits add sixty-eight square kilometres of sea at the outer edge and turn nine hundred and eighty-eight into internal waters. Drop the test of direction and the outer rocks come into reach: eighteen hundred added, forty-six hundred enclosed.
The control carries the error of every network above it
A local survey is adjusted with its control stations held fixed, and the error ellipses it reports come from its own observations alone. On a stated three-order network a local station reports 11 millimetres and sits 59 from where the national origin puts it: 52 of them from the national order and 26 from the regional. Its own residuals can see almost none of that. What it does own outright is its lengths, and nothing of those is inherited.
Three numbers are all a local survey can check of its control
Adjust a local network on its own lines, then fit it onto its three control stations, and the misfit at the control is the only place the control's error can show. It is exactly the excess the held adjustment's variance factor already carried, counted on three degrees of freedom instead of fifteen, and it catches a disturbed monument 79 times in a hundred where the held report catches it 49. Give the fit a scale and it hides 98 per cent of that excess, and lends the control's error to every length.
Naming the control station that moved takes a fifth
A local survey tied to distant control can tell that a monument has moved, and with three control stations it cannot say which. Each station gives the local lines one number they know well — its distance — and a shift of the whole control uses up two of them, so three stations leave one and four leave two. Five evenly spaced name a station moved five centimetres 83 times in a hundred; three name it 20. Better lines cannot stand in for the missing stations, and every station added also exposes more of the control's own published error.
Written in numbers, the archipelago rule still leaves eighteen
The article on archipelagic baselines gives its tolerances as numbers: no segment over a hundred nautical miles except three per cent of them, which may reach 125, and water between one and nine times the land. On a stated archipelago 8,799 closed systems keep the main islands inside, the ratio passes all of them and the length test eighteen, and the eighteen still differ by 27,585 square kilometres. Thin the land and the ratio does the deciding; the lower bound decides nothing; and the three-per-cent exception can be used only by a system padded to thirty-four segments.
Honest about its control, the test needs eight centimetres where it needed five
A test that holds the control exact reads the control's published error as movement, and at five stations flags a good survey sixteen times in a hundred. Carry the published covariance into the adjustment and the false alarms fall to half a per cent — and the smallest move the test can find grows from 48 millimetres to 77. A five-centimetre move is then named 29 times in a hundred instead of 63, the naming rate of perfect control takes twenty-four stations, and above ten centimetres the two tests agree. What the honest test buys is a flag that means what it says.
The archipelago rule moves water between regimes, not between states
The eighteen baseline systems the archipelago article allows on a stated archipelago differ by 27,585 square kilometres of water inside the lines. Measured outward, as the Convention measures every zone, the difference almost vanishes: whatever a system takes inside it gives up from the zone outside, and water inside plus zone outside differs by 993 square kilometres, seven hundredths of a per cent. Against a straight neighbouring coast on any side, the equidistance line does not move at all, because a baseline's nearest point to a straight coast is always an island. Only a neighbour facing the middle of a gap feels the choice — 2,112 square kilometres at thirty kilometres off, 688 at 250.
One survey cannot tell an optimistic report from a moved mark
A local survey that carries its control's published covariance can estimate, from its own residuals, how far that covariance understates the truth. With five control stations the estimate has a standard deviation of 3.8 times the published scale, and a single monument moved five centimetres adds about as much again — four times what a report twice too optimistic would add. Run at the scale it estimated for itself, the honest test stops flooding good surveys with alarms, and names a moved mark seven times in a hundred instead of twenty-five: the move inflates the estimate and the estimate absorbs the move. The report can be checked only across surveys, and telling one twice too optimistic from an honest one takes about two hundred of them with five stations each, or fifty with twelve.
An agency can map where its published covariance is optimistic
The local surveys tied to a regional network each estimate, badly, how far the network's published covariance understates the truth, and the agency that receives them can pool what they say. When the understatement is confined to one part of the region, pooling it into one number dilutes it: a patch covering 15 per cent of the region with errors four times their published size reads as 1.30 for the whole. Counted place by place, the same surveys find the patch — two agencies in three find it with 400 surveys and nearly every one with 800, flagging almost nothing outside it. Fed back as one number the scale removes little of the false alarms inside the patch and costs nearly a third of the test's power outside; fed back as a map it cuts the false alarms inside from 20 per cent to 3 and costs little outside.
Copied out to sea, a coast keeps its length and loses its bays
Before 1930 a territorial sea was drawn by following the coast in all its sinuosities, and the phrase fits two constructions. Moved twelve miles seaward, a stated coast of five bays gives up 240 square kilometres against the envelope of arcs, most of it over the narrowest bay; moved along its own normals, it ties a loop over every bay. On a fractal coast charted at finer detail the copy grows by a quarter at every level, to 480 kilometres, while the envelope settles at 133.0.
46 essays in this field, the first 16 of them shown with their opening figure.